# Effective Darmon's program for the generalised Fermat equation

Martin Azon

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Martin Azon. Effective Darmon's program for the generalised Fermat equation. 2025. hal-04996061

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# Effective Darmon's program for the generalised Fermat equation

Martin AZON

**Abstract** — We follow the ideas of Darmon's program for solving infinite families of generalised Fermat equations of signatures $(p, p, r)$ and $(r, r, p)$, where $r$ is a fixed prime, and $p$ is varying. We do so by introducing a common framework for both signatures, allowing for a uniform treatment for the two families of equations. We analyse in detail the geometry of Frey hyperelliptic curves, and the reduction types of the Néron models of their Jacobians. We then study the associated 2-dimensional Galois representations: modularity, irreducibility, and level lowering. In order to illustrate the effectiveness of our results, we solve several examples of families of equations of signatures $(p, p, 5)$ and $(5, 5, p)$.

**Keywords:** Generalised Fermat equation, Darmon's program, Frey hyperelliptic curves.

**Classification:** 11D41 11G30 11F80 11Y50

## Contents

<table><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>2</td><td>Background and notation</td><td>7</td></tr><tr><td>2.1</td><td>Notation</td><td>7</td></tr><tr><td>2.2</td><td>Assumptions on Diophantine equations and their solutions</td><td>7</td></tr><tr><td>2.3</td><td>λ-adic and Weil-Deligne representations</td><td>8</td></tr><tr><td>2.4</td><td>Abelian varieties with real multiplication</td><td>10</td></tr><tr><td>2.5</td><td>Background on hyperelliptic curves</td><td>13</td></tr><tr><td>3</td><td>Construction of Frey representations and curves</td><td>16</td></tr><tr><td>3.1</td><td>Geometric construction of Frey representations</td><td>16</td></tr><tr><td>3.2</td><td>Frey curves attached to putative solutions</td><td>18</td></tr><tr><td>3.3</td><td>Common framework to both signatures</td><td>20</td></tr><tr><td>4</td><td>Reduction types of the Néron models ($\mathcal{J}_r^{\pm}$)($\delta_K$)</td><td>23</td></tr><tr><td>4.1</td><td>Roots of the defining polynomials $g_r^{\pm}$</td><td>24</td></tr><tr><td>4.2</td><td>Local behaviour at even places</td><td>27</td></tr><tr><td>4.3</td><td>Cluster pictures of $C_r^{\pm}$ at odd places</td><td>31</td></tr><tr><td>4.4</td><td>Ramification indices and discriminants</td><td>36</td></tr><tr><td>4.5</td><td>Conclusion of the proof of Theorems 4.2 and 4.3</td><td>40</td></tr><tr><td>5</td><td>Main properties of the 2-dimensional representations $\rho_{(\mathcal{J}_r^{\pm})}(\delta_K), \lambda$</td><td>43</td></tr><tr><td>5.1</td><td>Modularity</td><td>43</td></tr><tr><td>5.2</td><td>Conductor and inertial local types</td><td>45</td></tr><tr><td>5.3</td><td>Level lowering</td><td>49</td></tr><tr><td>6</td><td>Solving families of GFEs of signatures $(p, p, r)$ and $(r, r, p)$</td><td>51</td></tr><tr><td>6.1</td><td>Specialising $s_0$ and $\delta_Q$ for solving specific families of equations</td><td>51</td></tr><tr><td>6.2</td><td>Discarding isomorphisms by comparing traces of Frobenius</td><td>52</td></tr><tr><td>6.3</td><td>Computational aspects of the elimination step</td><td>54</td></tr></table>
# 1 Introduction

## The generalised Fermat equation

Wiles proof of Fermat's last theorem [Wil95] was one of the major breakthroughs in the recent history of number theory. Since then, there has been great progress in the study of modularity of Galois representations, leading to new Diophantine results. Building upon ideas of Frey, Serre, Ribet among others, many mathematicians have explored how to apply techniques from the study of Galois representations to the resolution of Diophantine equations. In particular, of great interest is the generalised Fermat equation (GFE)

$$ (\mathcal{E}_{p,q,r}) \qquad Ax^p + By^q = Cz^r. $$

Here we denote by $A, B, C$ three non-zero pairwise coprime integers, and by $p, q, r$ three integers greater or equal than 2. The triple $(p, q, r)$ is called the signature of the equation $(\mathcal{E}_{p,q,r})$. A solution $(a, b, c) \in \mathbb{Z}^3$ to $(\mathcal{E}_{p,q,r})$ is called non-trivial if $abc \neq 0$, and primitive if $\gcd(a, b, c) = 1$.

Darmon and Granville proved that, if we fix $A, B, C, p, q, r$ and we further assume that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r} < 1$, then equation $(\mathcal{E}_{p,q,r})$ has only finitely many non-trivial primitive solutions (see [DG95]). In the last few decades, several instances of $(\mathcal{E}_{p,q,r})$ have been solved for fixed numerical values of every considered parameter (we refer the reader to [WG24] for an overview of the current state of the art). On a different direction, various authors have studied infinite families of equations of the shape of $(\mathcal{E}_{p,q,r})$, where one allows some of the parameters $p, q, r$ in the signature to vary.

In this paper, we study the infinite families of generalised Fermat equations $(\mathcal{E}_{p,p,r})_p$ and $(\mathcal{E}_{r,r,p})_p$, of respective signatures $(p, p, r)$ and $(r, r, p)$. We fix, once and for all, a prime number $r$ and three non-zero pairwise coprime integers $A, B, C$. The parameter $p$ ranges through an infinite family of prime numbers, e.g., the prime numbers greater than some fixed quantity (in this case we talk about an asymptotic result). Even if the roles of $r$ and $p$ might seem to be symmetric, the families of equations $(\mathcal{E}_{p,p,r})_p$ and $(\mathcal{E}_{r,r,p})_p$ are distinct. For any fixed value of $p$, the equations above are given by

$$ (\mathcal{E}_{p,p,r}) \qquad Ax^p + By^p = Cz^r, $$

$$ (\mathcal{E}_{r,r,p}) \qquad Ax^r + By^r = Cz^p. $$

Our goal in this paper is to develop the necessary theoretical background to solve the families of equations $(\mathcal{E}_{p,p,r})_p$ and $(\mathcal{E}_{r,r,p})_p$ for given numerical values of $r, A, B, C$. We develop the ideas of Darmon's program (which will be explained below) to establish effective results regarding the solutions to such equations. To illustrate the explicitness of our discussion, we specify $r = 5$ and solve different instances of the families of equations $(\mathcal{E}_{p,p,5})_p$ and $(\mathcal{E}_{5,5,p})_p$. More precisely, we prove the following asymptotic results.

**Theorem 1.1 (= Theorems 6.2 + 6.5).** Let $p > 71$ be any prime number.

1) There are no primitive non-trivial solutions $(a, b, c) \in \mathbb{Z}^3$ to the generalised Fermat equation

$$ 7x^p + y^p = 3z^5 $$

that satisfy $10 \mid ab$.

2) For any $i \in [1..4]$ and $j \in \{3, 4\}$, there are no primitive non-trivial solutions to the generalised Fermat equation

$$ 7x^p + 2^i 5^j y^p = 3z^5. $$

**Theorem 1.2 (= Theorems 6.3 + 6.6).** Let $p > 41$ be any prime number.

1) There are no primitive non-trivial solutions $(a, b, c) \in \mathbb{Z}^3$ to the generalised Fermat equation

$$ x^5 + 7y^5 = z^p $$

that satisfy $10 \mid c$.
2) For any $i \in [1..4]$ and $j \in [2..4]$, there are no primitive non-trivial solutions to the generalised Fermat equation

$$x^5 + 7y^5 = 2^i 5^j z^p.$$

We stress out the fact that the equations above are the first examples of GFEs for $r \ge 5$ and $|AB| \ne 1$ that have been solved (even under 2-adic and 5-adic conditions). In section 6, the reader will find explanations, both theoretical and computational, on why we treated such examples.

## The modular method

The approach that we follow here for establishing Theorems 1.1 and 1.2 is the so-called *modular method*, which was pioneered by Frey, Serre, Ribet, and Wiles, among others. It allows to solve instances of infinite families of equations with one varying parameter in the signature, and may be summarised as follows:

**Step 1: Construct a Galois representation.** Attach to any putative solution a 2-dimensional representation $\rho: \text{Gal}(\overline{K}/K) \to \text{GL}_2(\overline{\mathbb{Q}}_p)$. Here, $K$ is a totally real number field, and $p$ is the varying parameter in the signature. We require to explicitly describe the ramification of $\rho$, which should behave “well” in terms of the putative solution.

**Step 2: Modularity / Level lowering.** Prove modularity of $\rho$, and irreducibility of the residual representation $\bar{\rho}$. Applying level lowering results, show that $\bar{\rho}$ arises from a Hilbert newform over $K$ of parallel weight 2 whose level is independent of the solution.

**Step 3: Elimination.** Compute all newforms predicted in the previous step, and prove that none of them gives rise to $\bar{\rho}$.

In the case of Fermat's last theorem, Wiles' main contribution was to show modularity of semistable elliptic curves defined over $\mathbb{Q}$, which concerns Step 2 above. His work opened the door for the era of modularity lifting theorems, a domain that has seen remarkable progress in the last few decades. Even if Step 2 uses a heavy machinery, many modularity results are well understood nowadays, and they are often sufficient for Diophantine applications.

On the other hand, Steps 1 and 3 are quite challenging to establish in full generality. In most of the already treated examples, the representation $\rho$ given by Step 1 arises from the action of $\text{Gal}(\overline{K}/K)$ on the $p$-adic Tate module of an elliptic curve $E(a, b, c)/K$, usually called a *Frey curve*. There is no generic recipe to construct Frey elliptic curves, and most of the known cases in the literature are obtained using *ad hoc* arguments. On top of that, the lack of theoretical techniques for distinguishing residual Galois representations makes Step 3 difficult in general. The latter requires, among other things, numerical computations of spaces of Hilbert newforms, whose dimensions grow very quickly with the size of the parameters in the equation. This algorithmic challenge is one of the main bottlenecks for applying the modular method in many situations.

## Darmon's program: strengthening the modular method

In [Dar00], Darmon described a program that aims at solving some of the difficulties raised by the modular method. We refer the curious reader to [CK25] for a detailed survey on this program. Among other questions, he studied how to successfully perform Step 1 in the modular method. One of his main contributions was to define what a suitable Frey object should be, and which properties should be satisfied by the representation $\bar{\rho}$ introduced above. Its core idea is the following:

**Main idea.** For each signature $(p, q, r)$, construct a so-called Frey representation $\bar{\rho}_s: G_{K(s)} \to \text{GL}_2(\overline{\mathbb{F}}_p)$ having “nice” ramification properties. Here $K(s)$ is the rational function field over a totally real field $K$. The representation $\bar{\rho}$ from above is then obtained by specialising the indeterminate $s$ at some algebraic number $s_0$ whose value depends on the considered equation $(\mathcal{E}_{p,q,r})$ and its putative solution.
Darmon classified and constructed Frey representations for every signature. As in the classical modular method, his examples arise from the geometry of abelian varieties. More specifically, he introduced curves of genus $\ge 1$, defined over function fields, whose Jacobians have real multiplication (hereafter abbreviated as RM). On a different direction, Darmon also envisioned how to use different Frey objects to propagate modularity from one another. We will give further details about this in subsection 5.1.

For the signature $(p, p, r)$, Darmon defines two curves $C_r^{\pm}(s)/\mathbb{Q}(s)$ that are hyperelliptic. For the signature $(r, r, p)$, he constructed two superelliptic curves defined over $\mathbb{Q}(s)$. Billerey, Chen, Dieulefait and Freitas introduce in [BCDF23] a hyperelliptic curve $C_r(s)/\mathbb{Q}(s)$ whose Jacobian gives rise to a Frey representation of signature $(r, r, p)$. It is a quadratic twist of Darmon's hyperelliptic curve $C_r^-(s)$ for the signature $(p, p, r)$ (see §3.2.2 for further details). In [Pac25], Pacetti uses the theory of hypergeometric motives to explain and give heuristics about the intimate relationship between all these geometric objects. We note that, when $r \ge 5$ (and away from the case $|AB| = 1$ for the signature $(r, r, p)$), these are the only known Frey objects, and, in particular, there are no known available elliptic curves.

The above discussion illustrates that the construction of Frey representations is not unique. Curves having different geometric natures may give rise to the same Frey representation. Most of the arithmetic properties of the latter can be read off from the geometry of the underlying curve, so understanding in detail this geometry is essential. After specialising the indeterminate $s$ at some $s_0 \in \mathbb{Q}$, one obtains a curve defined over a number field. Not all kinds of curves are equally understood. For some of them (like hyperelliptic ones), many effective and computational techniques have appeared in the last few years. For instance, the theory of cluster pictures [DDMM23], developed by Dokchitser, Dokchitser, Maistret and Morgan provides combinatorial tools to understand the local behaviour of a hyperelliptic curve at odd places of bad reduction. We refer the curious reader to [ACIK$^{+24}$], where Curcó-Iranzo, Khawaja, Maistret, Mocanu and the author use the machinery of clusters to compute conductor exponents for Frey hyperelliptic curves.

## Our contribution and results

A major contribution of this paper is introducing a common framework to simultaneously study the generalised Fermat equations $(\mathcal{E}_{p,p,r})$ and $(\mathcal{E}_{r,r,p})$, of respective signatures $(p, p, r)$ and $(r, r, p)$. The core idea is the following: to obtain Frey hyperelliptic curves defined over $K$ for these signatures, we consider a single curve defined over $K(s)$, that we specialise at different algebraic numbers.

Following [Dar00], we attach two Frey hyperelliptic curves $C_r^{\pm}(a, b, c)/\mathbb{Q}$ to a putative solution $(a, b, c)$ of $(\mathcal{E}_{p,p,r})$. Similarly, following [BCDF23] and generalising a construction by Kraus, we attach a Frey hyperelliptic curve $C_r(a, b, c)/\mathbb{Q}$ to a putative solution to $(\mathcal{E}_{r,r,p})$. The former ones are quadratic twists of specialisations of the curves $C_r^{\pm}(s)/\mathbb{Q}(s)$. The latter one is a quadratic twist of a specialisation of $C_r(s)/\mathbb{Q}(s)$. But as explained above, the curve $C_r(s)$ is a quadratic twist of $C_r^-(s)$, so the curve $C_r(a, b, c)$ is also a quadratic twist of a specialisation of $C_r^-(s)$. This elementary observation, which is made precise in the proposition below, is the starting point for the whole discussion in this paper.

**Proposition 1.3 (= Proposition 3.18 + Lemma 3.25).** The following properties hold.

1) Assume that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{p,p,r})$. There exists some $s_0 \in \mathbb{Q}$ and some $\delta_{\mathbb{Q}} \in \mathbb{Z}$ depending on $(a, b, c)$ and $(\mathcal{E}_{p,p,r})$, such that $C_r^{\pm}(a, b, c)$ is the quadratic twist by $\delta_{\mathbb{Q}}$ of the specialisation $C_r^{\pm}(s_0)$.

2) Assume that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{r,r,p})$. There exists some $s_0 \in \mathbb{Q}$ and some $\delta_{\mathbb{Q}} \in \mathbb{Z}$ depending on $(a, b, c)$ and $(\mathcal{E}_{r,r,p})$, such that $C_r(a, b, c)$ is the quadratic twist by $\delta_{\mathbb{Q}}$ of the specialisation $C_r^-(s_0)$.

In Table 1, we display the corresponding values of $s_0$ and $\delta_{\mathbb{Q}}$ for each of the signatures. Moreover, there are various hypotheses commonly satisfied for both choices of $s_0$ and $\delta_{\mathbb{Q}}$ (cf. §3.3).
**Proposition 1.3** provides a common framework for a uniform study of the two GFEs $(\mathcal{E}_{p,r,r})$, $(\mathcal{E}_{r,r,p})$. In subsection 3.3, we give axiomatic definitions for $s_0$, $\delta_Q$, and we list the mentioned hypotheses that they should satisfy. Throughout this paper, we state all our results in terms of $s_0$ and $\delta_Q$. Nevertheless, we leave our own "roadcrumb trail": after every important statement, we present the corresponding result for both generalised Fermat equations, a putative solution and the associated Frey object. Even if the construction of $C_r^{\pm}(s_0)$ depends (tautologically) on $s_0$, most of its properties are encoded in the behaviour of $s_0(s_0 - 1)$. This is beneficial for the sake of effectiveness: for instance, the choice of $s_0$ for the signature $(r, r, p)$ depends in a complicated way on the coefficients $A, B, C$ and the solution $(a, b, c)$. Nevertheless, Table 1 shows that $s_0(s_0 - 1)$ has a simpler expression in terms of the mentioned parameters.

Throughout the paper, we let $\mathcal{K} := \mathbb{Q}(\zeta_r)^+$ be the maximal totally real subfield of the cyclotomic $\mathbb{Q}(\zeta_r)$. Consider the base-changed curves $C_r^{\pm} := C_r^{\pm}(s_0)^{(\delta_Q)} \times_Q \mathcal{K}$: their Jacobians $J_r^{\pm} := \text{Jac}(C_r^{\pm})$ have real multiplication RM by $\mathcal{K}$ (see Theorem 3.5). Following the work of Ribet [Rib76], we construct 2-dimensional representations $\rho_{J_r^{\pm}, \lambda} : \text{Gal}(\bar{\mathcal{K}}/\mathcal{K}) \to \text{GL}_2(\mathcal{K}_{\lambda})$, where $\lambda$ is a finite place of $\mathcal{K}$ (see §2.4.1). When $\lambda$ divides the prime number $p$ appearing in the signature, we will use the reductions of these representations (and their twists) to perform the modular method as depicted above.

Serre, Tate [ST68], and Grothendieck [SGA72] illustrated that much of the arithmetic information of $\rho_{J_r^{\pm}, \lambda}$ is given by the geometric behaviour of $J_r^{\pm}/\mathcal{K}$. With this in mind, we perform a careful analysis of the reduction types of their Néron models. Moreover, we introduce a new parameter $\delta_{\mathcal{K}} \in \mathcal{O}_{\mathcal{K}}$, and consider the twisted Jacobian $(J_r^{\pm})^{(\delta_{\mathcal{K}})}$. The purpose of this is to obtain an abelian variety having better reduction properties (for some choices of $\delta_{\mathcal{K}}$, $(J_r^{\pm})^{(\delta_{\mathcal{K}})}$ is semistable, whereas $J_r^{\pm}$ is not).

**Theorem 1.4** (= Theorems 4.2 + 4.3). Let $q$ be a rational prime, $\mathfrak{q}$ a finite place of $\mathcal{K}$ dividing $q$, and define $\nu_q$ to be the $q$-adic valuation of $2^4 s_0(s_0 - 1)$. The decision trees depicted in Figures 1 and 2 describe the reduction type of the Néron models of $(J_r^{\pm})^{(\delta_{\mathcal{K}})}$ at $\mathfrak{q}$ in terms of $\nu_q$, $\delta_Q$ and $\delta_{\mathcal{K}}$.

In Example 4.5, we illustrate how this theorem applies when considering the specific choices of $s_0$, $\delta_Q$ for the signatures $(p, p, r)$ and $(r, r, p)$. To establish Theorem 1.4, we first exhibit algebraic expressions for the roots of the defining polynomials of $C_r^{\pm}(s_0)$, and we deduce from this the discriminants of the models and their twists (see Example 4.15 for the particular arithmetic statements). Knowing the places of bad reduction, we make use of the following “geometric trichotomy”: if $A$ is an abelian variety defined over a local field which has RM, then its Néron model has either good, totally toric, or totally unipotent reduction (cf. Proposition 2.19). This phenomenon, satisfied in particular by $(J_r^{\pm})^{(\delta_{\mathcal{K}})}$, simplifies the analysis of the reduction types of its Néron model (hence the proof of Theorem 1.4).

At even places, we study the geometry of different models of $(C_r^{\pm})^{(\delta_{\mathcal{K}})}$, if they are semistable, and, if not, over which extension they become so. Our analysis relies on some 2-adic conditions on $s_0(s_0 - 1)$: it covers the “generic” cases, and is therefore sufficient for our Diophantine purposes. At odd places, we use the machinery of cluster pictures [DDMM23], in the style of [ACIK+24]. We proceed to compute the cluster pictures of the curves $C_r^{\pm}/\mathcal{K}$ at every odd place of bad reduction. After this, we study the ramification properties of the splitting field of the defining polynomials. Finally, we use the criteria on clusters (Theorems 2.39 and 2.40), to check if $(J_r^{\pm})^{(\delta_{\mathcal{K}})}$ have good or bad semistable reduction (hence totally toric): otherwise they have totally unipotent reduction. We note that our approach here slightly differs from the one in [ACIK+24], as we do not use clusters to compute tame conductors, but only the reduction types of the Néron models.

Having understood the geometry of the (twisted) curves $(C_r^{\pm})^{(\delta_{\mathcal{K}})}$ and their Jacobians, we move onto the study of the 2-dimensional representations $\rho_{(J_r^{\pm})^{(\delta_{\mathcal{K}})}, \lambda}$. These form a compatible system of Galois representations. Since we aim at performing the modular method with $\rho_{J_r^{\pm}, \lambda}$ and its twist $\rho_{(J_r^{\pm})^{(\delta_{\mathcal{K}})}, \lambda}$, it is essential to know that:

**Theorem 1.5** (= Theorems 5.4 + 5.5). The compatible system of representations $(\rho_{J_r^-, \lambda})_{\lambda}$ arises from a Hilbert newform defined over $\mathcal{K}$. If $\nu_r(s_0(s_0 - 1)) > 2$, the same holds for $(\rho_{J_r^+, \lambda})_{\lambda}$.
Our proof of this result builds upon Darmon's panoramic view for propagating modularity (cf. [Dar00, Figure 1]). We show that the particular representation $\rho_{J_r^{\pm}, \tau}$ is modular, where $\tau$ is the unique place of $\mathcal{K}$ above $r$. Our proof involves deep theorems, such as Serre's modularity conjecture and modularity lifting theorems. At different points, we make use of the knowledge of the reduction types from Theorem 1.4 to establish desired properties about the considered representations.

Modularity theorems state that the level of the newform giving rise to the compatible system $(\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda})_{\lambda}$ equals its global conductor. It is therefore crucial to understand this conductor in detail, as it will be a key input for level lowering and the elimination step. We provide an explicit description of the Artin conductor of the system at every finite place:

**Theorem 1.6 (= Theorems 5.10 + 5.11).** Let $\mathfrak{q}$ be a finite place of $\mathcal{K}$. The Artin conductor of $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$ restricted to the decomposition group $D_{\mathfrak{q}} \simeq \text{Gal}(\overline{\mathcal{K}_{\mathfrak{q}}}/\mathcal{K}_{\mathfrak{q}})$ is described in Tables 2 and 3.

In Example 5.15, we illustrate how to recover from Theorem 1.6 the Artin conductors for the particular choices of $s_0, \delta_{\mathbb{Q}}$ displayed in Table 1. The proof of this theorem is largely based on the description of the reduction types given in Theorem 1.4. Indeed, Grothendieck [SGA72] proved that the tame part of the Artin conductor is encoded in the reduction type of the Néron model. At even places, we restrict ourselves to the cases of tame potential semistable reduction, in which there is no wild conductor. At odd places, the theory of cluster pictures yields the wild conductor by means of algebraic number theory. The content of subsections 4.4 and 5.2, where this is discussed, is largely inspired by [ACIK+24], and provides a generalisation of some of its results.

Along the way, we describe the inertial local types of the Weil-Deligne representations attached to $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$. This local study of the action of inertia is a key datum for proving absolute irreducibility of the residual representations. Building upon the work of [BCDF23, §7] about finiteness of Galois representations, we finally establish our desired level lowering result (Theorem 5.22). For any prime number $p$ and any place $\mathfrak{p}|p$, it asserts the existence of a newform $g$ such that $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)}, \mathfrak{p}} \simeq \bar{\rho}_{g, \mathfrak{g}}$ for some place $\mathfrak{P}|p$ in the field of coefficients $\mathcal{K}_g$. The key feature of $g$ is that, when we choose $s_0, \delta_{\mathbb{Q}}$ as in Table 1, the primes dividing the level of $g$ are do not depend on the putative solution, and are only expressed in terms of the parameters of the GFE.

We finally discuss the elimination step, in order to prove Theorems 1.1 and 1.2. As explained above, this requires an algorithmic implementation to discard isomorphisms like $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)}, \mathfrak{p}} \simeq \bar{\rho}_{g, \mathfrak{g}}$. We do so by comparing traces of Frobenius under these representations. We specify the values of the exponent $r$ and the coefficients $A, B, C$ as in the equations in Theorems 1.1 and 1.2, hence giving numerical values for the level of $g$. We then use the Magma software [BCP97] to compute spaces of Hilbert newforms, and eliminate Galois orbits of newforms by discarding isomorphisms of representations. We also discuss some computational aspects of our approach. To write our code, we apply functional programming, so that our functions may be reused as much as possible. All the electronic resources and supporting files for this paper are available at [Azo25].

## Acknowledgments

The present paper is based on the work that I have done during my doctoral studies at Université Clermont Auvergne. I wholeheartedly thank Nicolas Billerey for proposing this research topic, and for helpful comments on earlier versions of this manuscript. Special thanks go to Mar Curcó-Iranzo, Maleeha Khawaja, Céline Maistret and Diana Mocanu for fruitful discussions about clusters; and to Enric Florit for helpful discussions about endomorphism algebras. I would also like to thank Ariel Pacetti: a part of this work answers a question he raised. Last, but not least, many thanks go to Lucas Villagra Torcomian and Pedro José Cazorla García for interesting conversations.
# 2 Background and notation

## 2.1 Notation

For any field $K$, we let $\bar{K}$ be an algebraic closure of $K$, $K^{\text{sep}}$ be a separable closure included in $\bar{K}$. We let $G_K := \text{Gal}(K^{\text{sep}}/K)$ be the absolute Galois group of $K$, endowed with the profinite topology. If $v$ is a valuation defined on $K^{\text{sep}}$, we will say that $v$ is normalised with respect to $K$ if $v(K^\times) = \mathbb{Z}$.

For a fixed rational prime $q$, we let $\mathbb{Q}_q$ be the field of $q$-adic numbers, and $v_q$ be the valuation on $\overline{\mathbb{Q}}_q$ normalised with respect to $\mathbb{Q}_q$. Given a number field $\mathcal{F}$, we denote by $\mathcal{O}_{\mathcal{F}}$ its ring of integers. For a prime ideal $\mathfrak{q}$ of $\mathcal{O}_{\mathcal{F}}$, we let $\mathcal{F}_{\mathfrak{q}}$ be the completion of $\mathcal{F}$ at $\mathfrak{q}$, and we let $v_{\mathfrak{q}}$ be the valuation on $\overline{\mathcal{F}}_{\mathfrak{q}}$ normalised with respect to $\mathcal{F}_{\mathfrak{q}}$. We say that $\mathfrak{q}$ is even (resp. odd) if $\mathfrak{q} \mid 2$ (resp. $\mathfrak{q} \nmid 2$).

If $(K, \mathfrak{m})$ is a local field, endowed with a normalised valuation $v_{\mathfrak{m}}$, we write $\mathcal{O}_{\mathfrak{m}}$ for the ring of integers of $K$, and $\mathbb{F}_{\mathfrak{m}}$ for its residue field. We still denote by the symbol $v_{\mathfrak{m}}$ the valuation defined on $K^{\text{sep}}$ that matches $v_{\mathfrak{m}}$ over $K$. We let $K^{\text{unr}}$ be the maximal unramified extension of $K$, $I_K = \text{Gal}(K^{\text{sep}}/K^{\text{unr}})$ its inertia group, and $I_K^t$ (resp. $I_K^w$) the tame (resp. wild) inertia group. We let $\text{Frob}_{\mathfrak{m}} \in G_{\mathbb{F}_{\mathfrak{m}}}$ be the arithmetic Frobenius, and $\varphi_{\mathfrak{m}} \in G_K$ be a geometric Frobenius lift: its reduction $\varphi_{\mathfrak{m}}$ in $G_{\mathbb{F}_{\mathfrak{m}}}$ is $\text{Frob}_{\mathfrak{m}}^{-1}$. We let $W_K$ be the Weil group of $K$, which consists of the elements of $G_K$ that act on $\mathbb{F}_{\mathfrak{m}}$ as a power of $\text{Frob}_{\mathfrak{m}}$. The Weil group fits into the short exact sequence

$$1 \to I_K \to W_K \to \text{Frob}_{\mathfrak{m}}^{\mathbb{Z}} \to 1.$$

If $\mathcal{F}$ is a number field and $\mathfrak{q}$ is a finite place of $\mathcal{F}$, we will simply denote by $D_{\mathfrak{q}}$ the decomposition group of $G_{\mathcal{F}}$ at $\mathfrak{q}$ and by $I_{\mathfrak{q}}$ its inertia subgroup. We denote by $W_{\mathfrak{q}}$ for the Weil group of $D_{\mathfrak{q}}$.

We fix a prime number $r \ge 5$. Let $\zeta_r \in \overline{\mathbb{Q}}$ be a primitive $r$-th root of unity, and let $\mathbb{Q}(\zeta_r)$ be the $r$-th cyclotomic field. For any $0 \le j \le \frac{r-1}{2}$, we let $\omega_j := \zeta_r^j + \zeta_r^{-j}$, $\tau_j := \zeta_r^j - \zeta_r^{-j}$ and, for simplicity, we write $\omega := \omega_1$, and $\tau := \tau_1$. We denote by $\mathcal{K} := \mathbb{Q}(\omega)$ the maximal totally real subfield of $\mathbb{Q}(\zeta_r)$. The extension $\mathcal{K}/\mathbb{Q}$ has degree $\frac{r-1}{2}$, and is defined by the irreducible polynomial

$$h_r(x) := \prod_{j=1}^{\frac{r-1}{2}} (x - \omega_j) \in \mathbb{Z}[x]. \qquad (2.1)$$

Moreover, the extension $\mathcal{K}/\mathbb{Q}$ is totally ramified at $r$, and unramified at any other rational prime. We denote by $\mathfrak{t} = (2 - \omega)$ the unique prime ideal of $\mathcal{O}_{\mathcal{K}}$ dividing $r$.

## 2.2 Assumptions on Diophantine equations and their solutions

We now discuss the main assumptions that we will be doing on the parameters of the generalised Fermat equations under consideration, and on their putative solutions.

**Definition 2.1.** We let $r \ge 5$ be a fixed prime number, and $A, B, C \in \mathbb{Z} \setminus \{0\}$ be three pairwise coprime integers, that are free of $r$-th powers. We denote by $p$ any prime number greater than $r$.

The generalised Fermat equations of signatures $(p, p, 3)$ and $(3, 3, p)$ have already been studied in the literature. We refer the curious reader to [BVY04], where Bennett, Vatsal and Yazdani study the former equation; and to [BD13], where Bennett and Dahmen study the latter. Moreover, the case $r = p$ was also studied by Kraus in [Kra97], and by Dieulefait and Soto in [DS21].

Restricting to the case $p > r$ is not problematic, as this is the generic case. The assumptions on $A, B, C$ are natural to consider for proving that $(\mathcal{E}_{p,p,r})$ and $(\mathcal{E}_{r,r,p})$ do not have non-trivial primitive solutions. Indeed, supposing that $A, B, C$ are pairwise coprime is costless, as one could simplify the equation if that was not the case. For the GFE $(\mathcal{E}_{p,p,r}) : Ax^p + By^p = Cz^r$, one can perfectly assume that $C$ is free of $r$-th powers, and that $A, B$ are free of $p$-th powers. Similarly, for $(\mathcal{E}_{r,r,p}) : Ax^r + By^r = Cz^p$, one can assume that $A$ and $B$ are free of $r$-th powers, and $C$ free of $p$-th powers. If, for any of the signatures, this was not satisfied, one could replace the coefficients by some others being
free of r-th and p-th powers. Proving the non-existence of solutions for the newly obtained equation implies the non-existence of solutions for the original one.

Assuming that all the coefficients are free of r-th powers implies a loss of generality. Nevertheless, it allows for a uniform treatment of the families of equations $(\mathcal{E}_{p,p,r})_p$ and $(\mathcal{E}_{r,r,p})_p$, as the coefficients are necessarily independent of $p$. Moreover, under this (restrictive) hypothesis we have:

**Lemma 2.2.** Let $r, p, A, B, C$ be as in Definition 2.1.

1) If $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{p,p,r})$, then $Aa^p, Bb^p$ and $Cc^r$ are pairwise coprime.

2) If $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{r,r,p})$, then $Aa^r, Bb^r$ and $Cc^p$ are pairwise coprime.

*Proof.* We prove the first statement, as the second one is treated in the exact same way. Assume that $Aa^p + Bb^p = Cc^r$. If any prime number divides two among terms among $Aa^p, Bb^p$ and $Cc^r$, then it necessarily divides the third one two. Assume by contradiction that such a prime $q$ exists. Two different cases appear.

* If $q \mid A$, then $q \nmid B, C$, as $A, B, C$ are pairwise coprime, so $q \mid b^p, c^r$. We assumed that $A$ is free of r-th powers, so $v_q(Aa^p) = v_q(A) < r < v_q(Bb^p)$ (the last inequality is due to $p > r$). We have

$$r \le v_q(Cc^r) = v_q(Aa^p + Bb^p) = \min(v_q(Aa^p), v_q(Bb^p)) = v_q(Aa^p) < r,$$

hence a contradiction. The case $q \mid B$ is treated in the same way.

* If $q \nmid AB$, then $q \nmid a^p, b^p$, and since $a, b, c$ are coprime, then $q \nmid c$. In this case we have

$$v_q(C) = v_q(Cc^r) = v_q(Aa^p + Bb^p) \ge \min(v_q(Aa^p), v_q(Bb^p)) = \min(pv_q(a), pv_q(b)) \ge p > r.$$

This contradicts the fact that $C$ is free of r-th powers. The desired result follows. □

## 2.3 λ-adic and Weil-Deligne representations

We now recall the essentials facts about λ-adic and Weil-Deligne representations that will be used throughout this article. We refer the reader to [Del73, §8] or [Tat79] for details, and to [Roh94] for a motivation towards the study of elliptic curves and abelian varieties.

Let $(K, \mathfrak{m})$ be a local field, we adopt the notation introduced in §2.1. Fix a prime number $\ell \ne \text{char}(\mathbb{F}_\mathfrak{m})$. Let $E_\lambda$ be the completion of a number field $E$ at a finite place $\lambda$ dividing $\ell$.

**Definition 2.3.** A λ-adic representation of $G_K$ is a continuous homomorphism $\rho : G_K \to \text{GL}(V)$, where $V$ is finite dimensional $E_\lambda$-vector space endowed with the λ-adic topology. We denote by $\text{Rep}_{E_\lambda}(G_K)$ the category of λ-adic representations of $G_K$.

Examples of λ-adic representations naturally arise in the étale cohomology of smooth proper varieties, but also associated to modular forms. We refer the curious reader to [Tay04] for a survey.

**Definition 2.4.** Let $V$ be a vector space over a field $F$ of characteristic zero. We define an $F$-linear Weil-Deligne (WD) representation as a pair $(\rho, N)$, where $\rho : W_K \to \text{GL}(V)$ is a homomorphism continuous with respect to the discrete topology on $V$, and $N$ is an endomorphism of $V$, called the monodromy operator, satisfying the equality $\rho N \rho^{-1} = \omega_K N$. Here $\omega_K : W_K \to F^\times$ denotes the unramified character of $W_K$ such that $\omega_K(\varphi_\mathfrak{m}) = |\mathbb{F}_\mathfrak{m}|^{-1}$.

A morphism of WD-representations is a morphism between the underlying representations that commutes with the monodromy operators. We denote by $\text{Rep}_F(W'_K)$ the category of $F$-linear WD-representations.

**Remark 2.5.** The continuity of a WD-representation $(\rho, N)$ implies that $\rho(I_K)$ is finite. Moreover, the equality satisfied by the monodromy operator $N$ implies that it is a nilpotent endomorphism.
The following result establishes the link between $\lambda$-adic representations of $G_K$ and Weil-Deligne representations. Building on Grothendieck's $\ell$-adic monodromy theorem, one can attach a WD-representation to any $\lambda$-adic representation of $G_K$. This construction is completely explicit, and the curious reader can find details in [Del73, §8], [ST68, Appendix].

**Theorem 2.6.** There is a fully faithful functor

$$ W : \mathrm{Rep}_{E_\lambda}(G_K) \to \mathrm{Rep}_{E_\lambda}(W'_K) \\ \rho_\lambda \mapsto (W(\rho_\lambda), N_{\rho_\lambda}) $$

**Remark 2.7.** For any $\lambda$-adic representation $\rho_\lambda \in \mathrm{Rep}_{E_\lambda}(G_K)$, $\rho_\lambda(I_K)$ is finite if and only if $N_{\rho_\lambda} = 0$. Whenever this is the case, we have $W(\rho_\lambda) = (\rho_\lambda, 0)$.

The interest of considering WD-representations is that their definition uses only the discrete topology on $V$. Therefore, WD-representations are well adapted to modify the field of coefficients of $V$, namely shifting from $\mathbb{Q}_\ell$ or $E_\lambda$ to $\mathbb{C}$.

**Definition 2.8.** For any $\rho_\lambda \in \mathrm{Rep}_{E_\lambda}(G_K)$ and any embedding $\iota : E_\lambda \hookrightarrow \mathbb{C}$, we denote by $W_\iota(\rho_\lambda)$ the complex WD-representation obtained by extending the scalars of $W(\rho_\lambda)$ through $\iota$.

In this article, we will be mainly interested in 2-dimensional complex WD-representations. These can be explicitly classified, and we refer the reader to [Tat79, DFV24] for a detailed discussion.

**Proposition 2.9.** Let $(\rho, N) \in \mathrm{Rep}_C(W'_K)$ be a 2-dimensional complex WD-representation. Then $(\rho, N)$ arises up to isomorphism from one of the following three possibilities.

1) Steinberg representations: There is a continuous character $\chi : W_K \to \mathbb{C}^\times$ with open kernel such that $\rho = \chi \otimes (\omega_K \oplus 1)$, and $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$.

2) Principal series: There are two continuous characters $\chi_1, \chi_2 : W_K \to \mathbb{C}^\times$ with open kernel satisfying $\chi_1\chi_2^{-1} \neq \omega_K^{\pm 1}$, such that $\rho = \chi_1 \oplus \chi_2$, and $N = 0$.

3) Supercuspidal representations: $\rho$ is an irreducible 2-dimensional representation of $W_K$, and $N = 0$. We say that $\rho$ is non-exceptional if its projective image is dihedral. Otherwise, it is called exceptional, and it has projective image $A_4$ or $S_4$.

**Remark 2.10.** Any non-exceptional supercuspidal WD-representation is the induction of a character of a quadratic extension of $K$. For further details about properties of supercuspidal representations, we refer the reader to [DFV24, §2.3].

When considering $\lambda$-adic and WD-representations, one is particularly interested in the action of the inertia group $I_K$. For this purpose, we introduce:

**Definition 2.11.** An inertial local type is an equivalence class $[\rho, N]$ of WD-representations under the relation $(\rho, N) \sim (\rho', N')$ if and only if $(\rho|_{I_K}, N) \sim (\rho'|_{I_K}, N')$.

**Remark 2.12.** There is an explicit classification of local inertial types (see [DFV24, Proposition 2.4.1]), which is compatible with the classification given in Proposition 2.9. We will say that $[\rho, N]$ is Steinberg, principal series or supercuspidal if it is the restriction of such a WD-representation.

For any real number $u \ge -1$, we let $G_K^u$ be the absolute $u$-th ramification group of $K$ in upper numbering (see [Ulm16, §3]). These groups produce a filtration of $G_K$: we have $G_K^{-1} = G_K$, $G_K^0 = I_K$ and $\bigcup_{u>0} G_K^u = I_K^w$. For any $\lambda$-adic representation $\rho_\lambda : G_K \to V$, and any subgroup $G < G_K$, we let $V^G$ be the subspace of $V$ fixed by $\rho(G)$.
**Definition 2.13.** Let $\rho_\lambda : G_K \to \text{GL}(V)$ be a $\lambda$-adic representation. We define the tame and wild (or Swan) conductor of $\rho_\lambda$ as

$$ \mathfrak{n}_{\text{tame}}(\rho_{\lambda}) := \int_{-1}^{0} \text{codim}_{E_{\lambda}} V^{G_{K}^{u}} du \quad \text{and} \quad \mathfrak{n}_{\text{wild}}(\rho_{\lambda}) := \int_{0}^{\infty} \text{codim}_{E_{\lambda}} V^{G_{K}^{u}} du. $$

We define the Artin conductor of $\rho_\lambda$ by $\mathfrak{n}(\rho_\lambda) := \mathfrak{n}_{\text{tame}}(\rho_\lambda) + \mathfrak{n}_{\text{wild}}(\rho_\lambda)$.

**Remark 2.14.** One can show that the Artin conductor $\mathfrak{n}(\rho_\lambda)$ is an integer (see [Ser79, VI, §2]). Similarly, one can define the conductor of a WD-representation. The important feature of this definition, is that, for any $\rho_\lambda \in \text{Rep}_{E_\lambda}(G_K)$, the conductor of $\text{W}(\rho_\lambda)$ matches the Artin conductor $\mathfrak{n}(\rho_\lambda)$ introduced above (see [Ulm16, §8]).

**Lemma 2.15.** Let $L/K$ be a tame extension with ramification index $e_{L/K}$. Let $\rho_\lambda : G_K \to \text{GL}(V)$ be a $\lambda$-adic representation. The wild conductor of the restriction $\rho_\lambda|_{G_L}$ is given by

$$ \mathfrak{n}_{\text{wild}}(\rho_{\lambda}|_{G_L}) = e_{L/K} \mathfrak{n}_{\text{wild}}(\rho_{\lambda}). $$

*Proof.* This is a straightforward generalisation of [ACIK$^{+}$24, Lemma 2.2]. □

## 2.4 Abelian varieties with real multiplication

In this section we discuss various properties of abelian varieties with real multiplication. Our main references are [Rib76] and [Mel22].

The following discussion does not require the ground field to be local. That is why, we let $L$ be any field, and consider an abelian variety $A/L$ of dimension $g$.

**Definition 2.16.** Let $F$ be a totally real field of degree $g$ over $\mathbb{Q}$. We say that $A$ has real multiplication by $F$ (or simply RM) if there exists an isomorphism $F \simeq \text{End}_L(A) \otimes \mathbb{Q}$.

**Remark 2.17.** Abelian varieties with RM are instances of so-called abelian varieties of GL$_2$-type. Some of the properties below rely only on the fact of $A$ being of GL$_2$-type. Nevertheless, $F$ being totally real has stronger consequences (see Remark 2.20 below). Moreover, the abelian varieties considered in this article will have RM, so we focus on this case.

**Lemma 2.18.** Let $d \in L^\times \setminus (L^\times)^2$, and $\chi_d : G_L \to \{\pm 1\}$ be the character corresponding to $L(\sqrt{d})$. If $A/L$ has RM by $F$, then $A^{(d)}$, its quadratic twist by $\chi_d$, also has RM by $F$.

*Proof.* The character $\chi_d$ acts on $A^{(d)}$ (and thus on $A^{(d)} \times_L L(\sqrt{d})$) as $-1 \in \text{Aut}_L(A^{(d)})$. But the action of $F$ on $\text{End}_L(A)$ commutes with the automorphism $-1 \in \text{Aut}_L(A)$, so the action of $F$ on $A \otimes_L L(\sqrt{d}) \simeq A^{(d)} \otimes_L L(\sqrt{d})$ is actually defined over $L$. □

### 2.4.1 Decomposition of the Tate module

Fix a prime number $\ell \neq \text{char}(L)$. Let $T_\ell(A) := \lim_{\leftarrow n \geq 1} A[\ell^n]$ be the $\ell$-adic Tate module of $A$, and $V_\ell(A) := T_\ell(A) \otimes \mathbb{Q}_\ell$. It is well-known that $V_\ell(A)$ is a $\mathbb{Q}_\ell$-vector space of dimension $2g$ where the absolute Galois group $G_L$ acts continuously, giving rise to a representation

$$ \rho_{A, \ell} : G_L \to \text{Aut}_{\mathbb{Q}_\ell}(V_\ell(A)) \simeq \text{GL}_{2g}(\mathbb{Q}_\ell). $$

Assume now that $A$ has RM by a totally real field $F$, and define $F_\ell := F \otimes \mathbb{Q}_\ell$. This is a semisimple $\mathbb{Q}_\ell$-algebra that acts on $V_\ell(A)$, and the action of $G_L$ on $V_\ell(A)$ is $F_\ell$-linear. For any place $\lambda$ of $F$ dividing $\ell$, define $V_\lambda(A) := V_\ell(A) \otimes_{\mathbb{Q}_\ell} F_\lambda$. In [Rib76, II §1], Ribet proves that $V_\ell(A)$ is a free $F_\ell$-module of
rank 2, so $V_\lambda(A)$ is a 2-dimensional $F_\lambda$-vector space. Again, the action of $G_L$ on $V_\lambda(A)$ is $F_\lambda$-linear, so we obtain a continuous representation

$$ \rho_{A, \lambda} : G_L \to \mathrm{Aut}_{F_\lambda}(V_\lambda(A)) \simeq \mathrm{GL}_2(F_\lambda). $$

Ribet shows in [Rib76] that the $\rho_{A, \lambda}$'s form a weakly compatible system of Galois representations in the sense of [Bö13, Definition 3.12]. By the work of Fontaine [Fon94], this is moreover a strictly compatible system (see [ACIK+24, Appendix] for further details).

Denote by $\mathrm{Res}_{F_\lambda/\mathbb{Q}_\ell}(V_\lambda(A))$ the restriction of scalars of $V_\lambda$ to $\mathbb{Q}_\ell$, and by $\mathrm{Res}_{F_\lambda/\mathbb{Q}_\ell}(\rho_{A, \lambda})$ the associated representation of $G_L$. The isomorphism of $\mathbb{Q}_\ell$-vector spaces $F_\ell \simeq \prod_{\lambda|\ell} F_\lambda$ induces an isomorphism of $\mathbb{Q}_\ell$-representations

$$ \rho_{A, \ell} \simeq \bigoplus_{\lambda|\ell} \mathrm{Res}_{F_\lambda/\mathbb{Q}_\ell}(\rho_{A, \lambda}). \qquad (2.2) $$

Finally, we define $\bar{\rho}_{A, \lambda} : G_L \to \mathrm{GL}_2(\mathbb{F}_\lambda)$ to be the representation obtained by considering the action of $G_L$ on the semi-simplification of the module $T_\ell(A) \otimes_{\mathcal{O}_F} \mathbb{F}_\lambda$ (see [Rib76, §II.2]).

### 2.4.2 Reduction types of the Néron model

We now come back to the setting of §2.1, by assuming that $L = K$ is a local field, and recover all the notation introduced above.

Let $A/K$ be an abelian variety (not necessarily having RM). Let $\mathcal{A} \to \mathrm{Spec}(\mathcal{O}_m)$ be the Néron model of $A/K$ (see [BLR90] for a detailed discussion). Denote by $\mathcal{A}_m$ its special fiber, and $\mathcal{A}_m^0$ the connected component of the identity in it. By Chevalley's theorem (see [Mil17]), there exists a torus $T/\mathbb{F}_m$, a unipotent group $U/\mathbb{F}_m$ and an abelian variety $B/\mathbb{F}_m$ that fit in the short exact sequence

$$ 1 \to T \times U \to \mathcal{A}_m^0 \to B \to 1. \qquad (2.3) $$

The datum of $T, U$ and $B$ is called the *reduction type* of $\mathcal{A}$ at $m$. We say that $\mathcal{A}$ is semistable if the unipotent group $U$ in (2.3) is trivial. The semistable reduction theorem [SGA72, Exposé IX] states that any abelian variety over a local field has potential semistable reduction.

Abelian varieties with real multiplication have restricted possibilities for their reduction types.

**Proposition 2.19.** Assume that $A$ has RM. Then $\mathcal{A}_m^0$ is either an abelian variety, a torus or a unipotent group, meaning that exactly one among $T, U$ and $B$ as in (2.3) is non-trivial.

*Proof.* This is [Rib76, Proposition 3.6.1] for $\mathcal{A}$ semistable, and [Mel22, Proposition 2.6] in general. $\square$

**Remark 2.20.** Ribet proved in [Rib76, §3.6] that any abelian variety of $\mathrm{GL}_2$-type having semistable reduction has either good or totally toric reduction. However, Proposition 2.19 does not hold if we only assume $A/K$ to be of $\mathrm{GL}_2$-type without having RM. The following example, which was communicated to the author by Enric Florit, illustrates so.

Consider the Jacobian of the modular curve $X_1(13)/\mathbb{Q}$ (labelled on the LMFDB [LMF25] as 169.a.169.1). Its endomorphism algebra is $\mathrm{End}_{\mathbb{Q}}(\mathrm{Jac}(X_1(13))) \otimes \mathbb{Q} \simeq \mathbb{Q}(\sqrt{-3})$, so $\mathrm{Jac}(X_1(13))/\mathbb{Q}$ is of $\mathrm{GL}_2$-type, but does not have RM. Looking at the cluster picture of $X_1(13)$ at $p = 13$, one can check that the reduction type of the Néron model of $\mathrm{Jac}(X_1(13))$ is an extension of an elliptic curve by a unipotent group of dimension 1 (cf. [BBB+22, §5]). Thus, Proposition 2.19 does not hold for $\mathrm{Jac}(X_1(13))/\mathbb{Q}$.

**Definition 2.21.** If $A$ has RM, we say that $\mathcal{A}$ has good (resp. toric, or unipotent) reduction at $m$ if $\mathcal{A}_m^0$ is an abelian variety (resp. a torus, or unipotent).
**Remark 2.22.** In general, the Néron model of an abelian variety has “mixed reduction” types, meaning that two or more among T, U and B as in (2.3) are non-trivial. Whenever $A_q^0$ is a torus (resp. unipotent), we say that $A$ has **totally toric** (resp. **totally unipotent**) reduction. In this paper we will only consider abelian varieties with RM so, to simplify the terminology, we drop the adverb “totally”.

As explained in [SGA72, Exposé IX], any abelian variety (having RM is not necessary) over a local field attains semistable reduction over a finite extension of $K$. With this in mind, we introduce.

**Definition 2.23.** Let $A/K$ be an abelian variety, consider $M/K^{\text{unr}}$ be the minimal extension such that $A \times_K M$ has semistable reduction. We define the semistability defect of $A/K$ as $\text{sd}_{A/K} := [M : K^{\text{unr}}]$.

**Remark 2.24.** Assume that $A/K$ has potential good reduction. We can characterise the field extension $M/K^{\text{unr}}$ from Definition 2.23 (and thus the semistability defect of $A/K$) as follows:

* $M$ is the subfield of $K^{\text{sep}}$ fixed by $\ker(\rho_{A, \ell}|_{I_K})$.

* For any $m \ge 3$ coprime to the residue characteristic of $K$, we have $M = K(A[m])$ (see [ST68]).

### 2.4.3 Conductors and associated WD-representations

Until the end of this subsection, we assume that $A/K$ has RM. Fix an embedding $i : \mathbb{Q}_{\ell} \hookrightarrow \mathbb{C}$. In (2.2), we decomposed $\rho_{A, \ell}$ as the direct sum of $\text{Res}_{F_{\lambda}/\mathbb{Q}_{\ell}}(\rho_{A, \lambda})$, where $\lambda$ ranges through the places of $F$ dividing $\ell$. Considering the associated complex WD-representations, we get

$$W_i(\rho_{A, \ell}) \simeq \bigoplus_{\lambda | \ell} \bigoplus_{\ell : F_{\lambda} \to \mathbb{C}} W_{\ell}(\rho_{A, \lambda}), \qquad (2.4)$$

where the second direct sum on the RHS is indexed by embeddings of $F_{\lambda}$ into $\mathbb{C}$ that extend $i$.

Knowing the reduction type of $A$, we can describe the inertial local type of $W_{\ell}(\rho_{A, \lambda})$. The Néron-Ogg-Shafarevich criterion [ST68] states that $A$ has good reduction at $\mathfrak{m}$ if and only $\rho_{A, \ell}(I_K)$ is trivial. Therefore, the Artin conductor of $\rho_{A, \lambda}$ and the inertial local type of $W_{\ell}(\rho_{A, \lambda})$ are non-trivial if and only if $A$ has bad reduction at $\mathfrak{m}$.

If $A/K$ has potential good reduction, let $M/K^{\text{unr}}$ be the extension cut out by $\rho_{A, \ell}|_{I_K}$, so that $\text{Gal}(M/K^{\text{unr}}) \simeq \rho_{A, \ell}(I_K)$. and $[M : K^{\text{unr}}]$ is the semistability defect of $A/K$. Let also $I^t(M/K)$ be the tame part of $\text{Gal}(M/K^{\text{unr}}) \simeq I(M/K)$.

**Proposition 2.25.** All the WD-representations $W_{\ell}(\rho_{A, \lambda})$ are Aut($\mathbb{C}$)-conjugate.

1) If $A$ has potential toric reduction, then every $W_{\ell}(\rho_{A, \lambda})$ is Steinberg.

2) If $A$ has potential good reduction, then every $W_{\ell}(\rho_{A, \lambda})$ is semisimple, and $\det(W_{\ell}(\rho_{A, \lambda})) = \omega_K^{-1}$. Moreover, $W_{\ell}(\rho_{A, \lambda})$ is a principal series $\Leftrightarrow M/K^{\text{unr}}$ is abelian $\Leftrightarrow |I^t(M/K)|$ divides $|\mathbb{F}_{\mathfrak{m}}| - 1$.

*Proof.* See Propositions 3.4 and 3.6, Theorem 4.1 and §5 in [Mel22]. □

We now focus on the Artin conductor of $\rho_{A, \lambda}$. The content of [ACIK+24, Appendix] shows that the conductor of $W_{\ell}(\rho_{A, \lambda})$ does not depend on $\lambda$, and implies the equalities

$$\mathfrak{n}_{\text{tame}}(\rho_{A, \ell}) = [F : \mathbb{Q}] \mathfrak{n}_{\text{tame}}(\rho_{A, \lambda}), \quad \text{and} \quad \mathfrak{n}_{\text{wild}}(\rho_{A, \ell}) = [F : \mathbb{Q}] \mathfrak{n}_{\text{wild}}(\rho_{A, \lambda}) \qquad (2.5)$$

We can compute the tame part of the conductor from the reduction type of the Néron model.

**Proposition 2.26.** The tame part of the conductor of $\rho_{A, \lambda}$ is given by

$$\mathfrak{n}_{\text{tame}}(\rho_{A, \lambda}) = \begin{cases} 1 & \text{if } A \text{ has toric reduction at } \mathfrak{m}, \\ 2 & \text{if } A \text{ has unipotent reduction at } \mathfrak{m}. \end{cases}$$
Proof. Let $T$ and $U$ be the torus and the unipotent group as in (2.3). Grothendieck proved in [SGA72, Exposé IX, §4] that $n_{\text{tame}}(\rho_{A,\ell}) = \dim T + 2 \dim U$. If $\mathcal{A}$ has toric (resp. unipotent) reduction at $\mathfrak{m}$, then $n_{\text{tame}}(\rho_{A,\ell}) = g$ (resp. $2g$). But we have $[F : \mathbb{Q}] = g$, so (2.5) gives the desired result. $\square$

Concerning the wild part of the conductor, we have the following result.

**Lemma 2.27.** If $\mathcal{A}$ attains semistable reduction over a tame extension of $K$, then $n_{\text{wild}}(\rho_{A,\ell}) = 0$. If we further assume that $A/K$ has RM, then $n_{\text{wild}}(\rho_{A,\lambda}) = 0$.

Proof. Let $L$ be a tame extension of $K$ where $\mathcal{A}$ attains semistable reduction. By [SGA72, Exposé IX, §4], we have $n_{\text{wild}}(\rho_{A,\ell}|_{G_L}) = 0$. The result follows by applying Lemma 2.15 and (2.5). $\square$

## 2.5 Background on hyperelliptic curves

We now recall some well-known facts about hyperelliptic curves and their models. Moreover, we briefly present the theory of cluster pictures, which allows to study the local arithmetic of hyperelliptic curves at odd places of bad reduction.

### 2.5.1 Hyperelliptic curves and defining equations

For now, we let $L$ be any field. In this subsection we discuss the main properties of hyperelliptic curves. The reader can find more details on the topic in [Liu02], [Loc94], [Liu96].

**Definition 2.28.** A hyperelliptic curve $C/L$ is a smooth projective curve defined over $L$, such that there exists a finite separable morphism $C \to \mathbb{P}_L^1$ of degree 2.

Let $C/L$ be a hyperelliptic curve, and write $L(C)$ for its function field. Let $\sigma$ be the generator of $\text{Gal}(L(C)/L(\mathbb{P}^1))$. It induces an automorphism of $C$ of order 2, which we still denote by $\sigma$, and call the hyperelliptic involution of $C$. If $C$ has genus $\ge 2$, then $\sigma$ is unique, and the degree 2 morphism $C \to \mathbb{P}_L^1$ is unique up to automorphism. From now on we assume that this is indeed the case, and denote by $g \ge 2$ the genus of $C$. We refer to [Sil86, Sil94] for further details on the elliptic case $g = 1$.

The curve $C$ can be described by a hyperelliptic equation. One can always choose two functions $x \in L(C)^{\langle\sigma\rangle}$ and $y \in L(C)$ such that $\{1, y\}$ is a basis of the integral closure of $L[x]$ in $L(C)$ (see [Liu96, §1]). Then $C$ is described on an affine subset by a so-called Weierstrass equation

$$ (\mathcal{H}) \qquad y^2 + Q(x)y = P(x), $$

where $Q(x), P(x) \in L[x]$, and $\deg Q \le g+1$, $\deg P \le 2g+2$. When $\text{char}(L) \ne 2$, one can choose the function $y$ in such a way that $Q = 0$. When $Q = 0$ and the leading coefficient of $P$ is a non-zero square in $L$, then $C(L) \ne \emptyset$. Indeed $C$ has one or two $L$-points at infinity (i.e., in the chart where $1/x \ne 0$), depending on whether $\deg P$ is odd or even. Following [Liu96], we introduce:

**Definition 2.29.** Let $F(x) = 4P(x) + Q(x)^2$, and denote by $c_F$ the leading coefficient of $F$. When $\text{char}(L) \ne 2$, the discriminant $\Delta(\mathcal{H})$ of equation (H) is defined by

$$ \Delta(\mathcal{H}) = \begin{cases} 2^{-4(g+1)} \text{disc}(F) & \text{if } \deg F = 2g+2, \\ 2^{-4(g+1)} c_F^2 \text{disc}(F) & \text{if } \deg F = 2g+1. \end{cases} $$

In particular, when $Q = 0$ and $P$ is monic, we have $\Delta(\mathcal{H}) = 2^{4g} \text{disc}(P)$, whether $\deg P$ is odd or not. The curve described by (H) is smooth if and only if $\Delta(\mathcal{H}) \ne 0$ (see [Loc94, Theorem 1.7]).

**Definition 2.30.** Given a hyperelliptic equation $(\mathcal{H}) : y^2 + Q(x)y = P(x)$ describing a curve $C/L$, we define the Weierstrass model of $C$ associated to $(\mathcal{H})$ as the glueing of the two open affine schemes

$$ \text{Spec}(L[x, y]/(y^2 + Q(x)y = P(x))) \quad \text{and} \quad \text{Spec}(L[w, z]/(z^2 + w^{g+1}Q(w^{-1})w = w^{2g+2}P(w^{-1}))) $$

along the identification $w = 1/x$, $z = y/x^{g+1}$. We define its discriminant as $\Delta(\mathcal{H})$. If we let $\mathcal{W}$ be the model associated to $(\mathcal{H})$, we will write $\mathcal{W} : y^2 + Q(x)y = P(x)$ to state that $\mathcal{W}$ is described by $(\mathcal{H})$.
A Weierstrass equation defining the hyperelliptic curve $C$ is not unique, so neither is a Weierstrass model of $C$. We now describe the admissible changes of variables to obtain a new hyperelliptic equation for $C$, and the behaviour of the discriminant under such transformations.

**Lemma 2.31.** Let $(\mathcal{H}) : y^2 + Q(x)y = P(x)$ and $(\tilde{\mathcal{H}}) : Y^2 + \tilde{Q}(X)Y = \tilde{P}(X)$ be two hyperelliptic equations describing the curve $C/L$. The coordinates $(x, y)$ and $(X, Y)$ are related by a change of variables

$$x = \frac{aX + b}{cX + d}, \quad \text{and} \quad y = \frac{eY + R(X)}{(cX + d)^{g+1}},$$

with $a, b, c, d, e \in L$, $R(X) \in L[X]$, and $ad - bc, e \neq 0$. We have the equality between discriminants

$$\Delta(\tilde{\mathcal{H}}) = e^{-4(2g+1)}(ad - bc)^{2(g+1)(2g+1)}\Delta(\mathcal{H}).$$

*Proof.* See [Loc94, §1] or [Liu96, §2]. □

**Definition 2.32.** Let $C$ be a hyperelliptic curve described by $(\mathcal{H}) : y^2 = P(x)$. Let $\delta \in L$. We define the quadratic twist of $C$ by $\delta$, which we denote by $C^{(\delta)}$, to be the hyperelliptic curve described by the affine equation $y^2 = \delta P(x)$. When $\deg P$ is odd, we can also describe $C^{(\delta)}$ by

$$Y^2 = \delta^{\deg P} P(X/\delta), \quad \text{where} \quad x = \frac{X}{\delta}, \quad y = \frac{Y}{\delta(1+\deg P)/2}. \qquad (2.6)$$

The curves $C$ and $C^{(\delta)}$ are isomorphic over $L(\sqrt{\delta})$, so if $\delta$ is a square in $L$, then $C \simeq C^{(\delta)}$ over $L$.

## 2.5.2 Local models and cluster pictures

We now discuss the behaviour of hyperelliptic curves defined over local fields. That is why, we recover the previous notation from §2.1 and assume that $L = K$ is local. We keep denoting by $C/K$ a hyperelliptic curve.

**Definition 2.33.** 1) A model of $C/K$ over $\mathcal{O}_m$ is a flat proper $\mathcal{O}_m$-scheme together with a $K$-isomorphism of its generic fiber with $C$.

2) We say that the Weierstrass model $\mathcal{W}$ of $C$ corresponding to $(\mathcal{H}) : y^2 + Q(x)y = P(x)$ is $\mathcal{O}_m$-integral if $P, Q \in \mathcal{O}_m[x]$. In this case, $\mathcal{W} \to \text{Spec}(\mathcal{O}_m)$ is a model in the sense above. We say that $\mathcal{W}$ is a minimal Weierstrass model if $v(\Delta(\mathcal{H}))$ is minimal among all integral Weierstrass models of $C$.

**Definition 2.34.** Let $\mathcal{X} \to \text{Spec}(\mathcal{O}_m)$ be a model of $C$ over $\mathcal{O}_m$, and let $\mathcal{X}_m$ be its special fiber.

1) We say that $\mathcal{X}$ has good reduction at $\mathfrak{m}$ if $\mathcal{X}_m$ is smooth.

2) We say that $\mathcal{X}$ has semistable reduction at $\mathfrak{m}$ if $\mathcal{X}_m$ is geometrically reduced and has only ordinary double points as singularities.

We say that $C/K$ has good (resp. semistable) reduction at $\mathfrak{m}$ if there is some model of $C$ over $\mathcal{O}_m$ having good (resp. semistable) reduction at $\mathfrak{m}$. Deligne and Mumford proved in [DM69] that $C/K$ has semistable reduction if and only if its Jacobian Jac$(C)/K$ has semistable reduction.

**Remark 2.35.** Suppose that $C/K$ is described by the equation $(\mathcal{H}) : y^2 + Q(x)y = P(x)$. Applying a change of variables as in Lemma 2.31 modifies the valuation of $\Delta(\mathcal{H})$ by adding a multiple of $2(2g+1)$. Therefore, if $v_m(\Delta(\mathcal{H})) \neq 0 \mod 2(2g+1)$, then $C$ does not have good reduction over $\mathcal{O}_m$.

Denote by $C \to \text{Spec}(\mathcal{O}_K)$ the minimal regular model of $C$, and by $\mathcal{C}_m$ its special fiber (see [Liu02, §8.3] for details about its construction and main properties). If $C(K) \neq \emptyset$, then $C$ is the minimal desingularization of the minimal Weierstrass model of $C$. In practice, if $\mathcal{W}$ is a minimal Weierstrass model, we construct $\mathcal{C}$ by successively blowing-up $\mathcal{W}$ along its singular points and normalising a minimal amount of times until we obtain a regular model that contains no exceptional divisors. Assume that the lcm of the multiplicities of the irreducible components of the special fiber is 1 (e.g. $C(K) \neq \emptyset$ or $\mathcal{C}_m$ is reduced). Then, the special fiber of the Néron model of the Jacobian Jac$(C)$ is $\text{Pic}^0(\mathcal{C}_m)$, so the reduction type of the Néron model of Jac$(C)$ is encoded in the geometry of $\mathcal{C}_m$.

The following criterion is very helpful for checking if a scheme is regular or not.
**Lemma 2.36.** Let $(A, \mathfrak{m})$ be a regular Noetherian local ring, and let $f \in \mathfrak{m} \setminus \{0\}$. Then $A/fA$ is regular if and only if $f \notin \mathfrak{m}^2$.

*Proof.* This is Corollary 2.12 in [Liu02, §4.2.2]. □

From now on, we assume that the residue characteristic of $K$ is odd. The theory of cluster pictures provides a combinatorial way to describe the special fiber of the minimal regular model, and to deduce arithmetic information concerning $C$ and $Jac(C)$. We refer to [DDMM23] for further details on the theory, or to [BBB+22] for a survey. Assume that $C/K$ is described by the Weierstrass equation

$$y^2 = P(x) = c_P \prod_{\gamma \in \mathcal{R}} (x - \gamma)$$

where $c_P \in K^\times$, $P(x) \in K[x]$ is separable, and $\mathcal{R}$ denotes the set of roots of $P$ in $\overline{K}$.

**Definition 2.37.** A cluster is a non-empty subset $\mathfrak{s} \subseteq \mathcal{R}$ of the form $\mathfrak{s} = D \cap \mathcal{R}$ for some disc $D = \{x \in \overline{K} \mid v_m(x - z) \ge d\}$, some $z \in \overline{K}$ and some $d \in \mathbb{Q}$. If $|\mathfrak{s}| > 1$, we say that $\mathfrak{s}$ is a proper cluster, and define its depth $d_{\mathfrak{s}}$ as the maximal $d$ for which $\mathfrak{s}$ is cut out by such a disc, i.e., $d_{\mathfrak{s}} = \min_{\gamma, \gamma' \in \mathfrak{s}} v_m(\gamma - \gamma')$. If $\mathfrak{s} \neq \mathcal{R}$, the parent cluster $P(\mathfrak{s})$ of $\mathfrak{s}$ is the smallest cluster with $\mathfrak{s} \subsetneq P(\mathfrak{s})$, and the relative depth of $\mathfrak{s}$ is $\delta_{\mathfrak{s}} = d_{\mathfrak{s}} - d_{P(\mathfrak{s})}$.

We refer to this data as the cluster picture of $C$.

We introduce some terminology concerning cluster pictures (see [BBB+22] for further details).

**Definition 2.38.** • If $\mathfrak{s}' \subsetneq \mathfrak{s}$ is a maximal subcluster, we say that $\mathfrak{s}'$ is a child of $\mathfrak{s}$. For two clusters $\mathfrak{s}_1, \mathfrak{s}_2$ write $\mathfrak{s}_1 \wedge \mathfrak{s}_2$ for the smallest cluster containing both of them.

• A cluster $\mathfrak{s}$ such that $|\mathfrak{s}|$ is odd (resp. even) is called an odd (resp. even) cluster.

• A cluster consisting of precisely two roots is called a twin.

• An even cluster whose children are all even is called übereven.

• A non-übereven cluster with a child of size $2g$ is called a cotwin.

• A cluster $\mathfrak{s}$ is called principal if one of the following holds

$$\begin{cases} |\mathfrak{s}| \neq 2g + 2 \text{ and } \mathfrak{s} \text{ is proper, not a twin or a cotwin;} \\ |\mathfrak{s}| = 2g + 2 \text{ and } \mathfrak{s} \text{ has more than 3 children.} \end{cases}$$

• Recall that $c_P$ denotes the leading coefficient of $P$. We define $\nu_{\mathfrak{s}} = v_m(c_P) + |\mathfrak{s}|d_{\mathfrak{s}} + \sum_{\gamma \notin \mathfrak{s}} d_{\{\gamma\} \wedge \mathfrak{s}}$.

Cluster pictures provide criteria to check if $C/K$ has good, or semistable reduction. The following two theorems are (part of) [DDMM23, Theorem 10.8].

**Theorem 2.39.** The hyperelliptic curve $C/K$ has good reduction if and only if the following three conditions are satisfied:

1) the field extension $K(\mathcal{R})/K$ is unramified,

2) every proper cluster has size at least $2g + 1$,

3) the (necessarily unique) principal cluster has $\nu_{\mathfrak{s}} \in 2\mathbb{Z}$.

**Theorem 2.40.** The hyperelliptic curve $C/K$ (equivalently $Jac(C)$) is semistable if and only if the following three conditions are satisfied:

1) the field extension $K(\mathcal{R})/K$ has ramification index at most 2,

2) every proper cluster is invariant under the action of the inertia group $I_K$,

3) every principal cluster $\mathfrak{s}$ has $d_{\mathfrak{s}} \in \mathbb{Z}$ and $\nu_{\mathfrak{s}} \in 2\mathbb{Z}$.

Moreover, [DDMM23, Theorem 11.3], which we recall below, gives the wild conductor of the $\ell$-adic representation arising from the action of $G_K$ on $V_\ell(Jac(C))$. Recall that $g$ denotes the genus of $C$.
**Theorem 2.41.** The wild conductor of $\rho_{Jac(C), \ell} : G_K \to \text{GL}_{2g}(\mathbb{Q}_\ell)$ is given by

$$ \mathfrak{n}_{\text{wild}}(\rho_{\text{Jac}(C), \ell}) = \sum_{\gamma \in \mathcal{R}/G_K} (v(\Delta(K(\gamma)/K)) - [K(\gamma) : K] + f_{K(\gamma)/K}). $$

The sum is taken over representatives of orbits in $\mathcal{R}$ under the action of $G_K$, and $\Delta(K(\gamma)/K)$ (resp. $f_{K(\gamma)/K}$) denotes the discriminant (resp. residue degree) of the extension $K(\gamma)/K$.

# 3 Construction of Frey representations and curves

We now recall the definition of Frey representations, and its construction for the specific cases of signatures $(p, p, r)$ and $(r, r, p)$. We will then obtain by specialisation Frey curves attached to putative solutions to the generalised Fermat equations of the mentioned signatures. We will conclude the section by presenting a common framework to deal with both signatures at once.

## 3.1 Geometric construction of Frey representations

In this section, we recall the construction of Frey representations of signature $(p, p, r)$ and $(r, r, p)$. Our main references are [Dar00] for the former signature, and [BCDF23] for the latter.

Recall that we introduced above two primes $r \ge 5$, $p > r$, and that we denote by $\mathcal{K} = \mathbb{Q}(\omega)$ the maximal totally real subfield of $\mathbb{Q}(\zeta_r)$. Let $\mathbb{F}$ be a finite field. Denote by $\mathbb{P}^1$ the projective line over $\overline{\mathcal{K}}$, with local coordinate $s$ and function field $\overline{\mathcal{K}}(s)$. Let $x \in \mathbb{P}^1(\mathbb{C})$, $x \ne 0, 1, \infty$, and consider the topological fundamental group $\pi_1(\mathbb{P}^1(\mathbb{C}) \setminus \{0, 1, \infty\}, x)$. Its profinite completion equals the Galois group $\text{Gal}(\Omega/\overline{\mathcal{K}}(s))$, where $\Omega \subset \overline{\mathcal{K}}(s)$ is the maximal extension of $\overline{\mathcal{K}}(s)$ unramified outside the points $0, 1, \infty$ (see [Ser08, §6.3]). Thus, any continuous representation of $\pi_1(\mathbb{P}^1(\mathbb{C}) \setminus \{0, 1, \infty\}, x)$, usually called a *monodromy* representation, extends to $\text{Gal}(\Omega/\overline{\mathcal{K}}(s))$, and thus to $G_{\overline{\mathcal{K}}(s)}$. Since the latter is a normal subgroup of $G_{\mathcal{K}(s)}$, we obtain in this way continuous representations of $G_{\mathcal{K}(s)}$. We refer to [MP24] for broader use of monodromy representations in the context of Darmon's program.

**Definition 3.1.** A Frey representation of signature $(p, p, r)$ (resp. $(r, r, p)$) is a Galois representation $\varrho : G_{\mathcal{K}(s)} \to \text{GL}_2(\mathbb{F})$ satisfying the following:

1) The restriction of $\varrho$ to $G_{\overline{\mathcal{K}}(s)}$ has trivial determinant and is irreducible. Denote its projectivization by $\overline{\varrho}^{\text{geom}} : G_{\overline{\mathcal{K}}(s)} \to \text{PSL}_2(\mathbb{F})$.

2) The homomorphism $\overline{\varrho}^{\text{geom}}$ is unramified outside $\{0, 1, \infty\}$.

3) $\overline{\varrho}^{\text{geom}}$ maps the inertia subgroups at $0, 1, \infty$ to subgroups of $\text{PSL}_2(\mathbb{F})$ of order $p, p, r$ (resp. $r, r, p$).

### 3.1.1 Frey representations of signature $(p, p, r)$

We now proceed to recall the geometric construction of Frey representations of signatures $(p, p, r)$. We refer to [Dar00, §1.3] for details. Recall that $h_r \in \mathbb{Z}[x]$ denotes the minimal polynomial of $\omega = \zeta_r + \zeta_r^{-1}$.

**Definition 3.2.** We define $C_r^-(s), C_r^+(s)/\mathbb{Q}(s)$ to be the hyperelliptic curves given by the equations

$$ (\mathcal{H}_{r,s}^-) : y^2 = g_{r,s}^-(x) \quad \text{and} \quad (\mathcal{H}_{r,s}^+) : y^2 = g_{r,s}^+(x) $$

respectively, where $g_{r,s}^-(x), g_{r,s}^+(x) \in \mathbb{Z}[x]$ are the polynomials defined by

$$ (3.1) \qquad g_{r,s}^-(x) := (-1)^{\frac{r-1}{2}} x h_r(2-x^2) + 2 - 4s, \quad \text{and} \quad g_{r,s}^+(x) := g_{r,s}^-(x)(x+2). $$

We also define $\text{Jac}(C_r^{\pm}(s))/\mathbb{Q}(s)$ to be the Jacobian of $C_r^{\pm}(s)$, and let $J_r^{\pm}(s) := \text{Jac}(C_r^{\pm}(s)) \times_{\mathbb{Q}(s)} \mathcal{K}(s)$.
**Remark 3.3.** We point out that the notation above slightly differs from the one used by Darmon. In [Dar00, §1.3], he defines $C_r^-(s)$ using the minimal polynomial $-\omega$, which he denotes by $g$. The polynomials $h_r$ and $g$ are related by the equality $h_r(x) = (-1)^{\frac{r-1}{2}} g(-x)$.

The polynomial $g_{r,s}^-$ has degree $r$, and $g_{r,s}^+$ has degree $r+1$, so both curves $C_r^\pm(s)$ have genus $\frac{r-1}{2}$. The polynomial $g_{r,s}^-$ can be expanded: using [BCDF23, Lemma 2.16], we rewrite $(\mathcal{H}_{r,s}^-)$ as

$$ (\mathcal{H}_{r,s}^-) \qquad y^2 = \sum_{k=0}^{\frac{r-1}{2}} (-1)^k c_k x^{r-2k} + 2 - 4s, \quad \text{where} \quad c_k := \frac{r}{r-k} \binom{r-k}{k} \in \mathbb{Z}. $$

Given the equality $g_{r,s}^+(x) = (x+2)g_{r,s}^-(x)$, we easily deduce expanded equations defining $C_r^+(s)$.

**Example 3.4.** For small values of $r$, the hyperelliptic curve $C_r^-(s)$ is described by

$$ (\mathcal{H}_{3,s}^-) \qquad y^2 = x^3 - 3x + 2 - 4s, $$

$$ (\mathcal{H}_{5,s}^-) \qquad y^2 = x^5 - 5x^3 + 5x + 2 - 4s, $$

$$ (\mathcal{H}_{7,s}^-) \qquad y^2 = x^7 - 7x^5 + 14x^3 - 7x + 2 - 4s, $$

$$ (\mathcal{H}_{11,s}^-) \qquad y^2 = x^{11} - 11x^9 + 44x^7 - 77x^5 + 55x^3 - 11x + 2 - 4s. $$

**Theorem 3.5.** The abelian varieties $J_r^\pm(s)/K(s)$ have RM by $K$, and more precisely

$$ \mathrm{End}_{K(s)}(J_r^\pm(s)) \simeq \mathcal{O}_K. $$

*Proof.* This follows from Theorem 1, Corollary 6 in [TTV91]. See also [DM00, Proposition 2.1]. □

Let now $\mathfrak{p}$ be a place of $K$ lying above $\mathfrak{p}$, $K_{\mathfrak{p}}$ be the completion of $K$ at $\mathfrak{p}$, and $\mathbb{F}_{\mathfrak{p}}$ its residue field. The discussion in §2.4.1 applies to $J_r^\pm(s)$, giving rise to 2-dimensional representations

$$ \rho_{J_r^\pm(s), \mathfrak{p}} : G_{K(s)} \to \mathrm{GL}_2(\mathcal{K}_{\mathfrak{p}}) \quad \text{and} \quad \bar{\rho}_{J_r^\pm(s), \mathfrak{p}} : G_{K(s)} \to \mathrm{GL}_2(\mathbb{F}_{\mathfrak{p}}) $$

**Theorem 3.6.** Both $\bar{\rho}_{J_r^+(s), \mathfrak{p}}$ and $\bar{\rho}_{J_r^-(s), \mathfrak{p}}$ are Frey representations of signature $(p, p, r)$. The former is even, and the latter is odd in the sense of [Dar00, Definition 1.4].

*Proof.* See [Dar00, Theorem 1.10]. □

### 3.1.2 Frey representations of signature $(r, r, p)$

We now introduce a Frey representation of signature $(r, r, p)$, building upon the construction of the curve $C_r^-(s)$ introduced above. In [Pac25], Pacetti explains how to recover this construction using the theory of hypergeometric motives. We will not discuss these topics here, and we refer the curious reader to the mentioned reference. The discussion below is inspired by the results in [BCDF23, §2.4].

In order to simplify the notation, let $\tilde{s} := 2 - 4s$ be a new indeterminate in $\bar{K}(s)$. With this notation, the constant term of $(\mathcal{H}_{r,s}^-)$ is simply $\tilde{s}$. Consider two new copies of the projective line over $\bar{K}$, which we denote by $\mathbb{P}_{\tilde{s}}^1$ and $\mathbb{P}_t^1$, with local coordinates $\tilde{s}$ and $t$. The definition of $\tilde{s}$ implies an equality of function fields $\bar{K}(s) = \bar{K}(\tilde{s})$. Let $\mathcal{B}/\bar{K}$ be the smooth projective curve defined by the affine equation

$$ t(t-1) = \frac{1}{\tilde{s}^2 - 4}, \quad \text{or equivalently,} \quad t(t-1) = \frac{1}{2^4 s(s-1)}. \qquad (3.2) $$

Denote by $\bar{K}(\mathcal{B})$ its function field. The projection onto the $s$ and $t$ coordinates give two morphisms $\mathcal{B} \to \mathbb{P}_s^1$ and $\mathcal{B} \to \mathbb{P}_t^1$ of degree two, having 0 and 1 as branching points. Therefore, $\bar{K}(\mathcal{B})$ is a quadratic extension of both function fields $\bar{K}(\tilde{s}) = \bar{K}(s)$ and $\bar{K}(t)$. Equality (3.2) shows that the points $0, 1 \in \mathbb{P}_s^1$ (or equivalently, $\pm 2 \in \mathbb{P}_s^1$) correspond to $\infty \in \mathbb{P}_t^1$, and similarly $0, 1 \in \mathbb{P}_t^1$ correspond to $\infty \in \mathbb{P}_s^1$ (and thus to $\infty \in \mathbb{P}_s^1$).
**Remark 3.7.** The authors of [BCDF23] already introduce the function field $\overline{\mathcal{K}}(\mathcal{B})$. However, their notation is different, as they introduce the curve $\mathcal{B}'/\overline{\mathcal{K}}$ described by $t(1-t) = 1/(\sigma^2+4)$. This difference between the considered models is due to the fact that the authors of [BCDF23] do not manipulate $C_r^-(s)$, but its quadratic twist by $i = \sqrt{-1}$. If we were to consider $\mathcal{B}$ and $\mathcal{B}'$ as curves over $\mathcal{K}$ (instead of $\overline{\mathcal{K}}$) defined by the same equations, their function fields would not be isomorphic.

Define $\tilde{\delta} := (2t-1)/\tilde{s} = (2t-1)/(2-4s) \in \overline{\mathcal{K}}(\mathcal{B})$. Using (3.2), one can check that $\tilde{s}^2 = \frac{(2t-1)^2}{t(t-1)}$, so $\tilde{\delta}^2 = t(t-1)$. Consider $C_r^-(s)^{(\tilde{\delta})}$, the quadratic twist by $\tilde{\delta}$ of the base change of $C_r^-(s)$ to $\mathcal{K}(\mathcal{B})$. Thanks to $(\mathcal{H}_{r,s}^-)$, we describe $C_r^-(s)^{(\tilde{\delta})}$ by the hyperelliptic equation

$$ (\mathcal{H}_{r,t}) \qquad y^2 = \sum_{k=0}^{\frac{r-1}{2}} (-1)^k c_k (t(t-1))^k x^{r-2k} + (t(t-1))^{\frac{r-1}{2}} (2t-1). $$

A priori, $C_r^-(s)^{(\tilde{\delta})}$ is defined over $\overline{\mathcal{K}}(\mathcal{B})$, but $(\mathcal{H}_{r,t})$ defines a model of $C_r^-(s)^{(\tilde{\delta})}$ over $\mathbb{Q}(t)$.

**Definition 3.8.** We let $C_r(t)/\mathbb{Q}(t)$ be the hyperelliptic curve given by the Weierstrass equation $(\mathcal{H}_{r,t})$. We also define Jac$(C_r(t))/\mathbb{Q}(t)$ to be the Jacobian of $C_r(t)$, and let $J_r(t) := Jac(C_r(t)) \times_{\mathbb{Q}(t)} \mathcal{K}(t)$.

**Theorem 3.9.** The abelian variety $J_r(t)/\mathcal{K}(t)$ has RM by $\mathcal{K}$.

*Proof.* This is [BCDF23, Theorem 2.38]: although it only states that $\mathcal{K} \hookrightarrow \text{End}_{\mathcal{K}(t)}(J_r(t)) \otimes \mathbb{Q}$, the arguments in its proof show that the inclusion is actually an isomorphism. $\square$

Recall that $\mathfrak{p}$ is a place of $\mathcal{K}$ lying above $p$. Again, the discussion in section 2.4 applies to $J_r(t)$, giving rise to 2-dimensional representations

$$ \rho_{J_r(t), \mathfrak{p}} : G_{\mathcal{K}(t)} \to \text{GL}_2(\mathcal{K}_{\mathfrak{p}}) \quad \text{and} \quad \bar{\rho}_{J_r(t), \mathfrak{p}} : G_{\mathcal{K}(t)} \to \text{GL}_2(\mathbb{F}_{\mathfrak{p}}) $$

**Theorem 3.10.** The representation $\bar{\rho}_{J_r(t), \mathfrak{p}}$ is a Frey representation of signature $(r, r, p)$.

*Proof.* This is a consequence of Theorem 3.6, and the fact that $0, 1 \in \mathbb{P}_s^1$ (resp. $\infty \in \mathbb{P}_s^1$) correspond to $\infty \in \mathbb{P}_t^1$ (resp. $0, 1 \in \mathbb{P}_t^1$). See Theorem 2.38 and Lemma 2.3 in [BCDF23] for details. $\square$

## 3.2 Frey curves attached to putative solutions

We now discuss how to associate Frey curves to putative solutions to the generalised Fermat equations $(\mathcal{E}_{p,p,r})$ and $(\mathcal{E}_{r,r,p})$. Those are obtained as specialisations of the curves giving rise to the corresponding Frey representations. We first construct such curves for the signature $(p, p, r)$, and then for the signature $(r, r, p)$. Both constructions fit into a single framework, which will allow for a uniform study of both signatures.

Recall that we introduced $r, p, A, B, C$ in Definition 2.1.

### 3.2.1 Frey curves for the signature $(p, p, r)$

The generalised Fermat equation of signature $(p, p, r)$, which we denote by $(\mathcal{E}_{p,p,r})$, is

$$ (\mathcal{E}_{p,p,r}) \qquad Ax^p + By^p = Cz^r. $$

Assume there exists a primitive non-trivial solution $(a, b, c) \in \mathbb{Z}^3$ to $(\mathcal{E}_{p,p,r})$. Following [Dar00], we construct two Frey curves associated to $(a, b, c)$ as follows. Fix $s_0 := Aa^p/Cc^r$, and let $C_r^{\pm}(Aa^p/Cc^r)$ be the specialization at $s = s_0$ of the curve $C_r^{\pm}(s)/\mathbb{Q}(s)$ introduced in §3.1.1.

**Definition 3.11.** We define $C_r^-(a, b, c)/\mathbb{Q}$ to be the quadratic twist by $Cc$ of $C_r^-(Aa^p/Cc^r)$, and $C_r^+(a, b, c) := C_r^+(Aa^p/Cc^r)$. We call these the *Frey curves* of signature $(p, p, r)$ attached to $(a, b, c)$.
We note that, since $g_{r,s_0}^+$ has even degree, one can apply a change of variables as in Lemma 2.31 to obtain an integral hyperelliptic equation for $C_r^+(s_0)$, whose RHS polynomial is monic. On the other hand, $\deg(g_{r,s_0}^-)$ is odd, so if one applies a change of variables as in Lemma 2.31 to get rid of the denominator of $2-4s_0$, the RHS polynomial cannot be monic. In order to have a monic polynomial on the RHS, one needs to consider a twist of the curve. This motivates the definition of $C_r^-(a, b, c)$ as a twist of the specialization, whereas $C_r^+(a, b, c)$ is the specialization itself.

Using $(\mathcal{H}_{r,s}^-)$, we describe the curves $C_r^{\pm}(a, b, c)$ by the equations

$$ (\mathcal{H}_r^-(a, b, c)) \qquad y^2 = \sum_{k=0}^{\frac{r-1}{2}} (-1)^k c_k (Cc)^{2k} x^{r-2k} + 2C^{r-1}(Bb^p - Aa^p), \\ (\mathcal{H}_r^+(a, b, c)) \qquad y^2 = \left( \sum_{k=0}^{\frac{r-1}{2}} (-1)^k c_k (Cc)^{2k} x^{r-2k} + 2C^{r-1}(Bb^p - Aa^p) \right) (x + 2Cc). $$

The reader can find more details about these curves in [Dar00] and [CK22] for the particular cases $A=B=C=1$, with an emphasis on the case $r=5$ in the second reference. When $r=3$, the elliptic curve $C_3^+(a, b, c)$ was studied by Bennett, Vatsal, Yazdani in [BVY04, §2]. However, the defining model therein is different, and the roles of $Aa^3$ and $Bb^3$ are switched compared to our presentation.

### 3.2.2 Frey curve for the signature $(r, r, p)$

The generalised Fermat equation of signature $(r, r, p)$, which we denote by $(\mathcal{E}_{r,r,p})$, is

$$ (\mathcal{E}_{r,r,p}) \qquad Ax^r + By^r = Cz^p. $$

Assume there exists a primitive non-trivial solution $(a, b, c) \in \mathbb{Z}^3$ to $(\mathcal{E}_{r,r,p})$. Following [BCDF23], we construct a Frey curve associated to $(a, b, c)$ as follows. Fix $t_0 := Aa^r/Cc^p$, and let $C_r(Aa^r/Cc^p)$ be the specialization at $t = t_0$ of the curve $C_r(t)/\mathbb{Q}(t)$ introduced in the §3.1.2. Thanks to $(\mathcal{H}_{r,t})$, we see that $C_r(Aa^r/Cc^p)$ is described by

$$ y^2 = \sum_{k=0}^{\frac{r-1}{2}} c_k \frac{(ABa^r b^r)^k}{(Cc^p)^{2k}} x^{r-2k} + \frac{(-ABa^r b^r)^{\frac{r-1}{2}} (Aa^r - Bb^r)}{(Cc^p)^r}. $$

**Definition 3.12.** We define $C_r(a, b, c)/\mathbb{Q}$ to be the quadratic twist by $-Cc^p/(-ab)^{\frac{r-1}{2}}$ of the specialization $C_r(Aa^r/Cc^p)$. We call it the *Frey curve* of signature $(r, r, p)$ attached to $(a, b, c)$.

Again, applying a change of variables as in Lemma 2.31, we describe $C_r(a, b, c)$ by

$$ (\mathcal{H}_r(a, b)) \qquad y^2 = \sum_{k=0}^{\frac{r-1}{2}} c_k (ABab)^k x^{r-2k} + (AB)^{\frac{r-1}{2}} (Bb^r - Aa^r). $$

This curve generalises a construction by Kraus. In the particular case of trivial coefficients, i.e., $A=B=C=1$, the reader can find details on this curve in [BCDF23]. When $r=3$, the elliptic curve $C_3(a, b, c)$ was studied in by Bennett and Dahmen ([BD13, §13]) for odd coefficients $A, B$ (we point out that our notation slightly differs from the one therein).

The following elementary remark will have huge Diophantine consequences:

**Remark 3.13.** Assume first that $(a, b, c)$ is a primitive non-trivial solution to the GFE $(\mathcal{E}_{r,r,p})$. The Frey curve attached to $(a, b, c)$ is presented by a model $(\mathcal{H}_r(a, b))$ that depends only on the coefficients $A, B$ and the members of the solution $a, b$.

Assume now that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{p,p,r})$. We have $Bb^p = Cc^r - Aa^p$. Using this equality, one can rewrite the defining equations $(\mathcal{H}_r^-(a, b, c))$ and $(\mathcal{H}_r^+(a, b, c))$, using only
the parameters A, C and the members of the solution a, c. For instance, the former hyperelliptic equation can be rewritten as

$$ (\mathcal{H}_r^-(a, c)) \qquad y^2 = \sum_{k=0}^{\frac{r-1}{2}} (-1)^k c_k (Cc)^{2k} x^{r-2k} + 2C^{r-1}(Cc^r - 2Aa^p). $$

One can also give a model $C_r^+(a, b, c)$ depending only on A, C, a, c. The Frey curves depending only on 2 among the 3 parameters will be used in §6 for solving many GFEs at the cost of only solving one.

### 3.3 Common framework to both signatures

By construction, the curve $C_r^-(a, b, c)$ is a quadratic twist of the specialisation $C_r^-(Aa^p/Cc^r)$. Recall from §3.1.2 that the curve $C_r(t)/\mathcal{K}(\mathcal{B})$ is a quadratic twist of $C_r^-(s)/\mathcal{K}(\mathcal{B})$. Therefore, $C_r(a, b, c)$ can also be obtained as a twist of a specialization of $C_r^-(s)/\mathcal{Q}(s)$. In this subsection we introduce a common framework covering both signatures, and we explain how to recover each of the Frey curves associated to solutions by correctly choosing the parameters introduced below.

**Definition 3.14.** We fix $s_0 \in \overline{\mathbb{Q}} \setminus \{0, 1\}$, and $\delta_{\mathbb{Q}} \in \overline{\mathbb{Z}} \setminus \{0\}$ such that $\delta_{\mathbb{Q}}^r(2 - 4s_0)$ and $\delta_{\mathbb{Q}}^2$ belong to $\mathbb{Z}$.

**Lemma 3.15.** We have $2^4\delta_{\mathbb{Q}}^{2r}s_0(s_0 - 1) \in \mathbb{Z}$, and $s_0(s_0 - 1) \in \mathbb{Q}$. If $\delta_{\mathbb{Q}} \in \mathbb{Z}$, then $s_0 \in \mathbb{Q}$.

*Proof.* By assumption, $\delta_{\mathbb{Q}}^r(2 - 4s_0) \in \mathbb{Z}$, so its square $2^2\delta_{\mathbb{Q}}^{2r}(4s_0(s_0 - 1) + 1)$ also belongs to $\mathbb{Z}$. But $\delta_{\mathbb{Q}}^2 \in \mathbb{Z}$, so $2^4\delta_{\mathbb{Q}}^{2r}s_0(s_0 - 1) = (\delta_{\mathbb{Q}}^r(2 - 4s_0))^2 - 4\delta_{\mathbb{Q}}^{2r}$ also belongs to $\mathbb{Z}$. Finally, dividing the last term by $2^4\delta_{\mathbb{Q}}^{2r}$, we deduce that $s_0(s_0 - 1) \in \mathbb{Q}$. The last claim follows easily. $\square$

**Definition 3.16.** Denote by $C_r^{\pm}(s_0)$ the specialisation of $C_r^{\pm}(s)/\mathbb{Q}(s)$ at $s = s_0$, and by $C_r^-(s_0)^{(\delta_{\mathbb{Q}})}$ the quadratic twist by $\delta_{\mathbb{Q}}$ of $C_r^-(s_0)$. After applying to $C_r^-(s_0)^{(\delta_{\mathbb{Q}})}$ and $C_r^+(s_0)$ changes of variables as in Lemma 2.31, we describe these curves by the hyperelliptic equations $(\mathcal{H}_{r,s_0}^{-(\delta_{\mathbb{Q}})}) : y^2 = g_r^-(x)$ and $(\mathcal{H}_{r,s_0}^{+(\delta_{\mathbb{Q}})}) : y^2 = g_r^+(x)$, where

$$ g_r^-(x) := \delta_{\mathbb{Q}}^r g_{r,s_0}(x/\delta_{\mathbb{Q}}) \in \mathbb{Q}[x], \quad \text{and} \quad g_r^+(x) := \delta_{\mathbb{Q}}^{r+1} g_{r,s_0}^+(x/\delta_{\mathbb{Q}}) \in \mathbb{Q}[x]. $$

Just as in $(\mathcal{H}_{r,s}^-)$, we can rewrite the defining equations above in terms of expanded sums

$$ (\mathcal{H}_{r,s_0}^{-(\delta_{\mathbb{Q}})}) \qquad y^2 = \sum_{k=0}^{\frac{r-1}{2}} (-1)^k c_k \delta_{\mathbb{Q}}^{2k} x^{r-2k} + \delta_{\mathbb{Q}}^r (2 - 4s_0), $$

$$ (\mathcal{H}_{r,s_0}^{+(\delta_{\mathbb{Q}})}) \qquad y^2 = \left( \sum_{k=0}^{\frac{r-1}{2}} (-1)^k c_k \delta_{\mathbb{Q}}^{2k} x^{r-2k} + \delta_{\mathbb{Q}}^r (2 - 4s_0) \right) (x + 2\delta_{\mathbb{Q}}). $$

**Remark 3.17.** As we will see in Proposition 3.18, in practice, $s_0$ has a non-trivial denominator. We introduce the parameter $\delta_{\mathbb{Q}}$ to obtain integral models for $C_r^-(s_0)^{(\delta_{\mathbb{Q}})}$ and $C_r^+(s_0)$. The assumptions $\delta_{\mathbb{Q}}^2, \delta_{\mathbb{Q}}^r(2 - 4s_0) \in \mathbb{Z}$ show that $g_r^- \in \mathbb{Z}[x]$, and $(\mathcal{H}_{r,s_0}^{-(\delta_{\mathbb{Q}})})$ defines a model of $C_r^-(s_0)^{(\delta_{\mathbb{Q}})}$ over $\mathbb{Z}$. Moreover, if $\delta_{\mathbb{Q}} \in \mathbb{Z}$, then $g_r^+ \in \mathbb{Z}[x]$, and $(\mathcal{H}_{r,s_0}^{+(\delta_{\mathbb{Q}})})$ also defines a model of $C_r^+(s_0)$ over $\mathbb{Z}$. From now on, we only consider the curve $C_r^+(s_0)$ when $\delta_{\mathbb{Q}}$ is an element of $\mathbb{Z}$.

**Proposition 3.18.** 1) Assume that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{p,r})$. Then the curve $C_r^-(a, b, c)$ (resp. $C_r^+(a, b, c)$) is the curve $C_r^-(s_0)^{(\delta_{\mathbb{Q}})}$ (resp. $C_r^+(s_0)$), described by $(\mathcal{H}_{r,s_0}^{-(\delta_{\mathbb{Q}})})$ (resp. $(\mathcal{H}_{r,s_0}^{+(\delta_{\mathbb{Q}})})$), with the specific choices

$$ s_0 = \frac{Aa^p}{Cc^r}, \quad \text{and} \quad \delta_{\mathbb{Q}} = Cc. $$
2) Assume that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{r,r,p})$, and let $z_0 := \sqrt{-ABab} \in \overline{\mathbb{Q}}$ be a square root of $-ABab$. Then the curve $C_r(a, b, c)/\overline{\mathbb{Q}}$ is the curve $C_r^-(s_0)^{(\delta_{\overline{\mathbb{Q}}})}$, described by $(\mathcal{H}_{r,s_0}^{-(\delta_{\overline{\mathbb{Q}}})})$, with the specific choices

$$s_0 = \frac{1}{2} + \frac{(AB)^{\frac{r-1}{2}}(Aa^r - Bb^r)}{4z_0^r}, \quad \text{and} \quad \delta_{\overline{\mathbb{Q}}} = z_0.$$

*Proof.* The first statement follows directly from the construction of $C_r^{\pm}(a, b, c)$ done in §3.2.1. Assume now that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{r,r,p})$. Let $t_0 = Aa^r/Cc^p$, consider $s_0$ as in the Proposition, and let $\tilde{s}_0 := 2 - 4s_0 = (AB)^{\frac{r-1}{2}}(Bb^r - Aa^r)/z_0^r$. Since $Aa^r + Bb^r = Cc^p$, we have the equalities

$$t_0(t_0 - 1) = \frac{-ABa^r b^r}{(Aa^r + Bb^r)^2} \quad \text{and} \quad \tilde{s}_0^2 = \frac{(Bb^r - Aa^r)^2}{-ABa^r b^r} = \frac{1}{t_0(t_0 - 1)} + 4,$$

so $(\tilde{s}_0, t_0)$ defines a point in $\mathcal{B}(\overline{\mathcal{K}})$. Recall that $C_r(t_0)$ is the quadratic twist of $C_r^-(s_0)$ by

$$\frac{2t_0 - 1}{\tilde{s}_0} = \frac{(Aa^r - Bb^r) z_0^r}{Cc^p (AB)^{\frac{r-1}{2}} (Bb^r - Aa^r)} = \frac{-z_0^r}{Cc^p (AB)^{\frac{r-1}{2}}}.$$

Moreover, $C_r(a, b, c)$ is the quadratic twist of $C_r(t_0)$ by $-Cc^p/(-ab)^{\frac{r-1}{2}}$, so composing twists, we conclude that $C_r(a, b, c)$ is the quadratic twist of $C_r^-(s_0)$ by $z_0$. $\square$

Whenever the element $\delta_{\overline{\mathbb{Q}}}$ does not belong to $\mathbb{Z}$, the curve $C_r^+(s_0)$ does not necessarily admit a model over $\mathbb{Z}$. For instance, given a putative non-trivial solution to $(\mathcal{E}_{r,r,p})$, the twisting parameter $\delta_{\overline{\mathbb{Q}}} = z_0$ is not an element of $\mathbb{Z}$ in general. This is why, when solving the GFE of signature $(r, r, p)$, one cannot employ $C_r^+(s_0)$, and only the curve $C_r^-(s_0)^{(\delta_{\overline{\mathbb{Q}}})}$ is available.

**Remark 3.19.** We note that, whenever $(a, b, c)$ is a solution to $(\mathcal{E}_{r,r,p})$, our choice of $z_0$ as in Proposition 3.18 differs from the one done in [BCDF23]. In loc. cit., it is suggested to choose $z_0 = \sqrt{-ABab}$, as the authors of [BCDF23] manipulate the quadratic twist by $i$ of $C_r^-(s_0)$. Choosing $z_0$ to be a square root of $-ABab$ allows one to work directly with $C_r^-(s_0)$, without having to twist by $i$.

**Remark 3.20.** In [CK25], the authors explain that one can attach a hyperelliptic curve to a putative solution to the generalised Fermat equation of signature $(2, r, p)$. Its construction relies on that of Kraus' hyperelliptic curve, introduced in §3.2.2. It would be interesting to study if the hyperelliptic curve associated to a solution to the equation of signature $(2, r, p)$ fits in the framework we introduce here, for some specific choice of $s_0$ and $\delta_{\overline{\mathbb{Q}}}$.

In order to solve the generalised Fermat equations $(\mathcal{E}_{p,p,r})$ and $(\mathcal{E}_{r,r,p})$, we will need to consider the base change to $\mathcal{K}$ of the curves introduced above.

**Definition 3.21.** 1) We let $C_r^-$ be the base change of $C_r^-(s_0)^{(\delta_{\overline{\mathbb{Q}}})}$ to $\mathcal{K}$, and $J_r^- := \text{Jac}(C_r^-)/\mathcal{K}$ be its Jacobian. We let $W_r^- \to \text{Spec}(\mathcal{O}_{\mathcal{K}})$ be the Weierstrass model of $C_r^-/\mathcal{K}$ defined by $(\mathcal{H}_{r,s_0}^{-(\delta_{\overline{\mathbb{Q}}})})$, and $\mathcal{J}_r^- \to \text{Spec}(\mathcal{O}_{\mathcal{K}})$ be the Néron model of $J_r^-/\mathcal{K}$.

2) If $\delta_{\overline{\mathbb{Q}}} \in \mathbb{Z}$, we let $C_r^+$ be the base change $C_r^+(s_0)$ to $\mathcal{K}$, and $J_r^+ := \text{Jac}(C_r^+)/\mathcal{K}$ be its Jacobian. We let $W_r^+ \to \text{Spec}(\mathcal{O}_{\mathcal{K}})$ be the Weierstrass model of $C_r^+/\mathcal{K}$ defined by $(\mathcal{H}_{r,s_0}^{+(\delta_{\overline{\mathbb{Q}}})})$, and $\mathcal{J}_r^+ \to \text{Spec}(\mathcal{O}_{\mathcal{K}})$ be the Néron model of $J_r^+/\mathcal{K}$.

Although the objects introduced in Definition 3.21 depend on the parameters $s_0$ and $\delta_{\overline{\mathbb{Q}}}$, we write them without any reference to the latter in order to simplify the notation. Whenever we make a specific choice for $s_0$ and $\delta_{\overline{\mathbb{Q}}}$, we will state it in an explicit way. The choices for $s_0$ and $\delta_{\overline{\mathbb{Q}}}$ done in Proposition 3.18 satisfy indeed the assumptions in Definition 3.14. By considering generic choices of the parameters $s_0, \delta_{\overline{\mathbb{Q}}}$, we can manipulate the Frey objects for both signatures $(r, r, p)$ and $(p, p, r)$ in a single framework.
**Definition 3.22.** Let $\delta_K$ be a non-zero square-free element of $\mathcal{O}_K$, divisible only by primes of $\mathcal{O}_K$ where $\mathcal{J}_r^{\pm}$ has bad reduction. We let $(C_r^{\pm})^{(\delta_K)}$ be the quadratic twist of $C_r^{\pm}$ by $\delta_K$. We describe these curves by the hyperelliptic equations

$$ (\mathcal{H}_r^{-})^{(\delta_K)} : y^2 = \delta_K g_r^{-}(x) \quad \text{and} \quad (\mathcal{H}_r^{+})^{(\delta_K)} : y^2 = \delta_K g_r^{+}(x) $$

We also let $W_r^{\pm}(\delta_K) \to \text{Spec}(\mathcal{O}_q)$ be the Weierstrass model associated to each of the equations above, $(J_r^{\pm})^{(\delta_K)}$ be the Jacobian of $(C_r^{\pm})^{(\delta_K)}$, and $(\mathcal{J}_r^{\pm})^{(\delta_K)} \to \text{Spec}(\mathcal{O}_K)$ be its Néron model.

**Remark 3.23.** Lemma 2.18 combined with Theorem 3.5 show that the quadratic twist $(J_r^{\pm})^{(\delta_K)}$ also has real multiplication by $K$. Therefore, all the discussion from §2.4 applies to $(J_r^{\pm})^{(\delta_K)}$ too.

The interest of introducing this extra twist by $\delta_K$ is to obtain new curves $(C_r^{\pm})^{(\delta_K)}$ having better reduction properties. When solving a Diophantine equation like $(\mathcal{E}_{p,p,r})$ or $(\mathcal{E}_{r,r,p})$, we view the parameters $s_0$ and $\delta_Q$ as fixed, associated to the equation and a putative solution. On the other hand, we treat $\delta_K$ as a variable parameter, whose value we will choose depending on the behaviour of $s_0$, $\delta_Q$. We search for values of $\delta_K$ such that, for any finite place $q$ of $K$, the conductor exponent of $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$ at $q$ is smaller or equal than the one of $\rho_{J_r^{\pm}, \lambda}$.

**Remark 3.24.** To conclude this section, we summarise in Table 1 all the important quantities that we will manipulate in the rest of the article for each of the considered signatures. As we will see in Corollary 4.13, the discriminants of the models $W_r^{\pm}$ depend on $2^4\delta_Q^{2r} s_0(s_0 - 1)$. In the column corresponding to the signature $(p, p, r)$ (resp. $(r, r, p)$), the triple $(a, b, c)$ denotes a primitive non-trivial solution to $(\mathcal{E}_{p,p,r})$ (resp. $(\mathcal{E}_{r,r,p})$).

<table><caption>Table 1: Table describing the explicit quantities introduced above for each of the signatures $(p, p, r)$ and $(r, r, p)$</caption><thead><tr><th>Signature</th><th>(p, p, r)</th><th>(r, r, p)</th></tr></thead><tbody><tr><td>s<sub>0</sub></td><td><sup>Aa<sup>p</sup></sup>&frasl;<sub>Cc<sup>r</sup></sub></td><td><sup>1</sup>&frasl;<sub>2</sub> + (<sup>(AB)</sup>&frasl;<sub>2</sub>)<sup>r-1</sup>(Aa<sup>r</sup> - Bb<sup>r</sup>) / (4z<sub>0</sub><sup>r</sup>)</td></tr><tr><td>&delta;<sub>Q</sub></td><td>Cc</td><td>z<sub>0</sub> := &radic;-ABab</td></tr><tr><td>&delta;<sub>Q</sub><sup>r</sup>(2 - 4s<sub>0</sub>)</td><td>2C<sup>r-1</sup>(Bb<sup>p</sup> - Aa<sup>p</sup>)</td><td>(AB)<sup>r-1</sup>(Bb<sup>r</sup> - Aa<sup>r</sup>)</td></tr><tr><td>s<sub>0</sub>(s<sub>0</sub> - 1)</td><td><sup>-Aa<sup>p</sup>Bb<sup>p</sup></sup>&frasl;<sub>(Cc<sup>r</sup>)<sup>2</sup></sub></td><td><sup>-(Cc<sup>p</sup>)<sup>2</sup></sup>&frasl;<sub>16Aa<sup>r</sup>Bb<sup>r</sup></sub></td></tr><tr><td>2<sup>4</sup>&delta;<sub>Q</sub><sup>2r</sup>s<sub>0</sub>(s<sub>0</sub> - 1)</td><td>-2<sup>4</sup>Aa<sup>p</sup>Bb<sup>p</sup>C<sup>2(r-1)</sup></td><td>(AB)<sup>r-1</sup>(Cc<sup>p</sup>)<sup>2</sup></td></tr></tbody></table>

Let $q$ be a rational prime. From now on, we assume that $s_0$ and $\delta_Q$ satisfy the hypotheses below.

**Hypothesis 1.** If $v_q(s_0(s_0 - 1)) \ge 0$, then $v_q(\delta_Q) = 0$, and $v_q(2 - 4s_0) \in \mathbb{Z}$.

**Hypothesis 2.** If $q \ge 3$, $v_q(s_0(s_0 - 1)) \le 0$ and $v_q(s_0(s_0 - 1)) \equiv 0 \bmod r$, then $v_q(\delta_Q^{2r} s_0(s_0 - 1)) = 0$.

**Hypothesis 3.** If $q = 2$ and $v_2(s_0(s_0 - 1)) \le 0$, then $v_2(s_0(s_0 - 1)) \notin \{-3, -1\}$.

**Lemma 3.25.** For each signature, the choice of $s_0$, $\delta_Q$ done in Proposition 3.18 satisfies Hypotheses 1, 2 and 3.

*Proof.* By Lemma 2.2, $Aa$, $Bb$ and $Cc$ are pairwise coprime, so Table 1 shows that Hypothesis 1 holds. Moreover, we assume that $A, B, C$ are free of $r$-th powers (cf. Definition 2.1).

1) Assume that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{p,p,r})$. If $q \ge 3$, $v_q(s_0(s_0 - 1)) \le 0$ and $v_q(s_0(s_0 - 1)) \equiv 0 \bmod r$, then $q \nmid c$. Table 1 shows that $q \nmid \delta_Q^{2r} s_0(s_0 - 1)$, so Hypothesis 2 is fulfilled. If $q = 2$ and $v_2(s_0(s_0 - 1)) \le 0$, then $v_2(s_0(s_0 - 1)) = 2v_2(Cc^r) \in 2\mathbb{Z}$, so $v_2(s_0(s_0 - 1)) \ne -3, -1$.
2) Assume now that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{r,r,p})$. If $q \ge 3$, $v_q(s_0(s_0 - 1)) \le 0$ and $v_q(s_0(s_0 - 1)) \equiv 0 \bmod r$, then $q \mid ab$. Again, Table 1 shows that $q \nmid \delta_{\mathbb{Q}}^{2r} s_0(s_0 - 1)$. If $q = 2$ and $v_2(s_0(s_0 - 1)) \le 0$, two cases appear. If $v_2(Cc^p) \ge 0$, then $v_2(s_0(s_0 - 1)) = 2v_2(Cc^p) - 4$, which is even. If $v_2(Cc^p) = 0$, then $v_2(s_0(s_0 - 1)) \le -4$. In both cases, we have $v_2(s_0(s_0 - 1)) \ne -3, -1$. $\square$

We will also assume that the following hypothesis holds.

**Hypothesis 4.** There is at least one prime number $q \ne r$ such that $v_q(2^4s_0(s_0 - 1)) > 0$.

**Hypothesis 5.** There is at least one prime number $q' \nmid 2r$ such that $v_{q'}(s_0(s_0 - 1)) < 0$ and $v_{q'}(s_0(s_0 - 1)) \ne 0 \bmod r$.

**Remark 3.26.** Hypotheses 4 and 5 are restrictive, and imply a loss of generality. However, assuming that they are not satisfied imposes huge restrictions on the coefficients $A, B, C$ and any primitive non-trivial solution $(a, b, c)$ to $(\mathcal{E}_{p,p,r})$ or $(\mathcal{E}_{r,r,p})$. For our arithmetic purposes, both of them will hold.

# 4 Reduction types of the Néron models $(\mathcal{J}_r^{\pm})^{(\delta_K)}$

In this section we study the geometry of the Néron models $(\mathcal{J}_r^{\pm})^{(\delta_K)}$, and their reduction types. The main results in here are Theorems 4.2 and 4.3, which describe how $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ reduce at any place $q$, depending on the behaviour of $s_0$, $\delta_{\mathbb{Q}}$ and $\delta_K$. Since the Jacobians $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ have RM by $\mathcal{K}$, they have either good, toric or unipotent reduction, and there is no phenomenon of mixed reduction (see Remark 2.22). Understanding these reduction types will help us compute Artin conductors of $\rho_{(\mathcal{J}_r^{\pm})^{(\delta_K)}, \lambda}$ (§5.2), but also to prove modularity of $\rho_{(\mathcal{J}_r^{\pm})^{(\delta_K)}, \lambda}$ (§5.1), and finiteness of the residue representations (§5.3).

Part of the content of this section is deeply inspired by [ACIK$^{+24}$], and the results presented below are a generalisation of the ones established therein. The author would like to thank again Mar Curcó-Iranzo, Maleeha Khawaja, Céline Maistret and Diana Mocanu for fruitful conversations.

At even places, under some 2-adic conditions, we study the geometry of different models of the curves $(C_r^{\pm})^{(\delta_K)}$, to deduce the reduction types of $(\mathcal{J}_r^{\pm})^{(\delta_K)}$. At odd places, we use the machinery of cluster pictures and the corresponding criteria for having good and semistable reduction.

As explained in §3.3, the parameters $s_0$ and $\delta_{\mathbb{Q}}$ are imposed by the choice of a Diophantine equation and a putative solution. However, the twisting parameter $\delta_K$ is not fixed a priori, and we search for values of $\delta_K$ that minimise the Artin conductor of the representations attached to $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ restricted to $G_{\mathcal{K}_q}$. As explained in the literature [ACIK$^{+24}$, BCDF23, CK22], a low semistability defect for $(\mathcal{J}_r^{\pm})^{(\delta_K)}/\mathcal{K}_q$ tends to lead to a small value of the Artin conductor of $\rho_{(\mathcal{J}_r^{\pm})^{(\delta_K)}, \lambda}$ at $q$. With this in mind, we aim at minimising the semistability defect of $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ among all twists of $\mathcal{J}_r^{\pm}$.

**Definition 4.1.** Let $q$ be a finite place of $\mathcal{O}_{\mathcal{K}}$ and $q \in \mathbb{Z}$ the prime below $q$. We define

$$v_q := v_q(2^4s_0(s_0 - 1))$$

Given an element $x \in \mathcal{O}_q^\times$, we say that the property SQ($x$) holds if $v_q(x)$ is even, and $x$ is a square mod $q^2$, i.e., there is some $\tau \in \mathcal{O}_q$ such that $v_q(x - \tau^2) \ge 2$.

**Theorem 4.2.** The reduction type of $(\mathcal{J}_r^{-})^{(\delta_K)}$ at $q$ is described in Figure 1.

**Theorem 4.3.** The reduction type of $(\mathcal{J}_r^{+})^{(\delta_K)}$ at $q$ is described in Figure 2.

**Remark 4.4.** Hypothesis 4 combined with Theorem 4.2 implies that $J_r^{-}$ has potential toric reduction at least at one finite place of $\mathcal{K}$ different from $r$. If the prime $q$ given in Hypothesis 4 is odd, the same holds for $J_r^{+}$. If the twisting parameter $\delta_K$ is conveniently chosen, then $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ has toric reduction at such place of $\mathcal{K}$. On the other hand, Hypothesis 5 being satisfied implies that $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ has unipotent reduction at least at one place not dividing $2r$.
<figure><img src="image_2.png" /><figcaption>Figure 1: Decision tree describing the reduction type of the Néron model ($\mathcal{J}_r^-(\delta_K)$) at $\mathfrak{q}$. At each node, follow the branch whose condition is satisfied by the considered parameters.</figcaption></figure>

**Example 4.5.** We can specialise the content of Theorems 4.2 and 4.3 for the specific choices of $s_0$, $\delta_Q$ done in §3.3 for each of the signatures $(p, p, r)$ and $(r, r, p)$. For instance, we describe the reduction types of the Néron models of the Jacobians considered in §3.2.1 and §3.2.2 at $\mathfrak{q} \nmid 2r$. Describing this reduction types for $\mathfrak{q} \mid 2$, or $\mathfrak{q} = r$ can be done in a similar way.

1) Assume that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{p,p,r})$. The choice of $s_0$, $\delta_Q$ done in Proposition 3.18 gives the equality $2^4 s_0(s_0 - 1) = \frac{-2^4 A a^p B b^p}{(Cc^r)^2}$. If $\mathfrak{q} \nmid 2r$, $q$ lies below $\mathfrak{q}$, and $\delta_K$ is chosen conveniently, then

$$ \text{the Néron model of } \text{Jac}(C_r^{\pm}(a, b, c) \times \mathcal{K})^{(\delta_K)} \text{ has } \begin{cases} \text{toric reduction at } \mathfrak{q} & \text{if } q \mid ABab, \\ \text{good reduction at } \mathfrak{q} & \text{if } q \mid c \text{ and } q \nmid C, \\ \text{unipotent reduction at } \mathfrak{q} & \text{if } q \mid C. \end{cases} $$

2) Assume that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{r,r,p})$. The choice of $s_0$, $\delta_Q$ done in Proposition 3.18 gives the equality $2^4 s_0(s_0 - 1) = \frac{(Cc^p)^2}{Aa^r Bb^r}$. If $\mathfrak{q} \nmid 2r$, $q$ lies below $\mathfrak{q}$, and $\delta_K$ is chosen conveniently, then

$$ \text{the Néron model of } \text{Jac}(C_r(a, b, c) \times \mathcal{K})^{(\delta_K)} \text{ has } \begin{cases} \text{toric reduction at } \mathfrak{q} & \text{if } q \mid Cc, \\ \text{good reduction at } \mathfrak{q} & \text{if } q \mid ab \text{ and } q \nmid AB, \\ \text{unipotent reduction at } \mathfrak{q} & \text{if } q \mid AB. \end{cases} $$

## 4.1 Roots of the defining polynomials $g_r^{\pm}$

In this subsection we exhibit algebraic expressions for the roots of $g_r^{\pm}(x) \in \mathbb{Z}[x]$, and we deduce the discriminants of the Weierstrass models $W_r^{\pm}(\delta_K) \to \text{Spec}(\mathcal{O}_K)$. Recall that we only consider the curve $C_r^+$ (and thus $g_r^+$) when $\delta_Q \in \mathbb{Z}$. In this case, the extra root of $g_r^+$ is $-2\delta_Q$, which belongs to $\mathbb{Z}$. Therefore, we focus on describing the roots of the polynomial $g_r^-(x)$.
<figure><img src="image_3.png" /><figcaption>Figure 2: Decision tree describing the reduction type of the Néron model ($\mathcal{J}_r^+(\delta_K)$) at $\mathfrak{q}$. At each node, follow the branch whose condition is satisfied by the considered parameters.</figcaption></figure>

All the content below is local, and concerns the curves $C_r^{\pm}$, which we consider as defined over $\mathcal{K}_q$. From now on, we treat $s_0$, $\delta_Q$, $\delta_K$ as belonging to $\overline{\mathbb{Q}}_q$, and we view $g_r^-$ as an element of $\mathbb{Q}_q[x]$, through the choice of an embedding $\overline{\mathbb{Q}} \hookrightarrow \overline{\mathbb{Q}}_q$.

**Definition 4.6.** Let $\sqrt{s_0}, \sqrt{s_0-1} \in \overline{\mathbb{Q}}_q$ be square roots of $s_0$ and $s_0-1$ respectively, and let $\sqrt{s_0(s_0-1)} := \sqrt{s_0}\sqrt{s_0-1}$. Define $\alpha_0 \in \overline{\mathbb{Q}}_q$ to be an r-th root of

$$ (\sqrt{s_0} + \sqrt{s_0-1})^2 = 2s_0 - 1 + 2\sqrt{s_0(s_0-1)}. $$

For any $j \in [0..r-1]$, define $\alpha_j := \zeta_r^j \alpha_0$, $\beta_j := 1/\alpha_j$ and $\gamma_j := \delta_Q(\alpha_j + \beta_j) \in \overline{\mathbb{Q}}_q$. For simplicity, we write $\gamma_{-j} := \gamma_{r-j}$. Moreover, we set $\gamma_r := -2\delta_Q$.

**Remark 4.7.** By definition of $\delta_Q$ and $s_0$, $\delta_Q\alpha_0$ is an r-th root of $\delta_Q^r(\sqrt{s_0} + \sqrt{s_0-1})^2$. By [Lan02, Section VI, Theorem 9.1], the polynomial $x^r - \delta_Q^r(\sqrt{s_0} + \sqrt{s_0-1})^2 \in \mathbb{Q}_q(\delta_Q^r\sqrt{s_0(s_0-1)})[x]$ is reducible if and only if it has a root in $\mathbb{Q}_q(\delta_Q^r\sqrt{s_0(s_0-1)})$. We choose the embedding $\overline{\mathbb{Q}} \hookrightarrow \overline{\mathbb{Q}}_q$ in such a way that, whenever the polynomial above is reducible, then $\delta_Q\alpha_0 \in \mathbb{Q}_q(\delta_Q^r\sqrt{s_0(s_0-1)})$ (see Proposition 4.38 for further details). Therefore, the extension $\mathbb{Q}_q(\delta_Q\alpha_0)/\mathbb{Q}_q(\delta_Q^r\sqrt{s_0(s_0-1)})$ has degree either 1 or r.

**Lemma 4.8.** For any $j \in [0..r-1]$, we have the following properties:

1) The element $\beta_j$ is an r-th root of $(\sqrt{s_0} - \sqrt{s_0-1})^2 = 2s_0 - 1 - 2\sqrt{s_0(s_0-1)}$.

2) $\alpha_j^r + \beta_j^r = 4s_0 - 2$.

3) $\alpha_j^r - \beta_j^r = 4\sqrt{s_0(s_0-1)}$.

4) $(\alpha_j^r + 1)^2 / \alpha_j^r = 4s_0$.

*Proof.* These are algebraic manipulations that follow from Definition 4.6. $\square$

**Proposition 4.9.** For any $j \in [0..r-1]$, $\gamma_j$ is a root of $g_r^-(x) \in \mathbb{Q}_q[x]$.
Proof. Fix $j \in [0..r-1]$. Since $g_r^-(x) = \delta_Q^r g_{r,s_0}(x/\delta_Q)$, it suffices to check that $\alpha_j + \beta_j$ is a root of $g_{r,s_0}$. Recall that $\alpha_j\beta_j = 1$ and $\alpha_j^r + \beta_j^r = 4s_0 - 2$. We apply [ACIK$^+$24, Lemma 2.14] to compute

$$g_{r,s_0}^{-}(\alpha_j + \beta_j) = (-\alpha_j\beta_j)^{\frac{r-1}{2}}(\alpha_j + \beta_j)h_r\left(2 - \frac{(\alpha_j + \beta_j)^2}{\alpha_j\beta_j}\right) + s_0 = \alpha_j^r + \beta_j^r + 2 - 4s_0 = 0. \quad \square$$

**Remark 4.10.** The notation used above is slightly different than the one employed in [ACIK$^+$24]. When $A=B=C=1$ and $(a,b,c)$ is a primitive non-trivial solution to $(\mathcal{E}_{p,p,r})$, we have $s_0 = a^p/c^r$, $\delta_Q = c$ with the choice of Proposition 3.18. We recover the elements that are denoted by $\alpha_j, \beta_j$ in [ACIK$^+$24, §4.1] as $\delta_Q\alpha_j, \delta_Q\beta_j$ in our notation.

**Proposition 4.11.** For any $0 \le j, k \le r-1$, we have the equalities

$$\gamma_k - \gamma_j = \delta_Q \zeta_r^k (1 - \zeta_r^{j-k}) (\alpha_0 - \zeta_r^{-j-k} \beta_0) \quad \text{and} \quad \gamma_j - \gamma_r = \frac{\delta_Q \zeta_r^j}{\alpha_0} (\alpha_0 + \zeta_r^{-j})^2.$$

For any $k \in [0..r-1]$, we have

$$\prod_{0 \le j \le r-1 \atop j \ne k} (\gamma_k - \gamma_j) = \frac{4r \delta_Q^{r-1} \sqrt{s_0(s_0-1)}}{\zeta_r^k (\alpha_0 - \zeta_r^{-2k} \beta_0)} \quad \text{and} \quad \prod_{j=0}^{r-1} (\gamma_j - \gamma_r) = 4\delta_Q^r s_0.$$

In particular, the $\gamma_j$'s are pairwise distinct.

*Proof.* The first identity is obtained by developing the right-hand side. Using such equality, we compute, for $k \in [0..r-1]$

$$\prod_{0 \le j \le r-1 \atop j \ne k} (\gamma_k - \gamma_j) = \prod_{0 \le j \le r-1 \atop j \ne k} \delta_Q \zeta_r^k (1 - \zeta_r^{j-k}) (\alpha_0 - \zeta_r^{-j-k} \beta_0) = \frac{\delta_Q^{r-1} \zeta_r^{k(r-1)} r (\alpha_0^r - \beta_0^r)}{\alpha_0 - \zeta_r^{-2k} \beta_0},$$

and the result follows from Lemma 4.8 (3). On the other hand, using $\beta_0 = 1/\alpha_0$, we compute

$$\frac{\delta_Q \zeta_r^j}{\alpha_0} (\alpha_0 + \zeta_r^{-j})^2 = \frac{\delta_Q \zeta_r^j}{\alpha_0} (\alpha_0^2 + 2 \zeta_r^{-j} \alpha_0 + \zeta_r^{-2j}) = \delta_Q (\zeta_r^j \alpha_0 + 2 + \zeta_r^{-j} \beta_0) = (\gamma_j - \gamma_r).$$

Now Lemma 4.8 (4) implies $\prod_{j=0}^{r-1} (\gamma_j - \gamma_r) = \delta_Q^r \alpha_0^{-r} (\alpha_0^r + 1)^2 = 4\delta_Q^r s_0$. Finally, the $\gamma_j$'s are pairwise distinct because we have $s_0 \ne 0, 1$, and $\delta_Q \ne 0$, so the products of the differences are non-zero. $\square$

**Definition 4.12.** We denote by $\mathcal{R} := \{\gamma_0, \dots, \gamma_{r-1}\}$ the set of roots of $g_r^-$, and by $\mathcal{R}^+ := \mathcal{R} \cup \{\gamma_r\}$ the set of roots of $g_r^+$.

**Corollary 4.13.** The discriminants of the Weierstrass models $W_r^{\pm}(\delta_\kappa)$ are given by

$$\Delta(W_r^{-(\delta_\kappa)}) = (-1)^{\frac{r-1}{2}} 2^{2(r-1)} r^r (2^4 \delta_Q^{2r} s_0(s_0-1))^{\frac{r-1}{2}} \delta_\kappa^{2r},$$

$$\Delta(W_r^{+(\delta_\kappa)}) = (-1)^{\frac{r-1}{2}} 2^{2(r-1)} r^r (2^4 \delta_Q^{2r} s_0(s_0-1))^{\frac{r-1}{2}} (4\delta_Q^r s_0)^2 \delta_\kappa^{2r}.$$

In particular, the places of bad reduction for $W_r^{\pm}(\delta_\kappa)$ divide 2, r and $\delta_Q^{2r} s_0(s_0 - 1)$.

*Proof.* We begin by computing the discriminants of $W_r^{\pm}$, which satisfy $\Delta(W_r^{\pm}) = 2^{4g} \text{disc}(g_r^{\pm})$, where $g = \frac{r-1}{2}$ is the genus of $C_r^{\pm}$, and $\text{disc}(g_r^{\pm})$ is the discriminant of $g_r^{\pm}$. First we have

$$\begin{aligned} \text{disc}(g_r^-) &= (-1)^{r-1} \prod_{0 \le j, k \le r-1 \atop j \ne k} (\gamma_k - \gamma_j) \\ &= (-1)^{\frac{r-1}{2}} \prod_{k=0}^{r-1} \frac{4\delta_Q^{r-1} r \sqrt{s_0(s_0-1)}}{\zeta_r^k (\alpha_0 - \zeta_r^{-2k} \beta_0)} = (-1)^{\frac{r-1}{2}} \frac{(4\delta_Q^{r-1} r \sqrt{s_0(s_0-1)})^r}{4\sqrt{s_0(s_0-1)}} \end{aligned}$$
so regrouping terms yields the displayed result. On the other hand, the discriminants of $g_r^-$ and $g_r^+$ are related by $\mathrm{disc}(g_r^+) = \mathrm{disc}(g_r^-) \prod_{j=0}^{r-1} (\gamma_j - \gamma_r)^2 = \mathrm{disc}(g_r^-) 4^2 \delta_Q^{2r} s_0^2$. The discriminants of $W_r^{\pm}(\delta_K)$ are obtained from those of $W_r^{\pm}$ by applying Lemma 2.31. $\square$

**Remark 4.14.** Recall that we assume that $s_0$, $\delta_Q$ satisfy Hypothesis 2. The latter implies that, when $\mathfrak{q}$ is odd, if $\nu_q \le 0$ and $\nu_q \equiv 0 \pmod r$, then $\mathfrak{q} \nmid \delta_Q^{2r} s_0(s_0 - 1)$. If, moreover, $v_q(\delta_K) = 0$, Corollary 4.13 shows that both curves $(C_r^{\pm})^{(\delta_K)}$ then have good reduction at $\mathfrak{q}$.

**Example 4.15.** 1) Assume that $(a, b, c)$ is a non-trivial primitive solution to the generalised Fermat equation $(\mathcal{E}_{p,p,r})$ of signature $(p, p, r)$. With the choice of $s_0$ done in Proposition 3.18, we obtain

$$ \alpha_0^r = \frac{1}{Cc^r} \left( \sqrt{Aa^p} + \sqrt{-Bb^p} \right)^2 \quad \text{and} \quad \beta_0^r = \frac{1}{Cc^r} \left( \sqrt{Aa^p} - \sqrt{-Bb^p} \right)^2. $$

Combining Corollary 4.13 and Remark 3.24, we describe the discriminants of $(\mathcal{H}_r^-(a, b, c))$ and $(\mathcal{H}_r^+(a, b, c))$, the equations describing the curves $C_r^\pm(a, b, c)$ (cf. §3.2.1)

$$ \Delta(\mathcal{H}_r^-(a, b, c)) = 2^{4(r-1)} r^r (Aa^p)^{\frac{r-1}{2}} (Bb^p)^{\frac{r-1}{2}} C^{\frac{(r-1)^2}{2}}, $$

$$ \Delta(\mathcal{H}_r^+(a, b, c)) = 2^{4r} r^r (Aa^p)^{\frac{r+3}{2}} (Bb^p)^{\frac{r-1}{2}} C^{\frac{(r-1)(r+3)}{2}}. $$

2) Assume now that $(a, b, c)$ is a non-trivial primitive solution to the generalised Fermat equation $(\mathcal{E}_{r,r,p})$ of signature $(r, r, p)$. With the choice of $s_0$ done in Proposition 3.18, we obtain

$$ \alpha_0^r = \frac{(AB)^{\frac{r-1}{2}} Aa^r}{z_0^r} \quad \text{and} \quad \beta_0^r = -\frac{(AB)^{\frac{r-1}{2}} Bb^r}{z_0^r}. $$

Just as above, Corollary 4.13 and Remark 3.24 yield the discriminant of $(\mathcal{H}_r(a, b))$, the equation defining $C_r(a, b, c)$ (see §3.2.2)

$$ \Delta(\mathcal{H}_r(a, b, c)) = (-1)^{\frac{r-1}{2}} 2^{2(r-1)} r^r (Cc^p)^{r-1} (AB)^{\frac{(r-1)^2}{2}}. $$

In the particular case of trivial coefficients $A = B = C = 1$, we recover the results of [Dar00, BCDF23, CK22, ACIK$^+$24] for both signatures.

## 4.2 Local behaviour at even places

Let $\mathfrak{q}$ be a finite place of $\mathcal{K}$ dividing 2. In this subsection we study the geometry of different models of $(C_r^{\pm})^{(\delta_K)}$, in order to deduce from this the reduction type of $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ at $\mathfrak{q}$. We let $\pi$ be a uniformizer of $\mathcal{O}_{\mathfrak{q}}$, and for any $x \in \mathcal{O}_{\mathfrak{q}}$, we denote by $\bar{x} \in \mathbb{F}_{\mathfrak{q}}$ the class of $x$ modulo $\mathfrak{q}$. Recall that we defined $\nu_2 := v_{\mathfrak{q}}(2^4 s_0(s_0 - 1)) \in \mathbb{Z}$ (here $q = 2$).

**Remark 4.16.** We know that $(C_r^{\pm})^{(\delta_K)}$ have potential semistable reduction, and since their Jacobians $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ have real multiplication by $\mathcal{K}$, they have either potential good or potential toric reduction. We are going to distinguish these two cases depending on $\nu_2$.

If $\nu_2 \ne 2$, the equality $(2-4s_0)^2 = 4(4s_0(s_0-1)+1)$ shows that $v_2(2-4s_0) = \min(\frac{\nu_2}{2}, 1)$. If $\nu_2 = 2$, then $4s_0(s_0-1)$ is a unit in $\mathbb{Z}_2$, so $v_2(4s_0(s_0-1)+1) \ge 1$, and therefore $v_2(2-4s_0) \ge \frac{3}{2}$. Moreover, Hypothesis 3 implies that $\nu_2 \notin \{1, 3\}$. We will treat separately the case $\nu_2 > 0$ (which implies that $v_2(2-4s_0) \ge 1$), and the case $\nu_2 \le 0$, (which yields $v_2(2-4s_0) = \nu_2/2 \le 0$).
### 4.2.1 The case of (potential) toric reduction

We first treat the case $v_2 > 0$, and we focus on the study of $(\mathcal{J}_r^{-})^{(\delta_K)}$.

**Proposition 4.17.** Assume that $v_2 > 0$. If $v_q(\delta_Q\delta_K) \in 2\mathbb{Z}$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ has toric reduction at $q$. Otherwise, it has unipotent reduction at $q$, and attains toric reduction over $K_q(\sqrt{\delta_Q\delta_K})$.

*Proof.* Assume first that $v_q(\delta_Q\delta_K)$ is even. Recall that $(C_r^{-})^{(\delta_K)}$ is the quadratic twist of $C_r^{-}(s_0)$ by $\delta_Q\delta_K$. Thus, up to applying to $W_r^{-}(\delta_K)$ a change of variables as in Lemma 2.31, we may assume that $\delta_Q, \delta_K \in \mathcal{O}_q^\times$. Liu's minimality criterion (see [Liu23, Proposition 3.3]) then shows that $W_r^{-}(\delta_K)$ is a minimal model for $(C_r^{-})^{(\delta_K)}$ at $q$. We are going to prove that the model $W_r^{-}(\delta_K)$ is regular, so that it is the minimal regular model of $(C_r^{-})^{(\delta_K)}$ (see §2.5.2). Moreover, we are going to see that it has bad semistable reduction, implying that $(\mathcal{J}_r^{-})^{(\delta_K)}$ has toric reduction at $q$.

The assumption $v_2 > 0$ implies that $v_q(2 - 4s_0) \ge 1$ (see Remark 4.16). Using the factorisation of the polynomial $h_r$ (see (2.1) and (3.1)), we describe the model $W_r^{-}(\delta_K)$ by the equation

$$ (W_r^{-}(\delta_K)) \qquad y^2 = \delta_K x \prod_{j=1}^{\frac{r-1}{2}} (x^2 + \delta_Q^2(\omega_j - 2)) + \delta_K \delta_Q^r(2 - 4s_0). $$

We note that, for any $k \in [1..r-1]$, we have $\omega_{2k} = \omega_k^2 - 2$. Let $F := K_q(\delta_Q\sqrt{4 - \omega_k^2})$: this is an unramified extension of $K_q$ which contains $\delta_Q\sqrt{4 - \omega_k^2} \in F$ for every $k$. Setting $k := j/2$ in the product above, we rewrite over $F$ the defining equation of $W_r^{-}(\delta_K)$ as

$$ (W_r^{-}(\delta_K) \times_{K_q} F) \qquad y^2 = \delta_K x \prod_{k=1}^{\frac{r-1}{2}} \left( x + \delta_Q \sqrt{4 - \omega_k^2} \right) \left( x - \delta_Q \sqrt{4 - \omega_k^2} \right) + \delta_K \delta_Q^r (2 - 4s_0). $$

Since $v_2(2 - 4s_0) \ge 1$, the special fiber of $W_r^{-}(\delta_K)$ is described by the equation

$$ ((W_r^{-}(\delta_K))_q) \qquad y^2 = \overline{\delta_K} x \prod_{k=1}^{\frac{r-1}{2}} (x + \overline{\delta_Q \omega_k})^2. $$

The projective curve $(W_r^{-}(\delta_K))_q$ is singular exactly at the closed points $\overline{P}_k := (-\overline{\delta_Q \omega_k}, 0)$, which are defined over $\mathbb{F}_q(\overline{\delta_Q})$. The model $W_r^{-}(\delta_K)$ is therefore regular at any closed point that does not reduce to some $\overline{P}_k$. Fix $1 \le k \le \frac{r-1}{2}$. Let $\mathfrak{m}_k^\pm := (x \pm \delta_Q\sqrt{4 - \omega_k^2}, y, \pi)$ be the two maximal ideals in $W_r^{-}(\delta_K) \times_{K_q} F$ whose corresponding closed points reduce to $\overline{P}_k$. (note that $\pi$ is also a uniformizer of $O_F$ as $F/K_q$ is unramified). The defining polynomial of $W_r^{-}(\delta_K) \times_{K_q} F$ belongs to $\mathfrak{m}_k^+$ but not to its square, as $x + \delta_Q\sqrt{4 - \omega_k^2}$ is the only factor in the product that belongs to $\mathfrak{m}_k^+$. Therefore Lemma 2.36 implies that $W_r^{-}(\delta_K) \times_{K_q} F$ is regular at the closed point corresponding to $\mathfrak{m}_k^+$, and the same reasoning applies to $\mathfrak{m}_k^-$. Thus, it is regular everywhere, and $F/K_q$ being unramified, $W_r^{-}(\delta_K)$ is regular too. The equation above shows that $(W_r^{-}(\delta_K))_q$ is geometrically reduced and all its singularities are ordinary double points. We conclude that $W_r^{-}(\delta_K)$ is semistable, and that $(\mathcal{J}_r^{-})^{(\delta_K)}$ has toric reduction at $q$.

Finally, assume that $v_q(\delta_Q\delta_K)$ is odd, so that $K_q(\sqrt{\delta_Q\delta_K})/K_q$ is ramified. We know from Definition 2.32 that $(C_r^{-})^{(\delta_K)} \simeq C_r^{-}(s_0) \times_{\mathbb{Q}_2} K_q$ over $K_q(\sqrt{\delta_Q\delta_K})$. The latter curve is obtained by setting $\delta_Q = \delta_K = 1$, so the discussion above shows that the Néron model of Jac$(C_r^{-}(s_0)) \times K_q$ has toric reduction. Thus, $(\mathcal{J}_r^{-})^{(\delta_K)}$ attains toric reduction over $K_q(\sqrt{\delta_Q\delta_K})$. Now write $\chi_{\delta_Q\delta_K}$ for the character of $G_{K_q}$ cut out by $K_q(\sqrt{\delta_Q\delta_K})$, and fix an odd prime number $\ell$. By Grothendieck's inertial criterion (see [SGA72, Exposé IX]), the action of $I_q$ on $V_\ell(\mathrm{Jac}(C_r^{-}(s_0)) \times K_q)$ is unipotent. But the action of $I_q$ on $V_\ell((J_r^{-})^{(\delta_K)})$ is given by the action above twisted by $\chi_{\delta_Q\delta_K}$. The latter being ramified, $I_q$ does not act unipotently on $V_\ell((J_r^{-})^{(\delta_K)})$, and thus $(\mathcal{J}_r^{-})^{(\delta_K)}/O_q$ is not semistable at $q$. $\square$
**Remark 4.18.** If $\nu_2 \ge 4$, then $v_2(s_0(s_0 - 1)) \ge 0$, and Hypothesis 1 implies that $v_2(\delta_Q) = 0$. Proposition 4.17 states that, in this case, $(\mathcal{J}_r^{-})^{(\delta_K)}$ has toric reduction as soon as $v_q(\delta_K) = 0$.

### 4.2.2 The case of (potential) good reduction

Assume now that $\nu_2 \le 0$. We first introduce some notation to simplify the discussion below.

**Definition 4.19.** We define $\mu := v_q(\delta_Q^r(2 - 4s_0)) \in \mathbb{Z}$, and $\kappa := \delta_Q^r(2 - 4s_0)/\pi^\mu$, $\delta_2 := \delta_K/\pi^{v_q(\delta_K)} \in \mathcal{O}_q^\times$.

**Remark 4.20.** If $\nu_2 \le 0$, Remark 4.16 shows that $v_2(2 - 4s_0) = \nu_2/2 \le 0$. In this case, with the notation above, we write the valuations of the discriminants given in Corollary 4.13

$$ \begin{aligned} v_q(\Delta(W_r^{-}(\delta_K))) &= (r-1)(2+\mu) + 2rv_q(\delta_K), \\ v_q(\Delta(W_r^{+}(\delta_K))) &= 2(r-1) + \mu(r+1) + 2rv_q(\delta_K). \end{aligned} $$

Until the end of this subsection, we let $L$ be a finite extension of $K_q$ with ramification index $r$. We denote by $\mathfrak{P}$ the maximal ideal of its ring of integers $\mathcal{O}_{\mathfrak{P}}$, $v_{\mathfrak{P}}$ a valuation on $\bar{L}$ normalised with respect to $L$, and $\pi_L$ a uniformizer of $\mathcal{O}_{\mathfrak{P}}$.

**Proposition 4.21.** Assume that $\nu_2 \le -4$. If $\mathfrak{SQ}(\delta_K \delta_Q^r(2 - 4s_0))$ holds, then $(C_r^{-})^{(\delta_K)}$ attains good reduction over $L$. Otherwise, it only attains good reduction over a quadratic ramified extension of $L$.

*Proof.* Assume first that $\mathfrak{SQ}(\delta_K \delta_Q^r(2 - 4s_0))$ holds. Then $\mu + v_q(\delta_K)$ is even, and there is some $\tau \in \mathcal{O}_q^\times$ such that $v_q(\delta_2\kappa - \tau^2) \ge 2$. Consider the model $\mathcal{G}_r^-$ of the base change $(C_r^{-})^{(\delta_K)} \times_{K_q} L$ obtained from $W_r^{-(\delta_K)} \times_{K_q} L$ by setting the change of variables

$$ x = \pi_L^{\mu+2} X, \quad \text{and} \quad y = \pi_L^{r(\mu+v_q(\delta_K))/2} (\pi_L^r Y + \tau). \qquad (4.1) $$

Recall that we describe $W_r^-$ by $(\mathcal{H}_{r,s_0}^{-(\delta_Q)})$, and that $W_r^{-(\delta_K)}$ is given by the latter equation with the RHS multiplied by $\delta_K$. Using the notation introduced in Definition 4.19, we describe $\mathcal{G}_r^-$ by

$$ (\mathcal{G}_r^-) \qquad Y^2 + \frac{2\tau}{\pi_L^r} Y = \delta_2 \sum_{k=0}^{\frac{r-1}{2}} (-1)^k c_k \left( \frac{\delta_Q}{\pi_L^{\mu+2}} \right)^{2k} X^{r-2k} + \frac{\delta_2\kappa - \tau^2}{\pi_L^{2r}}. $$

We have $2\tau/\pi_L^r \in \mathcal{O}_{\mathfrak{P}}^\times$. Since $\mathfrak{SQ}(\delta_K \delta_Q^r(2 - 4s_0))$ holds, the constant term belongs to $\mathcal{O}_q$. Moreover,

$$ v_{\mathfrak{P}}(\delta_Q / \pi_L^{\mu+2}) = rv_q(\delta_Q) - (v_q(\delta_Q^r(2 - 4s_0)) + 2) = -(v_q(2 - 4s_0) + 2) = -\frac{1}{2}(\nu_2 + 4) \ge 0, $$

as $\nu_2 \le -4$. We deduce that $\mathcal{G}_r^-$ describes an integral model of $(C_r^{-})^{(\delta_K)} \times_{K_q} L$. Recall that the curve $(C_r^{-})^{(\delta_K)}$ has genus $\frac{r-1}{2}$, so combining Lemma 2.31 and Remark 4.20, we compute

$$ \begin{aligned} v_{\mathfrak{P}}(\Delta(\mathcal{G}_r^-)) &= rv_q(\Delta(W_r^{-}(\delta_K))) - 4r \left( \frac{r(\mu + v_q(\delta_K))}{2} + r \right) + r(r+1)(\mu+2) \\ &= r(r-1)(2+\mu) + 2r^2v_q(\delta_K) - r(r-1)(2+\mu) - 2r^2v_q(\delta_K) = 0. \end{aligned} $$

We conclude that $\mathcal{G}_r^-$ has good reduction over $\mathcal{O}_{\mathfrak{P}}$, so $(C_r^{-})^{(\delta_K)}$ attains good reduction over $L$.

On the other hand, if $\mathfrak{SQ}(\delta_K \delta_Q^r(2 - 4s_0))$ does not hold, then $(\mathcal{J}_r^{-})^{(\delta_K)}$, the curve attains good reduction over a quadratic extension of $L$. A similar argument as in the proof of Proposition 4.17, using the action of inertia, shows that $(\mathcal{J}_r^{-})^{(\delta_K)} \times_{\mathcal{O}_L}$ has unipotent reduction. $\square$

**Proposition 4.22.** Assume that $\nu_2 \le 0$. If $\mathfrak{SQ}(\delta_K)$ holds, then $(C_r^{+})^{(\delta_K)}$ attains good reduction over $L$. Otherwise, it only attains good reduction over a quadratic ramified extension of $L$.
Proof. To assume that SQ($\delta_K$) holds means that $\delta_K = \delta_2 \in \mathcal{O}_q^\times$, and that there is some $\vartheta \in \mathcal{O}_q^\times$ such that $v_q(\delta_K - \vartheta^2) \ge 2$. Consider the model $\mathcal{G}_r^+$ of the base change $(C_r^+)^{(\delta_K)} \times_{K_q} L$ obtained from $\mathcal{W}_r^{(\delta_K)} \times_{K_q} L$ by setting the change of variables

$$x = \frac{\pi_L^{\mu-2}}{X}, \quad \text{and} \quad y = \pi_L^{(\mu-2)(\frac{r+1}{2})} \left( \frac{\pi_L^r Y + \vartheta}{X^{\frac{r+1}{2}}} \right). \qquad (4.2)$$

The model $\mathcal{W}_r^{(\delta_K)}$ is described by $(\mathcal{H}_{r,s_0}^{+}(\delta_Q))$, with the RHS multiplied by $\delta_K$. Using the notation introduced in Definition 4.19 and rearranging terms, we describe $\mathcal{G}_r^+$ by

$$(\mathcal{G}_r^+) (\pi_L^r Y + \vartheta)^2 = \delta_2 \left( \sum_{k=0}^{\frac{r-1}{2}} (-1)^k c_k \left( \frac{\delta_Q}{\pi_L^{\mu-2}} \right)^{2k} X^{2k} + \pi_L^{2r} \kappa X^r \right) \left( 1 + \frac{2\delta_Q}{\pi_L^{\mu-2}} X \right).$$

But $g_r^+$ is monic, so the constant term on the RHS is $\delta_2$. The equation above can be rewritten as

$$(\mathcal{G}_r^+) Y^2 + \frac{2\vartheta}{\pi_L^{2r}} Y = \frac{\delta_2 - \vartheta^2}{\pi_L^{2r}} + \frac{2\delta_Q}{\pi^{\mu-2+2r}} + \delta_2 \left( \sum_{k=1}^{\frac{r-1}{2}} (-1)^k \frac{c_k}{\pi_L^{2r}} \left( \frac{\delta_Q}{\pi_L^{\mu-2}} \right)^{2k} X^{2k} + \kappa X^r \right) \left( 1 + \frac{2\delta_Q}{\pi_L^{\mu-2}} X \right).$$

We have $2\vartheta/\pi_L^{2r} \in \mathcal{O}_p^\times$, and $v_p(\delta_2 - \vartheta^2) \ge 2r$ by definition of $\vartheta$. On top of that,

$$v_p(\delta_Q / \pi_L^{\mu-2}) = r v_q(\delta_Q) - (v_q(\delta_Q^r(2-4s_0)) - 2) = 2 - v_q(2-4s_0) = 2 - \nu_2/2 \ge 0,$$

where the last equality follows from the assumption $\nu_2 \le 0$ combined with Remark 4.16. It follows that $v_p(2\delta_Q/\pi_L^{\mu-2+2r}) \ge 0$, and the terms inside the sum also belong to $\mathcal{O}_p$, as the index $k \ge 1$. We deduce that $\mathcal{G}_r^+$ is integral over $\mathcal{O}_p$. Combining Lemma 2.31 with Remark 4.20, we compute

$$\begin{aligned} v_q(\Delta(\mathcal{G}_r^+)) &= r v_q(\Delta(\mathcal{W}_r^{+}(\delta_K))) - 4r \left( (\mu-2) \left( \frac{r+1}{2} \right) + r \right) + r(r+1)(\mu-2) \\ &= 2r(r-1) + \mu r(r+1) - \mu r(r+1) + 2r(r+1) - 4r^2 = 0. \end{aligned}$$

We conclude that $\mathcal{G}_r^+$ has good reduction over $\mathcal{O}_p$, so $(C_r^+)^{(\delta_K)}$ attains good reduction over $L$.

Again, if SQ($\delta_K$) does not hold, then $(C_r^+)^{(\delta_K)}$ attains good reduction over a quadratic ramified extension of $L$. The action of the inertia group $I_q$ on the $\ell$-adic Tate module of $(J_r^+)^{(\delta_K)}$ shows that $(\mathcal{J}_r^+)^{(\delta_K)} \times \mathcal{O}_L$ has unipotent reduction. $\square$

When some extra congruences are satisfied, the curves $(C_r^{\pm})^{(\delta_K)}$ have good reduction over $K_q$. Up to multiplying the uniformizer $\pi_L$ of $\mathcal{O}_p$ by a unit, we may assume that $\pi_L^r = \pi$.

**Corollary 4.23.** Assume that $\nu_2 \le -4$ and SQ($\delta_K \delta_Q^r(2-4s_0)$) holds. If $\nu_2 \equiv -4 \mod r$, then $(C_r^-)^{(\delta_K)}$ has good reduction over $K_q$, and otherwise it has bad reduction.

Proof. Since $\nu_2 \le -4 < 0$, the assumption $\nu_2 \equiv -4 \mod r$ is equivalent to $\mu \equiv -2 \mod r$. If this is the case, the model $(\mathcal{G}_r^-)$ is defined over $K_q$, and the change of variables given in (4.1) too, so $(C_r^-)^{(\delta_K)}$ has good reduction at $q$. If $\nu_2 \not\equiv -4 \mod r$, then $v_q(\Delta(\mathcal{W}_r^{-}(\delta_K))) \not\equiv 0 \mod 2r$ (cf. Remark 4.20). Since $(C_r^-)^{(\delta_K)}$ has genus $\frac{r-1}{2}$, Remark 2.35 implies that $(C_r^-)^{(\delta_K)}$ has bad reduction at $q$. $\square$

**Corollary 4.24.** Assume that $\nu_2 \le 0$ and SQ($\delta_K$) holds. If $\nu_2 \equiv 4 \mod r$, then $(C_r^+)^{(\delta_K)}$ has good reduction over $K_q$, and otherwise it has bad reduction.

Proof. Since $\nu_2 \le 0$, we have $\nu_2 \equiv 4 \mod r$ if and only if $\mu \equiv 2 \mod r$. If the latter holds, the model $\mathcal{G}_r^+$ and the change of variables given in (4.2) are defined over $K_q$. Thus, $(C_r^+)^{(\delta_K)}$ has good reduction at $q$. If $\nu_2 \not\equiv 4 \mod r$, then $v_q(\Delta(\mathcal{W}_r^+(\delta_K))) \not\equiv 0 \mod 2r$ by Remark 4.20. The curve $(C_r^+)^{(\delta_K)}$ also has genus $\frac{r-1}{2}$, so Remark 2.35 shows that $(C_r^+)^{(\delta_K)}$ has bad reduction at $q$. $\square$
**Corollary 4.25.** If $\nu_2 \le -4$, $\nu_2 \not\equiv -4 \mod r$ and $\mathrm{SQ}(\delta_K \delta_Q^r(2-4s_0))$ holds, then the semistability defect of $(J_r^{-})^{(\delta_K)}/K_q$ equals $r$. If $\nu_2 \le 0$, $\nu_2 \not\equiv 4 \mod r$ and $\mathrm{SQ}(\delta_K)$ holds, then the semistability defect of $(J_r^{+})^{(\delta_K)}/K_q$ equals $r$.

**Remark 4.26.** Our analysis of the reduction type of $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ at an even place $q$ is not complete. Numerical computations lead the author to believe that the following assertions hold. When $\nu_2 > 0$, the Néron model $(\mathcal{J}_r^{+})^{(\delta_K)}$ seems to have toric reduction at $q$. When $-4 \le \nu_2 \le 0$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ seems to have unipotent reduction at $q$, and to attain good reduction over a wildly ramified extension of $K_q$.

## 4.3 Cluster pictures of $C_r^{\pm}$ at odd places

We now describe the cluster pictures of $C_r^{\pm}$ at odd places of $K$. The curves $C_r^{+}(s_0)$ and $C_r^{-}(s_0)^{(\delta_Q)}$ are defined over $\mathbb{Q}$, and one could compute their cluster pictures over $\mathbb{Q}_q$, in the style of [ACIK$^{+24}$]. However, for the Diophantine applications we address, we only need to know the cluster pictures of the base-changed curves $C_r^{\pm}/K_q$. In particular, in §4.5 we will exploit the fact that the base changed Jacobians $J_r^{\pm}/K_q$ have RM by $K$, whereas those defined over $\mathbb{Q}_q$ do not. We note that the cluster pictures of the twisted curves $(C_r^{\pm})^{(\delta_K)}$ are the same ones as those of $C_r^{\pm}$. The only difference appears in the valuation of the leading coefficient of the defining polynomial.

From now on we assume that $q$ is an odd place of $K$, so that $q \ge 3$. Recall that we defined $\nu_q := v_q(2^4s_0(s_0-1)) = v_q(s_0(s_0-1))$. We begin by describing the $q$-adic valuation of $\alpha_0$ and $\beta_0$.

**Lemma 4.27.** We have the following properties:

1) If $\nu_q \ge 0$, then $v_q(\alpha_0) = v_q(\beta_0) = 0$.

2) If $\nu_q < 0$ then $v_q(s_0-1) = v_q(s_0) < 0$, and $\{v_q(\alpha_0), v_q(\beta_0)\} = \{\frac{1}{r}v_q(s_0), -\frac{1}{r}v_q(s_0)\}$.

*Proof.* 1) If $\nu_q > 0$ then exactly one among $v_q(s_0)$ and $v_q(s_0-1)$ is zero, and the statement follows from the equalities $\alpha_0^r = (\sqrt{s_0} + \sqrt{s_0-1})^2$ and $\beta_0^r = (\sqrt{s_0} - \sqrt{s_0-1})^2$. If $\nu_q = 0$, assume by contradiction that $v_q(\alpha_0) \ne 0$. We have $\beta_0 = 1/\alpha_0$, so up to switching $\alpha_0, \beta_0$, we may assume that $v_q(\alpha_0) > 0$ and $v_q(\beta_0) < 0$. Using Lemma 4.8 (3), we obtain

$$0 = \frac{1}{2} v_q(s_0(s_0-1)) = v_q(\alpha_0^r - \beta_0^r) = \min(r v_q(\alpha_0), r v_q(\beta_0)) = r v_q(\beta_0) < 0,$$

hence a contradiction. We conclude that $v_q(\alpha_0) = v_q(\beta_0) = 0$.

2) Assume that $\nu_q < 0$. At least one among $s_0$ and $s_0-1$ has negative valuation, and thus both of them have the same valuation. By definition of $\alpha_0, \beta_0$ and Lemma 4.8 (1), we have

$$v_q(\alpha_0^r) = v_q(s_0) + 2v_q\left(1 + \frac{\sqrt{s_0-1}}{\sqrt{s_0}}\right) \quad \text{and} \quad v_q(\beta_0^r) = v_q(s_0) + 2v_q\left(1 - \frac{\sqrt{s_0-1}}{\sqrt{s_0}}\right)$$

But $\sqrt{s_0-1}/\sqrt{s_0}$ is a square root of $1 - \frac{1}{s_0}$. Since $\frac{1}{s_0}$ belongs to $q$, we can compute the Taylor expansion of the square root of $1 - \frac{1}{s_0}$ in the ring of integers $\mathcal{O}_q$. There is some $\varepsilon \in \{\pm 1\}$ such that $\frac{\sqrt{s_0-1}}{\sqrt{s_0}} = \varepsilon(1 - \frac{1}{2s_0} + \dots)$ and further powers of $\frac{1}{s_0}$. We deduce that exactly one among

$$1 + \frac{\sqrt{s_0-1}}{\sqrt{s_0}} \quad \text{and} \quad 1 - \frac{\sqrt{s_0-1}}{\sqrt{s_0}}$$

has valuation 0 and the other has valuation $-v_q(s_0)$. Therefore, we conclude that exactly one among $\alpha_0^r$ and $\beta_0^r$ has $q$-adic valuation $v_q(s_0)$, and the other has valuation $-v_q(s_0)$. $\square$

Recall that we denote by $\mathfrak{r} = (2 - \omega)$ the unique prime of $\mathcal{O}_K$ dividing $r$. The valuation $v_{\mathfrak{r}}$ being normalised with respect to $K_{\mathfrak{r}}$, we have $v_{\mathfrak{r}}(1 - \zeta_r) = \frac{1}{2}$. As explained in Proposition 4.11, the difference of roots $\gamma_k - \gamma_j$ is divisible by $\alpha_0 - \zeta_r^{-j-k}\beta_0$. We now describe the valuation of the latter.
**Lemma 4.28.** We have the following properties.

1) If $\frac{1}{2}v_q(s_0(s_0-1)) > rv_q(1-\zeta_r)$, there is a unique $j_0 \in [0..r-1]$ such that $v_q(\alpha_0-\zeta_r^{j_0}\beta_0) > v_q(1-\zeta_r)$, and $v_q(\alpha_0 - \zeta_r^j\beta_0) = v_q(1-\zeta_r)$ for $j \in [0..r-1] \setminus \{j_0\}$. Moreover, if $\zeta_r \notin K_q$, then $j_0 = 0$.

2) If $\frac{1}{2}v_q(s_0(s_0-1)) \le rv_q(1-\zeta_r)$, then $v_q(\alpha_0 - \zeta_r^j\beta_0) = v_q(s_0(s_0-1))/2r$ for all $j \in [0..r-1]$.

*Proof.* Recall from Lemma 4.8 (3) that $\alpha_0^r - \beta_0^r = 4\sqrt{s_0(s_0-1)}$, so $v_q(\alpha_0^r - \beta_0^r) = \frac{1}{2}v_q(s_0(s_0-1))$.

1) If $\frac{1}{2}v_q(s_0(s_0-1)) > rv_q(1-\zeta_r) \ge 0$, we know from Lemma 4.27 (1) that $v_q(\alpha_0) = v_q(\beta_0) = 0$. The statement is a particular case of [ACIK$^{+24}$, Lemma 2.15], applied with $K = K_q$, $v = v_q$, $\alpha = \alpha_0$ and $\beta = -\beta_0$.

2) Assume that $\frac{1}{2}v_q(s_0(s_0-1)) \le rv_q(1-\zeta_r)$. We claim that the valuation of $\alpha_0 - \zeta_r^j\beta_0$ does not depend on $j$. Different possibilities arise.

2.a) If $v_q(\alpha_0 - \zeta_r^j\beta_0) \ge v_q(1-\zeta_r)$ for every $j \in [0..r-1]$, then we have

$$rv_q(1 - \zeta_r) \le \sum_{j=0}^{r-1} v_q(\alpha_0 - \zeta_r^j \beta_0) = v_q(\alpha_0^r - \beta_0^r) = \frac{1}{2} v_q(s_0(s_0-1)) \le rv_q(1 - \zeta_r).$$

Thus, $\frac{1}{2}v_q(s_0(s_0-1)) = rv_q(1-\zeta_r)$, and $v_q(\alpha_0 - \zeta_r^j\beta_0) = \frac{1}{r}v_q(\alpha_0^r - \beta_0^r)$ for all $j \in [0..r-1]$.

2.b) Suppose that there is some $j_1 \in [0..r-1]$ such that $v_q(\alpha_0 - \zeta_r^{j_1}\beta_0) < v_q(1-\zeta_r)$.

2.b.i) If $v_q(\beta_0) \ge 0$ then equality $\alpha_0 - \zeta_r^j\beta_0 = (\alpha_0 - \zeta_r^{j_1}\beta_0) + \zeta_r^{j_1}(1 - \zeta_r^{j-j_1})\beta_0$ shows that $v_q(\alpha_0 - \zeta_r^j\beta_0) = v_q(\alpha_0 - \zeta_r^{j_1}\beta_0)$.

2.b.ii) If $v_q(\beta_0) < 0$, equality $\alpha_0\beta_0 = 1$ implies that $v_q(\alpha_0) > 0$, and thus $v_q(\alpha_0 - \zeta_r^j\beta_0) = v_q(\beta_0)$.

In all cases, $v_q(\alpha_0 - \zeta_r^j\beta_0)$ is independent of $j$, and therefore equals $\frac{1}{r}v_q(\alpha_0^r - \beta_0^r) = \frac{1}{2r}v_q(s_0(s_0-1))$. $\square$

**Remark 4.29.** Lemma 4.28 describes the behaviour of $v_q(\alpha_0 - \zeta_r^j\beta_0)$ under certain conditions on $v_q(s_0(s_0-1))$. If $q \ne r$, the criterion is $v_q(s_0(s_0-1)) > 0$, and if $q = r$, the criterion is $v_q(s_0(s_0-1)) > r$. Since $v_q(1-\zeta_r) = 0$ if $q \ne r$, and $v_q(1-\zeta_r) = \frac{1}{2}$, the condition $\frac{1}{2}v_q(s_0(s_0-1)) > rv_q(1-\zeta_r)$ recovers both criteria above, whether $q = r$ or not.

**Definition 4.30.** If $q \ne r$ and $v_q(s_0(s_0-1)) > 0$, we let $j_0 \in [0..r-1]$ be as in Lemma 4.28 (1). If $q = r$ and $v_q(s_0(s_0-1)) > r$, we let $j_0 = 0$. Since $\zeta_r \notin K_q$, we have $v_q(\alpha_0 - \zeta_r^{j_0}\beta_0) > v_q(1-\zeta_r)$ and $v_q(\alpha_0 - \zeta_r^j\beta_0) = v_q(1-\zeta_r)$ for $j \ne j_0$, whether $q = r$ or not. We define $i_0 \in [0..r-1]$ to be such that $-2i_0 \equiv j_0 \mod r$.

**Theorem 4.31.** Let $q$ be an odd place of $K$ dividing $r\delta_{\mathbb{Q}}^{2r} s_0(s_0-1)$. Let $n := \frac{v_q(s_0(s_0-1))}{2}$ and $m := n - \frac{r}{2}$.

1) If $q \ne r$ and $v_q(s_0(s_0-1)) > 0$, then the cluster picture of $C_r^-/K_q$ is

<figure><img src="image_4.png" /></figure>

2) If $q \ne r$ and $v_q(s_0(s_0-1)) \le 0$, then the cluster picture of $C_r^-/K_q$ is

<figure><img src="image_5.png" /></figure>

3) If $q = r$ and $v_q(s_0(s_0-1)) > r$, then the cluster picture of $C_r^-/K_q$ is
1) Assume that $\frac{1}{2}v_q(s_0) > rv_q(1 - \zeta_r)$. Lemma 4.27 (1) yields $v_q(\alpha_0) = 0$, so $v_q(\alpha_0^r + 1) > rv_q(1 - \zeta_r)$. Lemma 2.15 from [ACIK$^+$24] applied with $K = K_q$, $v = v_q$, $\alpha = \alpha_0$, $\beta = 1$ states that there is a unique $k_0 \in [0..r-1]$ such that $v_q(\alpha_0 + \zeta_r^{k_0}) > v_q(1 - \zeta_r)$. We claim that $k_0 \equiv -i_0 \mod r$. Indeed

$$ (\alpha_0 + \zeta_r^{k_0}) - (\alpha_0 - \zeta_r^{-2i_0} \beta_0) = \frac{\zeta_r^{k_0}}{\alpha_0} (\alpha_0 + \zeta_r^{-2i_0 - k_0}) $$

gives $v_q(\alpha_0 + \zeta_r^{-2i_0 - k_0}) > v_q(1 - \zeta_r)$, so $-2i_0 - k_0 \equiv k_0 \mod r$, and therefore $k_0 \equiv -i_0 \mod r$.

2) Assume now that $\frac{1}{2}v_q(s_0) \le rv_q(1 - \zeta_r)$, so that $v_q((\alpha_0^r + 1)^2/\alpha_0^r) \le 2rv_q(1 - \zeta_r)$. We claim that the valuation of $(\alpha_0 + \zeta_r^{-j})^2/\alpha_0$ does not depend on $j$. Different cases arise.

2.a) If $v_q((\alpha_0 + \zeta_r^{-j})^2/\alpha_0) \ge 2v_q(1 - \zeta_r)$ for every $j \in [0..r-1]$, then we have

$$ 2rv_q(1 - \zeta_r) \le \sum_{j=0}^{r-1} v_q \left( \frac{1}{\alpha_0} (\alpha_0 + \zeta_r^{-j})^2 \right) = v_q \left( \frac{(\alpha_0^r + 1)^2}{\alpha_0^r} \right) = v_q(s_0) \le 2rv_q(1 - \zeta_r). $$

Thus, $v_q(s_0) = 2rv_q(1 - \zeta_r)$ and $v_q((\alpha_0 + \zeta_r^{-j})^2/\alpha_0) = \frac{1}{r}v_q((\alpha_0^r + 1)^2/\alpha_0^r)$ for all $j \in [0..r-1]$.

2.b) Suppose that there is some $k_1 \in [0..r-1]$ such that $v_q((\alpha_0 + \zeta_r^{-k_1})^2/\alpha_0) < 2v_q(1 - \zeta_r)$. Fix $j \in [0..r-1]$. Again, we treat different cases separately.

2.b.i) If $v_q(\alpha_0) = 0$, then $v_q(\alpha_0 + \zeta_r^{-k_1}) < v_q(1 - \zeta_r)$. Equality

$$ \alpha_0 + \zeta_r^{-j} = (\alpha_0 + \zeta_r^{-k_1}) + \zeta_r^{k_1} (1 - \zeta_r^{-j-k_1}) $$

implies that $v_q(\alpha_0 + \zeta_r^{-j}) = v_q(\alpha_0 + \zeta_r^{-k_1})$, so $v_q((\alpha_0 + \zeta_r^{-j})^2/\alpha_0) = 2v_q(\alpha_0 + \zeta_r^{-k_1})$.

2.b.ii) If $v_q(\alpha_0) > 0$, then $v_q(\alpha_0 + \zeta_r^{-j}) = 0$, so $v_q((\alpha_0 + \zeta_r^{-j})^2/\alpha_0) = -v_q(\alpha_0)$.

2.b.iii) If $v_q(\alpha_0) < 0$, then $v_q(\alpha_0 + \zeta_r^{-j}) = v_q(\alpha_0)$, and thus $v_q((\alpha_0 + \zeta_r^{-j})^2/\alpha_0) = v_q(\alpha_0)$.

In all cases, we see that $v_q((\alpha_0 + \zeta_r^{-j})^2/\alpha_0)$ is independent of $j$, and it is therefore equal to $\frac{1}{r}v_q((\alpha_0^r + 1)^2/\alpha_0^r) = \frac{1}{r}v_q(s_0)$. $\square$

We now proceed to describe the cluster picture of $C_r^+$ at $q$.

**Theorem 4.33.** Let $q$ be an odd place of $K$ dividing $r\delta_{\mathbb{Q}}^{2r} s_0(s_0-1)$. Let $n := \frac{v_q(s_0(s_0-1))}{2}$ and $m := n - \frac{r}{2}$.

1) If $q \ne r$ and $v_q(s_0) > 0$, then the cluster picture of $C_r^+/K_q$ is

<figure><img src="image_6.png" /></figure>

2) If $q \ne r$ and $v_q(s_0 - 1) > 0$, then the cluster picture of $C_r^+/K_q$ is

<figure><img src="image_7.png" /></figure>

3) If $q \ne r$ and $v_q(s_0(s_0 - 1)) \le 0$, then the cluster picture of $C_r^+/K_q$ is

<figure><img src="image_8.png" /></figure>

4) If $q = r$ and $v_q(s_0) > r$, then the cluster picture of $C_r^+/K_r$ is
<figure><img src="image_9.png" /></figure>

5) If $\mathbf{q} = \mathbf{r}$ and $v_{\mathbf{r}}(s_0 - 1) > r$, then the cluster picture of $C_{\mathbf{r}}^{+}/K_{\mathbf{r}}$ is

<figure><img src="image_10.png" /></figure>

6) If $\mathbf{q} = \mathbf{r}$ and $v_{\mathbf{r}}(s_0(s_0 - 1)) \le r$, then the cluster picture of $C_{\mathbf{r}}^{+}/K_{\mathbf{r}}$ is

<figure><img src="image_11.png" /></figure>

where $\tilde{m} := \frac{1}{2r}(v_{\mathbf{r}}(s_0 - 1) - v_{\mathbf{r}}(s_0))$.

*Proof.* By Theorem 4.31, it remains to compute the $\mathbf{q}$-adic valuations of $\gamma_j - \gamma_r$ for $j \in [0..r-1]$. We recall from Proposition 4.11 that $\gamma_j - \gamma_r = \frac{\delta_{\mathbb{Q}}\zeta_r^j}{\alpha_0}(\alpha_0 + \zeta_r^{-j})^2$.

1) Suppose that $\mathbf{q} \ne \mathbf{r}$ and $v_{\mathbf{q}}(s_0) > 0$. Hypothesis 1 yields $v_{\mathbf{q}}(\delta_{\mathbb{Q}}) = 0$. For $j \in [0..r-1] \setminus \{i_0\}$, we have $v_{\mathbf{q}}((\alpha_0 + \zeta_r^{-j})^2/\alpha_0) = 0$ by Lemma 4.32 (1), so $v_{\mathbf{q}}(\gamma_j - \gamma_r) = 0$. On the other hand, $v_{\mathbf{q}}((\alpha_0 + \zeta_r^{-i_0})^2/\alpha_0) = v_{\mathbf{q}}((\alpha_0^r + 1)^2/\alpha_0^r)$ which equals $v_{\mathbf{q}}(s_0)$ by Lemma 4.8 (4). It follows that $v_{\mathbf{q}}(\gamma_{i_0} - \gamma_r) = v_{\mathbf{q}}(s_0)$, so $\gamma_{i_0}$ and $\gamma_r$ lie in a twin of depth $v_{\mathbf{q}}(s_0) = 2n$.

2) Suppose that $\mathbf{q} \ne \mathbf{r}$ and $v_{\mathbf{q}}(s_0 - 1) > 0$. Then $v_{\mathbf{q}}(s_0) = 0$, and $v_{\mathbf{q}}(\delta_{\mathbb{Q}}) = 0$ by Hypothesis 1. Lemma 4.32 (2) states that for any $j \in [0..r-1]$, we have $v_{\mathbf{q}}((\alpha_0 + \zeta_r^{-j})^2/\alpha_0) = 0$, so $v_{\mathbf{q}}(\gamma_j - \gamma_r) = 0$. We conclude that the outer depth of the cluster picture is 0, and $\gamma_r$ is an isolated root in it.

3) Suppose that $\mathbf{q} \ne \mathbf{r}$ and $v_{\mathbf{q}}(s_0(s_0 - 1)) \le 0$. Lemma 4.32 (2) yields $v_{\mathbf{q}}((\alpha_0 + \zeta_r^{-j})^2/\alpha_0) = v_{\mathbf{q}}(s_0)/r$ for any $j \in [0..r-1]$, hence $v_{\mathbf{q}}(\gamma_j - \gamma_r) = v_{\mathbf{q}}(\delta_{\mathbb{Q}}) + v_{\mathbf{q}}(s_0)/r$. We conclude that all the roots of $g_r^+$ lie in a common cluster of depth $v_{\mathbf{q}}(\delta_{\mathbb{Q}}) + v_{\mathbf{q}}(s_0)/r = v_{\mathbf{q}}(\delta_{\mathbb{Q}}) + n/r$.

4) Suppose that $\mathbf{q} = \mathbf{r}$ and $v_{\mathbf{q}}(s_0) > r$. Hypothesis 1 gives $v_{\mathbf{r}}(\delta_{\mathbb{Q}}) = 0$, and Lemma 4.27 (1) yields $v_{\mathbf{r}}(\alpha_0) = 0$. Moreover, $i_0 = 0$ as $\zeta_r \notin K_{\mathbf{r}}$. Thus, we have $v_{\mathbf{r}}((\alpha_0 + \zeta_r^{-j})^2/\alpha_0) = 2v_{\mathbf{r}}(1 - \zeta_r)$ for any $j \in [1..r-1]$ (cf. Lemma 4.32 (1)), and $v_{\mathbf{r}}((\alpha_0 + 1)^2/\alpha_0) > 2v_{\mathbf{r}}(1 - \zeta_r)$. Thanks to Lemma 4.8 (4), we compute
$$v_{\mathbf{r}}(\alpha_0 + 1) = v_{\mathbf{r}}(\alpha_0^r + 1) - (r-1)v_{\mathbf{r}}(1 - \zeta_r) = \frac{1}{2}v_{\mathbf{r}}(s_0) + \frac{r-1}{2}.$$

It follows that $v_{\mathbf{r}}(\gamma_j - \gamma_r) = 1$ for any $j \ne 0$, and $v_{\mathbf{r}}(\gamma_0 - \gamma_r) = 1 + v_{\mathbf{r}}(s_0) - r$. Therefore, the outer depth remains 1, and $\gamma_0$ and $\gamma_r$ lie in a twin of relative depth $v_{\mathbf{r}}(s_0) - r = 2m$.

5) Suppose that $\mathbf{q} = \mathbf{r}$ and $v_{\mathbf{r}}(s_0 - 1) > r$, which implies $v_{\mathbf{r}}(s_0) = 0$, and $v_{\mathbf{r}}(\delta_{\mathbb{Q}}) = 0$ (cf. Hypothesis 1). For any $j \in [0..r-1]$ we have $v_{\mathbf{r}}((\alpha_0 + \zeta_r^{-j})^2/\alpha_0) = 0$, so $v_{\mathbf{r}}(\gamma_j - \gamma_r) = 0$. Therefore, $\gamma_r$ is an isolated root lying outside of the cluster picture of $g_r^-$, and the outer depth is 0.

6) Suppose that $\mathbf{q} = \mathbf{r}$ and $v_{\mathbf{r}}(s_0(s_0 - 1)) \le r$. Lemma 4.32 (2) yields $v_{\mathbf{r}}((\alpha_0 + \zeta_r^{-j})^2/\alpha_0) = v_{\mathbf{r}}(s_0)/r$ for all $j \in [0..r-1]$, so $v_{\mathbf{r}}(\gamma_j - \gamma_r) = v_{\mathbf{r}}(\delta_{\mathbb{Q}}) + v_{\mathbf{r}}(s_0)/r$. Fix $j, k \in [0..r-1]$ with $j \ne k$. Recall from Theorem 4.31 that $v_{\mathbf{r}}(\gamma_j - \gamma_k) = \frac{1}{2} + v_{\mathbf{r}}(\delta_{\mathbb{Q}}) + \frac{1}{2r}v_{\mathbf{r}}(s_0(s_0 - 1))$. We claim that

$$v_{\mathbf{r}}(\gamma_j - \gamma_k) \ge v_{\mathbf{r}}(\gamma_j - \gamma_r), \quad \text{or, equivalently,} \quad r + v_{\mathbf{r}}(s_0 - 1) \ge v_{\mathbf{r}}(s_0).$$

Let us prove that the latter is true by treating different cases separately.

6.a) If $0 < v_{\mathbf{r}}(s_0) \le r$, then $v_{\mathbf{r}}(s_0 - 1) = 0$ and the inequality becomes $v_{\mathbf{r}}(s_0) \le r$.

6.b) If $0 < v_{\mathbf{r}}(s_0 - 1) \le r$, then $v_{\mathbf{r}}(s_0) = 0$, and the inequality becomes $r + v_{\mathbf{r}}(s_0 - 1) \ge 0$.
6.c) If $v_r(s_0(s_0 - 1)) \le 0$, then we have $v_r(s_0) = v_r(s_0 - 1)$, and the inequality is simply $r \ge 0$.

We conclude that the outer depth of the cluster picture is $v_r(\delta_Q) + v_r(s_0)/r$, and $\gamma_r$ is an isolated root in it. The cluster containing the roots of $g_r^-$ has relative depth

$$v_r(\gamma_j - \gamma_k) - v_r(\gamma_j - \gamma_r) = \frac{1}{2} + \frac{1}{2r}(v_r(s_0 - 1) - v_r(s_0)) \ge 0. \quad \square$$

## 4.4 Ramification indices and discriminants

We now study the splitting field of the polynomial $g_r^-$, which we view as an element of $\mathbb{Q}_q[x]$. As explained in Definition 3.21, we only consider the curve $C_r^+$ when the parameter $\delta_Q$ belongs to $\mathbb{Z}$ (and thus to $\mathbb{Z}_q$). In that case, $\gamma_r = -2\delta_Q$ is a $\mathbb{Q}_q$ rational root, so $\mathbb{Q}_q(\mathcal{R})$ is also the splitting field of $g_r^+$.

We wish to understand the base-changed curve $C_r^{\pm}/\mathcal{K}_q$, so it could be reasonable to manipulate the polynomial $g_r^-$ as an element of $\mathcal{K}_q[x]$. However, for computational reasons, we prefer to consider that $g_r^-$ belongs to $\mathbb{Q}_q[x]$, and then deduce the desired results about $C_r^{\pm}/\mathcal{K}_q$.

### 4.4.1 Description of the splitting field

Recall that, for any $j \in [0..r-1]$, we defined $\omega_j = \zeta_r^j + \zeta_r^{-j}$, $\tau_j := \zeta_r^j - \zeta_r^{-j}$, $\omega = \omega_1$, and $\tau = \tau_1$. We view all these as elements of $\mathbb{Q}_q$, and we identify $\mathbb{Q}_q(\omega) \simeq \mathcal{K}_q$.

**Definition 4.34.** We let $\mathbb{Q}_q(\mathcal{R})$ be the splitting field of the polynomial $g_r^-(x) \in \mathbb{Q}_q[x]$. We let $\mathcal{K}_q(\mathcal{R})$ be the compositum of $\mathbb{Q}_q(\mathcal{R})$ and $\mathcal{K}_q$.

**Theorem 4.35.** The splitting field of $g_r^-$ is $\mathbb{Q}_q(\mathcal{R}) = \mathcal{K}_q(\gamma_0, \delta_Q \tau \sqrt{s_0(s_0-1)})$.

*Proof.* Let us begin with the reverse inclusion. We clearly have $\gamma_0 \in \mathbb{Q}_q(\mathcal{R})$, and the equality $\omega = (\gamma_1 + \gamma_{-1})/\gamma_0$ shows that $\mathcal{K}_q \subset \mathbb{Q}_q(\mathcal{R})$. It remains to prove that $\delta_Q \tau \sqrt{s_0(s_0-1)} \in \mathbb{Q}_q(\mathcal{R})$. Recall from Lemma 4.8 (3) that $\alpha_0^r - \beta_0^r = 4\sqrt{s_0(s_0-1)}$. One can check that, for any $j \in [0..r-1]$, we have $\delta_Q \tau_j(\alpha_0 - \beta_0) = (\gamma_j - \gamma_{-j}) \in \mathbb{Q}_q(\mathcal{R})$. We claim that $\phi_r(\alpha_0, -\beta_0)$ also belongs to $\mathbb{Q}_q(\mathcal{R})$. The equality

$$\delta_Q \tau \sqrt{s_0(s_0-1)} = \frac{\delta_Q \tau}{4} (\alpha_0^r - \beta_0^r) = \frac{\delta_Q \tau}{4} (\alpha_0 - \beta_0) \phi_r(\alpha_0, -\beta_0)$$

will then establish the desired inclusion. In order to prove the claim, note that, for any integer $n \ge 1$, the polynomial $X^{2n} + Y^{2n} \in \mathbb{Q}_q[X, Y]$ is symmetric, so it is a polynomial expression in $X + Y$ and $XY$. Specifying $X = \delta_Q \alpha_0$ and $Y = \delta_Q \beta_0$, we see that $(\delta_Q \alpha_0)^{2n} + (\delta_Q \beta_0)^{2n}$ is a polynomial expression in $\delta_Q(\alpha_0 + \beta_0) = \gamma_0$, and $\delta_Q^2 \alpha_0 \beta_0 = \delta_Q^2$. Since $\delta_Q^2 \in \mathbb{Z}$, it follows that $\alpha_0^{2n} + \beta_0^{2n} \in \mathbb{Q}_q(\gamma_0)$, and thus

$$\phi_r(\alpha_0, -\beta_0) = \sum_{j=0}^{r-1} \alpha_0^j \beta_0^{r-1-j} = 1^{r-1} + \sum_{j=0}^{(r-3)/2} 1^{2j} (\beta_0^{r-1-2j} + \alpha_0^{r-1-2j}) \in \mathbb{Q}_q(\gamma_0) \subset \mathbb{Q}_q(\mathcal{R}).$$

Let us show now that $\mathbb{Q}_q(\mathcal{R}) \subset \mathcal{K}_q(\gamma_0, \delta_Q \tau \sqrt{s_0(s_0-1)})$. Fix $j \in [0..r-1]$. One can check that $\gamma_j = (\omega_j \gamma_0 + \delta_Q \tau_j(\alpha_0 - \beta_0))/2$. Moreover, $\delta_Q \tau_j(\alpha_0 - \beta_0) = \frac{\delta_Q \tau_j}{4\tau} \tau \sqrt{s_0(s_0-1)} \phi_r(\alpha_0, -\beta_0)^{-1}$. But $\tau_j/\tau \in \mathbb{Q}_q(\omega) \simeq \mathcal{K}_q$, and by the discussion above, $\phi_r(\alpha_0, -\beta_0) \in \mathbb{Q}_q(\gamma_0)$, hence the result. $\square$

**Definition 4.36.** We let $\mathbb{Q}_1 := \mathbb{Q}_q(\delta_Q^r \sqrt{s_0(s_0-1)})$, $\mathbb{Q} := \mathcal{K}_q \cdot \mathbb{Q}_1$, and $\mathbb{Q}_{\tau} := \mathcal{K}_q(\delta_Q^r \tau \sqrt{s_0(s_0-1)})$.

**Remark 4.37.** Theorem 4.35 implies that $\mathcal{K}_q \subset \mathbb{Q}_q(\mathcal{R})$. Recall that $\delta_Q^{2r} s_0(s_0 - 1) \in \mathbb{Q}$, and we also have $\tau^2 = (\omega_2 - 2) \in \mathcal{O}_{\mathcal{K}}$. We deduce that $\mathbb{Q}_1/\mathbb{Q}_q$ is at most quadratic, and that $\mathbb{Q}, \mathbb{Q}_{\tau}$ are at most quadratic extensions of $\mathcal{K}_q$. With this notation, Theorem 4.35 simply states that $\mathbb{Q}_q(\mathcal{R}) = \mathbb{Q}_{\tau}(\gamma_0)$.

**Proposition 4.38.** If $g_r^-$ is reducible over $\mathbb{Q}_q$ then $\gamma_0 \in \mathbb{Q}_1$. If we further assume $\zeta_r \notin \mathbb{Q}_q$, then $\gamma_0 \in \mathbb{Q}_q$.
<figure><img src="image_12.png" /><figcaption>Figure 3: Diagram of field inclusions relating $\mathcal{Q}_1$ and $\mathcal{Q}_q(\gamma_0)$. The label on each line denotes the degree of the respective extension.</figcaption></figure>

*Proof.* Consider the diagram of field extensions depicted in Figure 3.

As we saw just above, $d_0 \le 2$. Similarly, we have $\beta_0 = \alpha_0^{-1}$, so $\gamma_0 = \delta_{\mathbb{Q}}\alpha_0 + \delta_{\mathbb{Q}}^2/(\delta_{\mathbb{Q}}\alpha_0)$. Therefore, the polynomial $X^2 - \gamma_0 X + \delta_{\mathbb{Q}}^2 \in \mathcal{Q}_q(\gamma_0)[X]$ vanishes at $\delta_{\mathbb{Q}}\alpha_0$, showing that $d'_0 \le 2$. Finally, the extension $\mathcal{Q}_q(\delta_{\mathbb{Q}}\alpha_0)/\mathcal{Q}_1$ is defined by the polynomial $x^r - \delta_{\mathbb{Q}}^r(\sqrt{s_0} + \sqrt{s_0-1})^2$, so Remark 4.7 implies that $n_0 \in \{1, r\}$. If $g_r^-$ is reducible over $\mathcal{Q}_q$, then $n'_0 < r$, and thus $n_0 d_0 = n'_0 d'_0 < 2r$. It follows that $n_0 < r$, and therefore $\mathcal{Q}_q(\delta_{\mathbb{Q}}\alpha_0) = \mathcal{Q}_1$, hence $\gamma_0 \in \mathcal{Q}_1$. If $\mathcal{Q}_1 = \mathcal{Q}_q$, the result follows.

Assume therefore that $\zeta_r \notin \mathcal{Q}_q$ and that $\mathcal{Q}_q \neq \mathcal{Q}_1$. Let $\sigma$ (resp. $N_{\mathcal{Q}_1/\mathcal{Q}_q}$, $\text{Tr}_{\mathcal{Q}_1/\mathcal{Q}_q}$) be the non-trivial automorphism (resp. the norm and trace maps) of the quadratic extension $\mathcal{Q}_1/\mathcal{Q}_q$. From Definition 4.6 we obtain the identities

$$ (\delta_{\mathbb{Q}}\alpha_0)^r = \delta_{\mathbb{Q}}^r(2s_0 - 1) + 2\delta_{\mathbb{Q}}^r\sqrt{s_0(s_0 - 1)} \quad \text{and,} \quad (\delta_{\mathbb{Q}}\beta_0)^r = \delta_{\mathbb{Q}}^r(2s_0 - 1) - 2\delta_{\mathbb{Q}}^r\sqrt{s_0(s_0 - 1)}. $$

But we know from §3.3 that $\delta_{\mathbb{Q}}^r(2s_0 - 1) \in \mathcal{Q}_q$, so $\sigma(\delta_{\mathbb{Q}}\alpha_0)^r = (\delta_{\mathbb{Q}}\beta_0)^r = \delta_{\mathbb{Q}}^{2r}/(\delta_{\mathbb{Q}}\alpha_0)^r$. Consequently $N_{\mathcal{Q}_1/\mathcal{Q}_q}(\delta_{\mathbb{Q}}\alpha_0)^r = \delta_{\mathbb{Q}}^{2r}$, and $N_{\mathcal{Q}_1/\mathcal{Q}_q}(\delta_{\mathbb{Q}}\alpha_0) = \delta_{\mathbb{Q}}^2$ as $\zeta_r \notin \mathcal{Q}_q$. Therefore $\sigma(\delta_{\mathbb{Q}}\alpha_0) = \delta_{\mathbb{Q}}\beta_0$, and we conclude that $\gamma_0 = \text{Tr}_{\mathcal{Q}_1/\mathcal{Q}_q}(\delta_{\mathbb{Q}}\alpha_0)$ indeed belongs to $\mathcal{Q}_q$. $\square$

**Remark 4.39.** Proposition 4.38 and Figure 3 show that the degrees $n_0$, $n'_0$ are equal and belong to $\{1, r\}$. Similarly, $d_0 = d'_0 \in \{1, 2\}$. By coprimality of degrees, we also obtain the equalities of ramification indices $e_{\mathcal{Q}_q(\gamma_0)/\mathcal{Q}_q} = e_{\mathcal{Q}_q(\delta_{\mathbb{Q}}\alpha_0)/\mathcal{Q}_1}$. Similarly, one can show that the extensions $K_q(\gamma_0)/K_q$ and $K_q(\delta_{\mathbb{Q}}\alpha_0)/\mathcal{Q}$ have same degree and same ramification index.

### 4.4.2 Reducibility criteria for the polynomial $g_r^-(x)$

We now give reducibility criteria concerning the polynomial $g_r^-(x) \in \mathcal{Q}_q[x]$.

**Remark 4.40.** Since $g_r^-$ has degree $r$ and $[K_q : \mathcal{Q}]$ has degree at most $\frac{r-1}{2}$, it is easy to check that $g_r^-$ is irreducible over $\mathcal{Q}_q$ if and only if it is irreducible over $K_q$. Moreover, we have $e_{\mathcal{Q}_r(\gamma_0)/\mathcal{Q}_r} = e_{K_r(\gamma_0)/K_r}$.

**Proposition 4.41.** If $\frac{1}{2}v_q(s_0(s_0 - 1)) > rv_q(1 - \zeta_r)$, then $g_r^-$ is reducible over $\mathcal{Q}_q$ and $\gamma_{i_0} \in \mathcal{Q}_q$.

*Proof.* As discussed in the beginning of §4.3, the cluster picture of $C_r^-(s_0)^{(\delta_{\mathbb{Q}})}/\mathcal{Q}_q$ is that of $C_r^-/K_q$, except that the depths are multiplied by the ramification index of $K_q/\mathcal{Q}_q$. We know that the absolute Galois group $G_{\mathcal{Q}_q}$ acts on clusters, preserving depths and containments ([BBB$^{+22}$, Remark 3.2]). When $v_q(s_0(s_0 - 1)) > rv_q(1 - \zeta_r)$ (whether $q = r$ or not), Theorem 4.31 implies that $\gamma_{i_0}$ is an isolated root in the cluster picture of $C_r^-(s_0)^{(\delta_{\mathbb{Q}})}/\mathcal{Q}_q$, hence fixed by all the elements of $G_{\mathcal{Q}_q}$. $\square$

**Proposition 4.42.** If $v_q(s_0) \le 0$ and $v_q(s_0) \ne 0 \bmod r$, then $g_r^-$ is irreducible over $\mathcal{Q}_q$.

*Proof.* From the equality $g_r^-(x) = \delta_{\mathbb{Q}}^r g_{r,s_0}^-(x/\delta_{\mathbb{Q}})$ we see that $g_r^-$ is irreducible over $\mathcal{Q}_q$ if and only if $g_{r,s_0}^-$ is so. The latter is given by $g_{r,s_0}^-(x) = \sum_{k=0}^{(r-1)/2} (-1)^k c_k x^{r-2k} + 2 - 4s_0$, with $c_k \in \mathbb{Z}$. Assuming that $v_q(s_0) \le 0$ implies that the Newton polygon of $g_{r,s_0}^-$ is made of a single line joining the points of coordinates $(0, v_q(s_0))$ and $(r, 0)$. The slope of such line is $v_q(s_0)/r$, so if $v_q(s_0) \ne 0 \bmod r$, then $g_{r,s_0}^-$ is irreducible over $\mathcal{Q}_q$ (see [Neu99, Chapter II §6]). $\square$
### 4.4.3 Ramification indices of the involved extensions

As explained in Remark 4.37, $K_q$ is a subfield of $Q_q(\mathcal{R})$. If one views $g_r^-$ as an element of $K_q[x]$, then its splitting field is $K_q(\mathcal{R})$. Therefore, when considering the base-changed curves $C_r^{\pm}/K_q$, it is important to know the ramification index of $K_q(\mathcal{R})/K_q$. As we saw above, $K_q(\mathcal{R})$ fits in the tower of extensions $K_q \subset Q_\tau \subset K_q(\mathcal{R})$. We now describe the ramification indices of the intermediary extensions.

**Lemma 4.43.** 1) If $q \neq \mathfrak{r}$, then $Q_\tau/K_q$ is ramified if and only if $v_q(\delta_{\mathbb{Q}}^{2r} s_0(s_0 - 1))$ is odd.

2) If $q = \mathfrak{r}$, then $Q_\tau/K_\mathfrak{r}$ is ramified if and only if $v_\mathfrak{r}(\delta_{\mathbb{Q}}^{2r} s_0(s_0 - 1))$ is even.

*Proof.* The quadratic extension $Q_\tau/K_q$ is defined by $x^2 - (\omega_2 - 2)\delta_{\mathbb{Q}}^{2r} s_0(s_0 - 1)$. It is ramified if and only if the discriminant of the polynomial above has odd valuation, i.e., $v_q((\omega_2 - 2)\delta_{\mathbb{Q}}^{2r} s_0(s_0 - 1))$ is odd. If $q \neq \mathfrak{r}$, we have $v_q(\omega_2 - 2) = 0$, and if $q = \mathfrak{r}$, $v_\mathfrak{r}(\omega_2 - 2) = 1$, hence the result. $\square$

**Remark 4.44.** Similarly, when $q = r$, one can check that the quadratic extensions $Q_1/\mathbb{Q}_r$ and $\mathbb{Q}_r(\delta_{\mathbb{Q}}\alpha_0)/\mathbb{Q}_r(\gamma_0)$ are ramified (cf. Remark 4.39) if and only if $v_\mathfrak{r}(\delta_{\mathbb{Q}}^{2r} s_0(s_0 - 1))$ is odd.

**Proposition 4.45.** If $q = \mathfrak{r}$, $v_\mathfrak{r}(s_0(s_0 - 1)) \le r$ and $v_\mathfrak{r}(s_0(s_0 - 1)) \equiv 0 \bmod r$, then $Q_\tau/K_\mathfrak{r}$ is ramified.

*Proof.* Recall that $K_\mathfrak{r}/\mathbb{Q}_r$ is totally ramified of degree $r-1/2$, and that $s_0(s_0 - 1)$ is a rational number, so $v_\mathfrak{r}(s_0(s_0 - 1)) = r-1/2 v_\mathfrak{r}(s_0(s_0 - 1))$. By coprimality of $r$ and $r-1/2$, we have $v_\mathfrak{r}(s_0(s_0 - 1)) \le 0$, and Hypothesis 2 implies that $v_\mathfrak{r}(\delta_{\mathbb{Q}}^{2r} s_0(s_0 - 1)) = 0$. The result follows from Lemma 4.43. $\square$

**Theorem 4.46.** Assume that $g_r^-$ is irreducible over $\mathbb{Q}_q$.

1) If $q \neq \mathfrak{r}$, then $K_q/\mathbb{Q}_\tau$ is (totally) ramified if and only if $v_q(s_0(s_0 - 1)) \neq 0 \bmod r$.

2) If $q = \mathfrak{r}$, then $K_\mathfrak{r}/\mathbb{Q}_\tau$ is (totally) ramified.

*Proof.* Recall that we defined $\mathcal{Q}$ and $\mathbb{Q}_\tau$ as $\mathcal{Q} = K_q(\delta_{\mathbb{Q}}\sqrt{s_0(s_0 - 1)})$ and $\mathbb{Q}_\tau = K_q(\delta_{\mathbb{Q}}\tau\sqrt{s_0(s_0 - 1)})$, so $K_q(\zeta_r, \delta_{\mathbb{Q}}\sqrt{s_0(s_0 - 1)}) = \mathcal{Q}(\zeta_r) = \mathbb{Q}_\tau(\zeta_r)$. We describe the different field inclusions in Figure 4.

As discussed in Remark 4.39, the extension $K_q(\gamma_0)/K_q$ has degree 1 or $r$, and is non-trivial if and only if $K_q(\delta_{\mathbb{Q}}\alpha_0)/\mathcal{Q}$ is non-trivial. If $g_r^-$ is irreducible over $\mathbb{Q}_q$ (or, equivalently, over $K_q$), then the extensions $K_q(\delta_{\mathbb{Q}}\alpha_0)/\mathcal{Q}$, $K_q(\zeta_r, \delta_{\mathbb{Q}}\alpha_0)/\mathcal{Q}(\zeta_r)$, and $K_q(\mathcal{R})/\mathbb{Q}_\tau$ are all non-trivial, hence of degree $r$.

<figure><img src="image_13.png" /><figcaption>Figure 4: Diagram of field inclusions when $g_r^-$ is irreducible over $K_q$. The degree of each extension divides the number appearing on the corresponding line.</figcaption></figure>

Studying the ramification indices that appear in Figure 4, the coprimality of 2 and $r$ gives the equalities

$$e_{K_q(\mathcal{R})/\mathbb{Q}_\tau} = e_{K_q(\zeta_r, \delta_{\mathbb{Q}}\alpha_0)/\mathcal{Q}(\zeta_r)} = e_{K_q(\delta_{\mathbb{Q}}\alpha_0)/\mathcal{Q}}.$$

As discussed in the proof of Proposition 4.38, the extension $K_q(\delta_{\mathbb{Q}}\alpha_0)/\mathcal{Q}$ is obtained by adjoining to $\mathcal{Q}$ the $r$-th root of $(\delta_{\mathbb{Q}}\alpha_0)^r = \delta_{\mathbb{Q}}^r(\sqrt{s_0 + \sqrt{s_0 - 1}})^2$. It follows that $K_q(\zeta_r, \delta_{\mathbb{Q}}\alpha_0)/\mathcal{Q}(\zeta_r)$ is a Kummer extension, obtained by adjoining to $\mathcal{Q}(\zeta_r)$ the $r$-th root of $\delta_{\mathbb{Q}}^r(\sqrt{s_0 + \sqrt{s_0 - 1}})^2$. Denote by $\mathfrak{P}$ the maximal ideal of the ring of integers of $\mathcal{Q}(\zeta_r)$. We treat different cases separately.
1) If $\mathfrak{q} \neq \mathfrak{r}$, then Theorem 6.3 in [Gra03, I §6] states that the Kummer extension $K_q(\zeta_r, \delta_Q\alpha_0)/Q(\zeta_r)$ is ramified if and only if $v_{\mathfrak{P}}((\delta_Q\alpha_0)^r) \neq 0 \bmod r$. But we assume that $g_r^-$ is irreducible over $K_q$, so Proposition 4.41 implies that $v_q(s_0(s_0-1)) \le 0$, hence $v_q(\alpha_0) = \pm v_q(s_0)/r$ by Lemma 4.27 (2). Since $r$ is odd and $e_{Q(\zeta_r)/K_q}$ is a power of 2, we deduce that $K_q(\zeta_r, \delta_Q\alpha_0)/Q(\zeta_r)$ is ramified if and only if $v_q(\delta_Q^r s_0) \neq 0 \bmod r \Leftrightarrow v_q(\delta_Q^{2r} s_0(s_0-1)) \neq 0 \bmod r \Leftrightarrow v_q(s_0(s_0-1)) \neq 0 \bmod r$.

2) If $\mathfrak{q} = \mathfrak{r}$, $v_{\mathfrak{r}}(s_0(s_0-1)) < 0$ and $v_{\mathfrak{r}}(s_0(s_0-1)) \neq 0 \bmod r$, then we have $v_{\mathfrak{P}}((\delta_Q\alpha_0)^r) \neq 0 \bmod r$, just as above. Therefore, [Gra03, I §6 Theorem 6.3] implies that $K_q(\zeta_r, \delta_Q\alpha_0)/Q(\zeta_r)$ is totally ramified, and so $e_{K_q(\mathcal{R})/\mathcal{Q}_r} = r$ too.

3) If $\mathfrak{q} = \mathfrak{r}$ and $0 \le v_{\mathfrak{r}}(s_0(s_0-1)) \le r$, then $v_{\mathfrak{r}}(\alpha_0) = 0$ by Lemma 4.27 (1), and $v_{\mathfrak{r}}(\delta_Q) = 0$ by Hypothesis 1, so $v_{\mathfrak{r}}((\delta_Q\alpha_0)^r) = 0$. We claim that $K_{\mathfrak{r}}(\delta_Q\alpha_0)/Q$ is totally ramified. Indeed, it is defined by $x^r - (\delta_Q\alpha_0)^r$, and $g_r^-$ being irreducible implies that $[K_{\mathfrak{r}}(\delta_Q\alpha_0) : Q] = r$. But this is not a Galois extension, as $\zeta_r \notin Q$, so it has to be ramified (cf. [Ser79]). We conclude that $e_{K_{\mathfrak{r}}(\delta_Q\alpha_0)/Q} = e_{K_{\mathfrak{r}}(\mathcal{R})/\mathcal{Q}_r} = r$.

4) If $v_{\mathfrak{r}}(s_0(s_0-1)) < 0$ and $v_{\mathfrak{r}}(s_0(s_0-1)) \equiv 0 \bmod r$, then Hypothesis 2 gives $v_{\mathfrak{r}}(\delta_Q^{2r} s_0(s_0-1)) = 0$. Thus, $v_{\mathfrak{r}}((\delta_Q\alpha_0)^r) = 0$, and we conclude just as in the previous case. $\square$

### 4.4.4 Discriminant of the totally ramified extension $\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r$

Theorem 4.46 shows that, when $g_r^-$ is irreducible over $\mathbb{Q}_r$, then $K_{\mathfrak{r}}(\mathcal{R})/K_{\mathfrak{r}}$ is not tame. As we will see in §5.2, the wild conductor at $\mathfrak{r}$ of the $\ell$-adic representation attached to $J_r^{\pm}/K_{\mathfrak{r}}$ is not trivial. Recall that $J_r^-/\mathbb{Q}_q$ is the base change of Jac$(C_r^-(s_0)^{(\delta_Q)})$ to $K_q$. We will deduce the value of the wild conductor at $\mathfrak{r}$ of $\rho_{(J_r^-)(\delta_{K_{\mathfrak{r}}}), \lambda}$ from that of the $\ell$-adic representation attached to Jac$(C_r^-(s_0)^{(\delta_Q)})/\mathbb{Q}_r$. In order to describe the latter, we compute now the $r$-adic valuation of $\Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r)$.

We begin with a more general statement, that we will use later in our specific context.

**Proposition 4.47.** Let $F/\mathbb{Q}_r$ be a finite extension such that $\zeta_r \notin F$, and let $v_F$ be a valuation on $\bar{F}$ normalised with respect to $F$. Let $u \in \mathcal{O}_F \setminus \mathcal{O}_F^r$, and let $M := F(u^{1/r})$. Then

$$v_F(\Delta(M/F)) = \begin{cases} r e_{F/\mathbb{Q}_r} & \text{if } v_F(u) \equiv 0 \bmod r, \\ r e_{F/\mathbb{Q}_r} + r - 1 & \text{if } v_F(u) \neq 0 \bmod r \end{cases}$$

*Proof.* Since $\zeta_r \notin F$, the extension $M/F$ has degree $r$ and is not Galois, so it is totally ramified. Let $\pi_F$ be a uniformizer of $\mathcal{O}_F$ and $\pi_M := \pi_F^{1/r}$, which is a uniformizer of $\mathcal{O}_M$.

1) Assume that $v_F(u) \equiv 0 \bmod r$. Up to multiplying $u$ by an $r$-th power of $\pi_F$, we may assume that $v_F(u) = 0$. The residue field $\mathcal{O}_F/\mathfrak{P}$ has size $r^f - 1$, so Fermat's little theorem yields $u^{r^{f-1}} \equiv 1 \bmod \mathfrak{P}$. Nevertheless, $u^{r^{f-1}} \neq 1 \bmod \mathfrak{P}^2$, as if that was the case, the strong version of Hensel's lemma would imply that $u$ is an $r$-th power in $\mathcal{O}_F$. It follows that $u^{r^{f-1}} - u^{1/r}$ is a uniformizer of $\mathcal{O}_M$, so $\mathcal{O}_M = \mathcal{O}_F[u^{r^{f-1}} - u^{1/r}] = \mathcal{O}_F[u^{1/r}]$. We conclude that

$$v_F(\Delta(M/F)) = v_F(\mathrm{disc}(x^r - u)) = v_F(\pm r^r u^{r-1}) = r e_{F/\mathbb{Q}_r}.$$

2) Assume now that $v_F(u) \neq 0 \bmod r$, and write $\eta := v_F(u)$. Since $M/F$ is totally ramified, $u^{1/r}/\pi_M^{\eta-1}$ is also a uniformizer of $\mathcal{O}_M$, and so $\mathcal{O}_M = \mathcal{O}_F[u^{1/r}/\pi_M^{\eta-1}]$ ([Ser79, Chapter III §6]). It follows that $\Delta(M/F)$ matches the discriminant of $x^r - u/\pi_M^{\eta-1}$. Taking valuations, we deduce that

$$v_F(\Delta(M/F)) = v_F\left(\pm r^r \left(u/\pi_F^{\eta-1}\right)^{r-1}\right) = r v_F(r) + (r-1) v_F\left(u/\pi_F^{\eta-1}\right) = r e_{F/\mathbb{Q}_r} + r - 1$$

$\square$
Recall that we write $v_r = v_r(s_0(s_0 - 1))$ (we are in the case $q = r$ odd).

**Theorem 4.48.** Assume that $g_r^-$ is irreducible over $\mathbb{Q}_r$. Then we have

$$v_r(\Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r)) = \begin{cases} r & \text{if } (\nu_r \le 0 \text{ and } \nu_r \equiv 0 \bmod r) \text{ or } (\nu_r = 2), \\ \frac{3r-1}{2} & \text{if } \nu_r = 1, \\ 2r-1 & \text{if } \nu_r < 0 \text{ and } \nu_r \not\equiv 0 \bmod r. \end{cases}$$

*Proof.* In order to compute $v_r(\Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r))$, we make use of the diagram introduced in Figure 3. With the notation of Remark 4.39, we have $d_0 = d'_0 \in \{1, 2\}$, and $n_0 = n'_0 = r$ as $g_r^-$ is irreducible over $\mathbb{Q}_r$. Moreover, both extensions $\mathbb{Q}_r(\delta_\mathbb{Q}\alpha_0)/\mathbb{Q}_1$ and $\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r$ are totally ramified (see Remarks 4.39 and 4.40). Corollary 2.10 in [Neu99, Chapter III §2] implies that

$$\Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r)^{d_0} = \Delta(\mathbb{Q}_1/\mathbb{Q}_r)^r \frac{\mathrm{N}_{\mathbb{Q}_1/\mathbb{Q}_r}(\Delta(\mathbb{Q}_r(\delta_\mathbb{Q}\alpha_0)/\mathbb{Q}_1))}{\mathrm{N}_{\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r}(\Delta(\mathbb{Q}_r(\delta_\mathbb{Q}\alpha_0)/\mathbb{Q}_r(\gamma_0)))}. \quad (4.3)$$

To simplify the notation, let $v_{\mathbb{Q}_1}$ (resp. $v_{\mathbb{Q}_r(\gamma_0)}$) be a valuation on $\overline{\mathbb{Q}}_r$ normalised with respect to $\mathbb{Q}_1$ (resp. $\mathbb{Q}_r(\gamma_0)$). We claim that the $r$-adic valuation of the discriminant of $\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r$ is given by

$$v_r(\Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r)) = \begin{cases} v_{\mathbb{Q}_1}(\Delta(\mathbb{Q}_r(\delta_\mathbb{Q}\alpha_0)/\mathbb{Q}_1)) & \text{if } \mathbb{Q}_1/\mathbb{Q}_r \text{ is unramified,} \\ \frac{r-1}{2} + \frac{1}{2} v_{\mathbb{Q}_1}(\Delta(\mathbb{Q}_r(\delta_\mathbb{Q}\alpha_0)/\mathbb{Q}_1)) & \text{if } \mathbb{Q}_1/\mathbb{Q}_r \text{ is ramified.} \end{cases} \quad (4.4)$$

Indeed, if $\mathbb{Q}_1/\mathbb{Q}_r$ is unramified, then $v_r(\Delta(\mathbb{Q}_1/\mathbb{Q}_r)) = v_{\mathbb{Q}_r(\gamma_0)}(\Delta(\mathbb{Q}_r(\delta_\mathbb{Q}\alpha_0)/\mathbb{Q}_r(\gamma_0))) = 0$, and the claim follows from (4.3). On the other hand, if $\mathbb{Q}_1/\mathbb{Q}_r$ is ramified, then $v_r(\Delta(\mathbb{Q}_1/\mathbb{Q}_r)) = 1$, and $v_{\mathbb{Q}_r(\gamma_0)}(\Delta(\mathbb{Q}_r(\delta_\mathbb{Q}\alpha_0)/\mathbb{Q}_r(\gamma_0))) = 1$. Since $\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r$ is totally ramified, and $v_r$ is normalised with respect to $\mathbb{Q}_r$, we deduce that $v_r(\mathrm{N}_{\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r}(\Delta(\mathbb{Q}_r(\delta_\mathbb{Q}\alpha_0)/\mathbb{Q}_r(\gamma_0)))) = 1$. Moreover, $\mathbb{Q}_1/\mathbb{Q}_r$ being ramified imposes $d_0 = 2$, so taking $r$-adic valuations in (4.3) and simplifying terms yields (4.4).

To conclude, we use Proposition 4.47 applied to $F = \mathbb{Q}_1$ and $u := (\delta_\mathbb{Q}\alpha_0)^r = \delta_\mathbb{Q}^r(\sqrt{s_0} + \sqrt{s_0 - 1})^2$. We treat different cases separately.

1) If $\nu_r = 2$, then $v_r(\delta_\mathbb{Q}) = 0$, and one among $v_r(s_0)$, $v_r(s_0 - 1)$ vanishes, so $v_r(u) = 0$. Moreover, Remark 4.44 shows that $\mathbb{Q}_1/\mathbb{Q}_r$ is unramified, so $v_r(\Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r)) = v_{\mathbb{Q}_1}(\Delta(\mathbb{Q}_r(\delta_\mathbb{Q}\alpha_0)/\mathbb{Q}_1)) = r$.

2) If $\nu_r = 1$, we have again $v_r(u) = 0$, but this time $\mathbb{Q}_1/\mathbb{Q}_r$ is ramified as $v_r(\delta_\mathbb{Q}^{2r}s_0(s_0 - 1)) = 1$ (see Remark 4.44). It follows that $v_r(\Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r)) = \frac{r-1}{2} + \frac{1}{2} = \frac{3r-1}{2}$.

3) If $\nu_r \equiv 0 \bmod r$, then by coprimality of $r$ and $\frac{r-1}{2}$ we have $\nu_r \le 0$. But then Hypothesis 2 yields $v_r(\delta_\mathbb{Q}^{2r}s_0(s_0 - 1)) = 0$, so $\mathbb{Q}_1/\mathbb{Q}_r$ is unramified. Thus $v_r(u) \equiv 0 \bmod r$, hence $v_r(\Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r)) = r$.

4) Finally, assume that $\nu_r \le 0$ and $\nu_r \not\equiv 0 \bmod r$. If $\mathbb{Q}_1/\mathbb{Q}_r$ is unramified, then Proposition 4.47 gives $v_r(\Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r)) = v_{\mathbb{Q}_1}(\Delta(\mathbb{Q}_r(\delta_\mathbb{Q}\alpha_0)/\mathbb{Q}_1)) = r + r - 1$. On the other hand, if $\mathbb{Q}_1/\mathbb{Q}_r$ is ramified, then $v_r(\Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r)) = \frac{r-1}{2} + \frac{1}{2}(2r + r - 1) = 2r - 1$. In both cases we get the desired result. $\square$

## 4.5 Conclusion of the proof of Theorems 4.2 and 4.3

We describe the reduction type of the Néron models $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ at $\mathfrak{q}$, and finally prove Theorems 4.2 and 4.3. Recall that we denote by $\mathfrak{q}$ the rational prime lying below $\mathfrak{q}$, and we let $\nu_q := v_q(2^4s_0(s_0 - 1))$.

*Proof of Theorem 4.2.* Corollary 4.13 states that the primes of bad reduction for $(\mathcal{J}_r^{-})^{(\delta_K)}$ divide 2, $r$ or $\delta_\mathbb{Q}^{2r}s_0(s_0 - 1)$. Fix such a $\mathfrak{q}$. Recall from §3.3 that $(\mathcal{J}_r^{-})^{(\delta_K)}$ has RM by $\mathcal{K}$, so when $(\mathcal{J}_r^{-})^{(\delta_K)}$ has bad reduction at $\mathfrak{q}$ and is not semistable, it has automatically unipotent reduction.

If $\mathfrak{q}$ is even, we use the results from §4.2. If $\mathfrak{q}$ is odd, we use the criteria on cluster pictures given in Theorems 2.39 and 2.40. By Definition 3.22, $(C_r^{-})^{(\delta_K)}$ is the quadratic twist by $\delta_K$ of $C_r^-$, and we describe it by the hyperelliptic equation $(\mathcal{H}_r^{-})^{(\delta_K)}: y^2 = \delta_K g_r^-(x)$. Note that its cluster picture at $\mathfrak{q}$ is the same as the one of $C_r^-$, but the valuation of the leading coefficient for $(C_r^{-})^{(\delta_K)}$ is $v_q(\delta_K)$.
1) Suppose that $\mathfrak{q} \mid 2$. If $\nu_2 > 0$, then Proposition 4.17 states that $(\mathcal{J}_r^{-})^{(\delta_K)}$ has toric reduction at $\mathfrak{q}$ if and only if $v_q(\delta_Q\delta_K)$ is even. On the other hand, assume that $\nu_2 \le -4$ and that $\mathrm{SQ}(\delta_K\delta_Q^r(2-4s_0))$ holds. If $\nu_2 \equiv -4 \mod r$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ has good reduction over $\mathcal{K}_q$, and otherwise it attains good reduction over any finite extension of $\mathcal{K}_q$ with ramification index $r$.

2) Suppose that $\mathfrak{q} = \mathfrak{r}$. We have the equality $\nu_r = \frac{r-1}{2}v_r(s_0(s_0-1))$.

2.a) Assume that $\nu_r > 2$, so that $v_r(s_0(s_0-1)) > r$. Every cluster has at most two odd children, and Propositions 4.38, 4.41 imply that $\mathcal{K}_r(\mathcal{R}) = \mathcal{Q}_\tau$, so $e_{\mathcal{K}_r(\mathcal{R})/\mathcal{K}_r} \le 2$. The only proper cluster strictly containing any twin is $\mathcal{R}$, and $d_{\mathcal{R}} = 1$. Therefore, [Bis22, Theorem 1.3 (iv)] implies that every twin is invariant under the action of $I_{\mathcal{K}_r}$, and clearly, $\mathcal{R}$ is $I_{\mathcal{K}_r}$-invariant too. Moreover, $\mathcal{R}$ is the only principal cluster: we have $d_{\mathcal{R}} = 1$, and $\nu_{\mathcal{R}} = v_r(\delta_{\mathcal{K}}) + rd_{\mathcal{R}} = v_r(\delta_{\mathcal{K}}) + r$. Thus, $(C_r^{-})^{(\delta_K)}$ satisfies the semistability criterion (so $(\mathcal{J}_r^{-})^{(\delta_K)}$ has toric reduction at $\mathfrak{r}$) if and only if $v_r(\delta_{\mathcal{K}}) = 1$.

2.b) Assume that $\nu_r \le 2$, so that $v_r(s_0(s_0-1)) \le r$. The only proper cluster is $\mathcal{R}$, which has size $2g+1$, so $(C_r^{-})^{(\delta_K)}$ has potential good reduction at $\mathfrak{r}$. The description of the ramification indices given in Lemma 4.43 and Theorem 4.46 shows that $e_{\mathcal{K}_r(\mathcal{R})/\mathcal{K}_r} > 1$. Theorem 2.39 implies that $(\mathcal{J}_r^{-})^{(\delta_K)}$ has unipotent reduction at $\mathfrak{r}$.

3) Suppose now that $\mathfrak{q} \nmid 2r$. Since $\mathcal{K}_q/\mathbb{Q}_q$ is unramified, we have $\nu_q = v_q(s_0(s_0-1))$.

3.a) Assume that $\nu_q > 0$. Every cluster has at most two odd children, and Propositions 4.38, 4.41 imply that $e_{\mathcal{K}_q(\mathcal{R})/\mathcal{K}_q} \le 2$. Again, the only proper cluster strictly containing any twin is $\mathcal{R}$ (which is invariant under $I_{\mathcal{K}_q}$), and $d_{\mathcal{R}} \in \mathbb{Z}$. Therefore, Theorem 1.3 (iv) from [Bis22] implies that every twin is $I_{\mathcal{K}_q}$-invariant. The only principal cluster is $\mathcal{R}$: we have $d_{\mathcal{R}} = 0$, and $\nu_{\mathcal{R}} = v_q(\delta_{\mathcal{K}}) + rd_{\mathcal{R}} = v_q(\delta_{\mathcal{K}})$. If $v_q(\delta_{\mathcal{K}}) = 0$ then $(\mathcal{J}_r^{-})^{(\delta_K)}$ has toric reduction at $\mathfrak{q}$, and otherwise it has unipotent reduction.

3.b) Assume that $\nu_q \le 0$ and $\nu_q \equiv 0 \mod r$. The only proper cluster is $\mathcal{R}$, which has size $2g+1$. Hypothesis 2 gives $v_q(\delta_Q^{2r}s_0(s_0-1)) = 0$. Lemma 4.43 and Theorem 4.46 imply that $\mathcal{K}_q(\mathcal{R})/\mathcal{K}_q$ is unramified. We finally compute

$$ \nu_{\mathcal{R}} = v_{\mathcal{q}}(\delta_{\mathcal{K}}) + r d_{\mathcal{R}} = v_{\mathcal{q}}(\delta_{\mathcal{K}}) + r v_{\mathcal{q}}(\delta_{\mathbb{Q}}) + \frac{1}{2} v_{\mathcal{q}}(s_0(s_0-1)) = v_{\mathcal{q}}(\delta_{\mathcal{K}} \delta_{\mathbb{Q}}^r (2-4s_0)). $$

The last equality follows from $(2-4s_0)^2 = 4(4s_0(s_0-1)+1)$. Thanks to Theorem 2.39, we conclude that if $v_q(\delta_{\mathcal{K}}\delta_{\mathbb{Q}}^r(2-4s_0)) \in 2\mathbb{Z}$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ has good reduction at $\mathfrak{q}$, and otherwise has unipotent reduction.

3.c) Assume that $\nu_q \le 0$ and $\nu_q \ne 0 \mod r$. The only proper cluster is $\mathcal{R}$, which has size $2g+1$, so $(C_r^{-})^{(\delta_K)}$ has potential good reduction at $\mathfrak{q}$. However, Proposition 4.42 states that $g_r^{-}$ is irreducible, and Theorem 4.46 implies that $r$ divides the ramification index of $\mathcal{K}_q(\mathcal{R})/\mathcal{K}_q$. It follows that $(\mathcal{J}_r^{-})^{(\delta_K)}$ has unipotent reduction at $\mathfrak{q}$. $\square$

We conclude the section by describing the reduction types of the Néron model $(\mathcal{J}_r^{+})^{(\delta_K)}$.

*Proof of Theorem 4.3.* Recall that, as we manipulate $C_r^+$, we assume that $\delta_Q \in \mathbb{Z}$, so $v_q(\delta_Q) \in \mathbb{Z}$, and $\mathcal{K}_q(\mathcal{R}) = \mathcal{K}_q(\mathcal{R}^+)$. We follow the same strategy as in the proof of Theorem 4.2. When $\mathfrak{q}$ is odd, note that the cluster picture of $(C_r^+)^{(\delta_K)}$ equals that of $C_r^+$, with valuation of the leading coefficient $v_q(\delta_K)$.

1) Suppose that $\mathfrak{q} \mid 2$, that $\nu_2 \le 0$, and that the property $\mathrm{SQ}(\delta_K)$ holds. If $\nu_2 \equiv 4 \mod r$, then $(\mathcal{J}_r^{+})^{(\delta_K)}$ has good reduction over $\mathcal{K}_q$, and otherwise it attains good reduction over any finite extension of $\mathcal{K}_q$ with ramification index $r$.

2) Suppose that $\mathfrak{q} = \mathfrak{r}$. We have the equality $\nu_r = \frac{r-1}{2}v_r(s_0(s_0-1))$.
2.a) Assume that $v_r(s_0) > 2$, so that $v_r(s_0) > r$. Every cluster has at most two odd children, and Propositions 4.38 and 4.41 give $e_{K_r(R)/K_r} \le 2$. The proper clusters are the twins, and the only cluster strictly containing any twin is $\mathcal{R}^+$, whose depth is an integer. Again, [Bis22, Theorem 1.3 (iv)] implies that every twin is invariant under the action of $I_{K_r}$, and $\mathcal{R}^+$ is $I_{K_r}$-invariant too. Moreover, $\mathcal{R}^+$ is the only principal cluster, and we have $\nu_{\mathcal{R}^+} = v_r(\delta_{\mathcal{K}}) + (r+1)d_{\mathcal{R}^+} = v_r(\delta_{\mathcal{K}}) + (r+1)$. We conclude that $(C_r^+)^{(\delta_{\mathcal{K}})}$ satisfies the semistability criterion if and only if $v_r(\delta_{\mathcal{K}}) = 0$.

2.b) Assume that $v_r(s_0 - 1) > 2$, so that $v_r(s_0 - 1) > r$. Just as in the previous case, we have $e_{K_r(R)/K_r} \le 2$. Since $g_r^-$ is reducible by Proposition 4.41, so $K_q(\mathcal{R}) = \mathcal{Q}_\tau$ and $e_{K_q(\mathcal{R})/K_q} \le 2$. We deduce from [Bis22] that every proper cluster is $I_{K_q}$-invariant. Finally, the only principal cluster is $\mathcal{R}^+$, and $\nu_{\mathcal{R}^+} = v_q(\delta_{\mathcal{K}})$. We conclude that $(C_r^+)^{(\delta_{\mathcal{K}})}$ satisfies the semistability criterion if and only if $v_q(\delta_{\mathcal{K}}) = 0$.

2.c) Assume that $\nu_r \le 2$ so that $v_r(s_0(s_0 - 1)) \le r$. The only proper clusters are $\mathcal{R}$ and $\mathcal{R}^+$, whose size is $\ge 2g+1$, so $(C_r^+)^{(\delta_{\mathcal{K}})}$ has potential good reduction at $\mathfrak{r}$. Again, Lemma 4.43 and Theorem 4.46 show that $K_r(\mathcal{R})/K_r$ is ramified. We deduce from Theorem 2.39 that $(\mathcal{J}_r^+)^{(\delta_{\mathcal{K}})}$ has unipotent reduction at $\mathfrak{r}$.

3) Suppose now that $\mathfrak{q} \nmid 2r$. Since $K_q/\mathbb{Q}_q$ is unramified, we have $\nu_q = v_q(s_0(s_0 - 1))$.

3.a) Assume that $\nu_q > 0$. Whether $v_q(s_0) > 0$ or $v_q(s_0 - 1) > 0$, every cluster has at most two odd children. Again, $g_r^-$ is reducible by Proposition 4.41, so $K_q(\mathcal{R}) = \mathcal{Q}_\tau$ and $e_{K_q(\mathcal{R})/K_q} \le 2$. We deduce from [Bis22] that every proper cluster is $I_{K_q}$-invariant. Finally, the only principal cluster is $\mathcal{R}^+$, and $\nu_{\mathcal{R}^+} = v_q(\delta_{\mathcal{K}})$. We conclude that $(C_r^+)^{(\delta_{\mathcal{K}})}$ satisfies the semistability criterion if and only if $v_q(\delta_{\mathcal{K}}) = 0$.

3.b) Assume that $\nu_q \le 0$ and $\nu_q \equiv 0 \bmod r$. The only proper cluster is $\mathcal{R}^+$, which has size $2g+2$, so $(C_r^+)^{(\delta_{\mathcal{K}})}$ has potential good reduction. Hypothesis 2 yields $v_q(\delta_{\mathbb{Q}}^{2r} s_0(s_0 - 1)) = 0$, so $\mathcal{Q}_\tau/\mathcal{K}_q$ is unramified, and $\mathcal{K}_q(\mathcal{R})/\mathcal{Q}_\tau$ too because $v_q(s_0(s_0 - 1)) \equiv 0 \bmod r$ (see Theorem 4.46). This time $\nu_{\mathcal{R}^+} = v_q(\delta_{\mathcal{K}}) + (r+1)(v_q(\delta_{\mathbb{Q}}) + v_q(s_0)/r)$: if $v_q(\delta_{\mathcal{K}}) = 0$, then $(\mathcal{J}_r^+)^{(\delta_{\mathcal{K}})}$ has good reduction at $\mathfrak{q}$, and otherwise it has unipotent reduction.

3.c) Assume that $\nu_q \le 0$ and $\nu_q \not\equiv 0 \bmod r$. Just as above, $(C_r^+)^{(\delta_{\mathcal{K}})}$ has potential good reduction, but now $r$ divides $e_{K_q(\mathcal{R})/K_q}$ by Theorem 4.46. We conclude that $(\mathcal{J}_r^+)^{(\delta_{\mathcal{K}})}$ does not have good reduction at $\mathfrak{q}$, and thus has unipotent reduction. $\square$

**Corollary 4.49.** Assume that $\mathfrak{q}$ is odd, that $(\mathcal{J}_r^{\pm})^{(\delta_{\mathcal{K}})}$ has unipotent and potential good reduction at $\mathfrak{q}$. If $e_{K_q(\mathcal{R})/K_q} v_q(\delta_{\mathcal{K}}(1 - \zeta_r)^r \delta_{\mathbb{Q}}^{2r} \sqrt{s_0(s_0 - 1)}) \in 2\mathbb{Z}$, then $(\mathcal{J}_r^{\pm})^{(\delta_{\mathcal{K}})}$ attains good reduction over $K_q(\mathcal{R})$. In particular, the semistability defect of $(\mathcal{J}_r^{\pm})^{(\delta_{\mathcal{K}})}$, which is described as follows.

$$1) \text{ If } \mathfrak{q} = \mathfrak{r}, \text{ then } \mathrm{sd}_{(\mathcal{J}_r^{\pm})^{(\delta_{\mathcal{K}})}/\mathcal{K}_r} = \begin{cases} 2 & \text{if } g_r^- \text{ is reducible over } \mathbb{Q}_r, \\ r & \text{if } g_r^- \text{ is irreducible over } \mathbb{Q}_r \text{ and } v_r(\delta_{\mathbb{Q}}^{2r} s_0(s_0 - 1)) \notin 2\mathbb{Z}, \\ 2r & \text{otherwise.} \end{cases}$$

$$2) \text{ If } \mathfrak{q} \nmid 2r, \text{ then } \mathrm{sd}_{(\mathcal{J}_r^{\pm})^{(\delta_{\mathcal{K}})}/\mathcal{K}_q} = \begin{cases} r & \text{if } v_q(\delta_{\mathbb{Q}}^{2r} s_0(s_0 - 1)) \in 2\mathbb{Z}, \\ 2r & \text{if } v_q(\delta_{\mathbb{Q}}^{2r} s_0(s_0 - 1)) \notin 2\mathbb{Z}. \end{cases}$$

*Proof.* We use the criterion on cluster pictures for good reduction (Theorem 2.39) to prove that $(\mathcal{J}_r^{\pm})^{(\delta_{\mathcal{K}})}$ attains good reduction over $K_q(\mathcal{R})$. The first two items from the mentioned theorem are clearly satisfied, and if $\delta_{\mathcal{K}}$ is chosen as in the statement, the third one too. By definition, the semistability defect equals the ramification index of the minimal extension of $K_q$ where $(\mathcal{J}_r^{\pm})^{(\delta_{\mathcal{K}})}$ attains good reduction. The last part of the statement follows from the description of the ramification indices $e_{\mathcal{Q}_\tau/\mathcal{K}_q}$ and $e_{K_q(\mathcal{R})/\mathcal{Q}_\tau}$ (see Lemma 4.43 and Theorem 4.46). $\square$
# 5 Main properties of the 2-dimensional representations $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$

In this section, we study the 2-dimensional representations $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$ that arise from the fact that $(J_r^{\pm})^{(\delta_K)}$ have real multiplication by $\mathcal{K}$. We begin by proving that the compatible system $(\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda})_{\lambda}$ is modular. After this, we will study its conductor and the inertial local types of the attached WD-representations at places of bad reduction. Finally, we will use level lowering results to obtain a newform giving rise to the representation $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$ whose level is supported at primes we control.

Throughout the section, we keep using the notation introduced in §3.3. Recall that we assume that $s_0$ satisfies Hypothesis 4. The latter, combined with Theorem 4.2 implies that $(\mathcal{J}_r^{-})^{(\delta_K)}$ has at least one place of potential toric reduction (see Remark 4.4).

## 5.1 Modularity

In this subsection, we prove that the compatible system of representations $(\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda})_{\lambda}$ arises from a Hilbert newform over $\mathcal{K}$ of parallel weight 2 and trivial character. We refer the reader to [Fre90] for an introduction on Hilbert modular forms. The discussion below is deeply inspired by Darmon's panorama for propagating modularity among Frey objects of different signatures. Indeed, we are going to use the Frey representation of signature $(r, r, r)$ (used for solving Fermat's last theorem) to establish modularity of $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$. The content below is a generalisation of some of the results in [BCDF23, §4] and [Dar00, §2].

**Remark 5.1.** If $\mathcal{K}(\sqrt{\delta_K}) = \mathcal{K}$, then $(J_r^{\pm})^{(\delta_K)} \simeq J_r^{\pm}$ over $\mathcal{K}$, so proving modularity of the latter implies modularity of the former. If $\mathcal{K}(\sqrt{\delta_K})/\mathcal{K}$ is non-trivial, denote by $\chi_{\delta_K}: G_{\mathcal{K}} \to \overline{\mathbb{Q}}_{\ell}^{\times}$ the associated character of $G_{\mathcal{K}}$. For any finite place $\lambda$ of $\mathcal{O}_{\mathcal{K}}$ of good reduction for $J_r^{\pm}$, we have an isomorphism

$$ \rho_{(J_r^{\pm})^{(\delta_K)}, \lambda} \simeq \rho_{J_r^{\pm}, \lambda} \otimes \chi_{\delta_K} \qquad (5.1) $$

In order to prove that $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$ is modular, it suffices to show that $\rho_{J_r^{\pm}, \lambda}$ is modular.

Recall that $\mathfrak{r}$ denotes the unique prime ideal of $\mathcal{O}_{\mathcal{K}}$ lying above $r$. Since $\mathcal{K}/\mathbb{Q}$ is totally ramified at $r$, the residue field $\mathcal{O}_{\mathfrak{r}}/\mathfrak{r}$ is isomorphic to $\mathbb{F}_r$. We are going to treat the particular case $\lambda = \mathfrak{r}$, and prove that $\bar{\rho}_{J_r^{\pm}, \mathfrak{r}}: G_{\mathcal{K}} \to \mathrm{GL}_2(\mathbb{F}_r)$ is modular.

**Lemma 5.2.** The representation $\bar{\rho}_{J_r^{-}, \mathfrak{r}}$ extends to an odd representation of $G_{\mathbb{Q}}$.

*Proof.* Recall that $J_r^{\pm}$ has RM by $\mathcal{K}$, so $V_r(J_r^{\pm})$ carries a structure of $\mathcal{K} \otimes_{\mathbb{Q}} \mathbb{Q}_r$-module. But $r$ is totally ramified in $\mathcal{K}/\mathbb{Q}$, so $\mathcal{K} \otimes_{\mathbb{Q}} \mathbb{Q}_r \simeq \mathcal{K}_{\mathfrak{r}}$. By definition, $C_r^{-}(s_0)^{(\delta_{\mathbb{Q}})}$ is defined over $\mathbb{Q}$, so the action of $G_{\mathcal{K}}$ on the $\mathcal{K}_{\mathfrak{r}}$-module $V_r(J_r^{\pm})$ extends to a semilinear $G_{\mathbb{Q}}$-action. Therefore, the action of $G_{\mathbb{Q}}$ on $\mathrm{T}_r(J_r^{\pm}) \otimes_{\mathcal{O}_{\mathfrak{r}}} \mathbb{F}_{\mathfrak{r}}$ is $\mathbb{F}_{\mathfrak{r}}$-linear, and restricts to the action of $G_{\mathcal{K}}$ given by $\bar{\rho}_{J_r^{-}, \mathfrak{r}}$. $\square$

**Proposition 5.3.** If $r \ge 5$, the restriction of $\bar{\rho}_{J_r^{-}, \mathfrak{r}}$ to $G_{\mathbb{Q}(\zeta_r)}$ is absolutely irreducible.

*Proof.* Specifying $p=r$, Theorem 3.6 implies that $\bar{\rho}_{J_r^{-}(s), \mathfrak{r}}: G_{\mathcal{K}(s)} \to \mathrm{GL}_2(\mathbb{F}_{\mathfrak{r}})$ is a Frey representation of signature $(r, r, r)$. Consider the Legendre elliptic curve $\mathcal{L}(s)/\mathcal{K}(s)$ described by $y^2 = x(x-1)(x-s)$. Darmon proves in [Dar00, §1.3] that $\bar{\rho}_{\mathcal{L}(s), r}: G_{\mathcal{K}(s)} \to \mathrm{GL}_2(\mathbb{F}_{\mathfrak{r}})$ is also a Frey representation of signature $(r, r, r)$, and that the latter is unique up to equivalence. Thus, there is a character $\varepsilon: G_{\mathcal{K}} \to \overline{\mathbb{F}}_{\mathfrak{r}}^{\times}$ giving the isomorphism

$$ \bar{\rho}_{J_r^{-}, \mathfrak{r}} \simeq \bar{\rho}_{\mathcal{L}(s_0), r} \otimes \varepsilon. \qquad (5.2) $$

Now $\det \bar{\rho}_{\mathcal{L}(s_0), r}$ and $\det \bar{\rho}_{J_r^{\pm}, \mathfrak{r}}$ are both the mod $r$ cyclotomic character ([BCDF23, Theorem 2.8]), so $\varepsilon|_{G_{\mathbb{Q}(\zeta_r)}}$ is trivial, and $\varepsilon$ has order at most 2. To establish the proposition, it suffices to show that the restriction of $\bar{\rho}_{\mathcal{L}(s_0), r}$ to $G_{\mathbb{Q}(\zeta_r)}$ is absolutely irreducible.

We claim that $\mathcal{L}(s_0)/\mathcal{K}$ does not have complex multiplication (CM). We are going to prove this claim by showing that $\mathcal{L}(s_0)$ does not have potential good reduction everywhere. Hypothesis 4 implies
that $J_r^-$ has potential toric reduction at least at one place $q \neq r$. Let $L/K_q$ be a finite extension where $J_r^-$ attains toric reduction, and write $I_L$ for its inertia group. Theorem 3.5 states that $\mathrm{End}_{K(s)}(J_r^-(s)) \simeq \mathcal{O}_K$, so if we restrict $\bar{\rho}_{J_r^-, r}$ to $G_L$, then $r$ is good in the sense of [Rib76, II §2]. Lemma 3.5.3 of loc. cit. states that there is an additive character $\psi: G_L \to \mathbb{F}_r$ such that

$$ \bar{\rho}_{J_r^-, r}|_{G_L} = \begin{pmatrix} \bar{\chi}_r & \psi \\ 0 & 1 \end{pmatrix}, \qquad (5.3) $$

where $\bar{\chi}_r: G_L \to \mathbb{F}_r^\times$ is the mod $r$ cyclotomic character. Since $(\mathcal{J}_r^-)^{(\delta_K)}$ has toric reduction at $q$, then $\psi$ has to be ramified. Now $\varepsilon$ is trivial on $G_{\mathbb{Q}(\zeta_r)}$ and $\mathbb{Q}(\zeta_r)/K$ is unramified outside of $r$, so $\varepsilon|_{I_L}$ is trivial, as $q \neq r$. Therefore, the restriction of $\bar{\rho}_{\mathcal{L}(s_0), r}$ to the inertia group $I_L$ is also described by the RHS of (5.3), whose action is unipotent and non-trivial. Grothendieck's inertial criterion implies then that $\mathcal{L}(s_0)$ has potential multiplicative reduction at $q$ (see also [Sil94, Chapter IV §10]).

To conclude the proof, note the equality $\bar{\rho}_{\mathcal{L}(s_0), r}(G_{\mathbb{Q}(\zeta_r)}) = \bar{\rho}_{\mathcal{L}(s_0), r}(G_{\mathbb{Q}}) \cap \mathrm{SL}_2(\mathbb{F}_r)$. Since $\mathcal{L}(s_0)$ has no CM and $r \ge 5$, Propositions 3.1 and 4.3 in [Naj24] imply that $\bar{\rho}_{\mathcal{L}(s_0), r}(G_{\mathbb{Q}(\zeta_r)}) = \mathrm{SL}_2(\mathbb{F}_r)$. Assume by contradiction that the restriction of $\bar{\rho}_{J_r^-, r}$ to $G_{\mathbb{Q}(\zeta_r)}$ is absolutely reducible. Then its image is isomorphic to a subgroup of upper triangular matrices, or to a non-split Cartan subgroup. In both cases, this image is solvable, contradicting the fact that $\mathrm{SL}_2(\mathbb{F}_r)$ is not so for $r \ge 5$. $\square$

**Theorem 5.4.** The compatible system of representations $(\rho_{J_r^-, \mathfrak{t}})_{\lambda}$ is modular, i.e., it arises from a Hilbert newform defined over $K$.

*Proof.* Lemma 5.2 states that $\bar{\rho}_{J_r^-, \mathfrak{t}}: G_K \to \mathrm{GL}_2(\mathbb{F}_r)$ extends to an odd representation $\bar{\rho}$ of $G_{\mathbb{Q}}$. By Proposition 5.3, $\bar{\rho}|_{G_{\mathbb{Q}(\zeta_r)}}$ is absolutely irreducible, so $\bar{\rho}$ is also absolutely irreducible. Serre's conjecture ([KW09a, KW09b]) implies that $\bar{\rho}$ is modular, and by cyclic base change, $\bar{\rho}_{J_r^-, \mathfrak{t}}$ is modular too ([Lan80]). Theorem 4.2 implies that $\rho_{J_r^-, \mathfrak{t}}$ is unramified almost everywhere. Moreover, $J_r^-$ is potentially semistable, so $\rho_{J_r^-, \mathfrak{t}}$ is de Rham, hence Hodge-Tate, with HT weights $\{0, 1\}$. Applying [KT17, Theorem 1.1], we conclude that $\rho_{J_r^-, \mathfrak{t}}$ is modular. $\square$

**Theorem 5.5.** If $v_r(s_0(s_0-1)) > 2$, the compatible system of representations $(\rho_{(J_r^+)^{(\delta_K)}, \lambda})_{\lambda}$ is modular.

*Proof.* Theorem 3.6 states that $\bar{\rho}_{J_r^+(s), \mathfrak{t}}$ is a Frey representation of signature $(r, r, r)$, that is even in the sense of [Dar00, §1.1]. But the only Frey representation (up to equivalence) of signature $(r, r, r)$ is $\bar{\rho}_{\mathcal{L}(s), r}$, which is odd in the sense of [Dar00]. Specialising at $s = s_0$, we deduce that $\bar{\rho}_{J_r^+, \mathfrak{t}}$ is reducible. The assumption $v_r(s_0(s_0-1)) > 2$ implies that $(\mathcal{J}_r^+)^{(\delta_K)}$ has toric reduction at $\mathfrak{t}$, so $V_r((J_r^+)^{(\delta_K)})$ is an ordinary representation of $G_K$. The result follows from [SW99, §4.5, Theorem A] (with $k=2$, and the field denoted by $F(\chi_1/\chi_2)$ being equal to $\mathbb{Q}(\zeta_r)$). $\square$

**Remark 5.6.** The work of Pan [Pan22] seems like a promising way to prove modularity of $\rho_{(J_r^+)^{(\delta_K)}, \lambda}$. A result in the style of [Pan22, Theorem 7.11] would allow to drop $r$-adic assumptions on $s_0(s_0-1)$ in Theorem 5.5 above. However, in order to apply the mentioned result of Pan, we would need $r$ to be completely split in $K$, which is not satisfied in our setting.

**Remark 5.7.** By the work of Carayol [Car86], the modularity of $(\rho_{J_r^{\pm}, \lambda})_{\lambda}$ provides another proof for the fact that it is strictly compatible system of Galois representations.

**Example 5.8.** When specialising the values of $s_0, \delta_{\mathbb{Q}}$ as in Proposition 3.18, one can deduce the modularity of the representations arising from the curve $C_r^-(a, b, c)$ for the signature $(p, p, r)$, and for $C_r(a, b, c)$ for the signature $(r, r, p)$. Regarding the representations arising from $C_r^+(a, b, c)$, Theorem 5.5 implies they are modular as soon as $v_r(Aa^pBb^p) > 2$.
## 5.2 Conductor and inertial local types

We now use the results from §2.4.3 and §4 to compute the Artin conductor of $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$ at every finite place of $\mathcal{K}$. Modularity theorems state that the level of the Hilbert newform giving rise to $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$ equals the global conductor of such representation. For Diophantine applications, it is therefore crucial to understand this conductor in detail. Along the process, we will describe the local inertial types of the complex Weil-Deligne representations associated to $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$. This will be helpful later when discussing level lowering in §5.3.

In order to simplify the discussion below, let us introduce some notation.

**Definition 5.9.** Fix two finite places $\lambda \neq q$ of $\mathcal{K}$ such that $q \mid 2r\delta_{\mathbb{Q}}^{2r}s_0(s_0-1)$, and $\lambda \nmid 2r\delta_{\mathbb{Q}}^{2r}s_0(s_0-1)$. We denote by $\mathfrak{n}_{\text{tame},q}^{\pm}$ and $\mathfrak{n}_{\text{wild},q}^{\pm}$ the tame and wild conductors of $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$ restricted to $D_q \simeq G_{\mathcal{K}q}$, and by $\mathfrak{n}_{q}^{\pm} := \mathfrak{n}_{\text{tame},q}^{\pm} + \mathfrak{n}_{\text{wild},q}^{\pm}$. We define the global conductor of $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$ as

$$ \mathcal{N}(\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}) := \prod_q q^{\mathfrak{n}_q^{\pm}}. $$

Strict compatibility of the system $(\rho_{J_r^{\pm}, \lambda})_{\lambda}$ implies that the Artin conductor at $q$ does not depend on $\lambda$, so neither does the global conductor $\mathcal{N}(\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda})$. It is convenient to choose $\lambda$ coprime with $q$, and we assume that this is indeed the case.

We now describe the Artin conductor $\mathfrak{n}_{q}^{\pm}$, depending on $q$ and the other involved parameters. Recall that we let $q$ denote the rational prime lying below $q$, and we write $\nu_q = v_q(2^4s_0(s_0-1))$. Again, our analysis is not complete, but it is sufficient for our Diophantine applications.

**Theorem 5.10.** The value of the Artin conductor $\mathfrak{n}_{q}^{-}$ is described in Table 2.

<table><caption>Table 2: Table describing the Artin conductor of the $\lambda$-adic representation $\rho_{(J_r^{\pm})^{(\delta_K)}, \lambda}$ in terms of $\nu_q, g_r^{-}$ and $\delta_K$. Whenever nothing is written concerning the polynomial $g_r^{-} \in \mathbb{Q}_q[x]$, no further conditions are required on it.</caption><thead><tr><th>Place q</th><th>Behaviour of v<sub>q</sub> and g<sub>r</sub><sup>-</sup> &isin; Q<sub>q</sub>[x]</th><th>Condition on &delta;<sub>K</sub></th><th>n<sub>q</sub><sup>-</sup></th></tr></thead><tbody><tr><td rowspan="3">q | 2</td><td>v<sub>2</sub> &gt; 0</td><td>v<sub>q</sub>(&delta;<sub>K</sub>) = 0</td><td>1</td></tr><tr><td>v<sub>2</sub> &le; -4 and v<sub>2</sub> &equiv; -4 mod r</td><td>SQ(&delta;<sub>K</sub>&delta;<sub>&#x2126;</sub><sup>r</sup>(2 - 4s<sub>0</sub>))</td><td>0</td></tr><tr><td>v<sub>2</sub> &le; -4 and v<sub>2</sub> &ne; -4 mod r</td><td>SQ(&delta;<sub>K</sub>&delta;<sub>&#x2126;</sub><sup>r</sup>(2 - 4s<sub>0</sub>))</td><td>2</td></tr><tr><td rowspan="6">q = r</td><td>v<sub>r</sub> &gt; 2</td><td>v<sub>r</sub>(&delta;<sub>K</sub>) = 1</td><td>1</td></tr><tr><td>v<sub>r</sub> &le; 2 and g<sub>r</sub><sup>-</sup> reducible</td><td>&empty;</td><td>2</td></tr><tr><td rowspan="3">g<sub>r</sub><sup>-</sup> irreducible</td><td>v<sub>r</sub> = 2 or v<sub>r</sub> &equiv; 0 mod r</td><td>&empty;</td><td>3</td></tr><tr><td>v<sub>r</sub> = 1</td><td>&empty;</td><td>2 + r/2</td></tr><tr><td>v<sub>r</sub> &le; 0 and v<sub>r</sub> &ne; 0 mod r</td><td>&empty;</td><td>2 + r</td></tr><tr><td rowspan="3">q &nmid; 2r</td><td>v<sub>q</sub> &gt; 0</td><td>v<sub>q</sub>(&delta;<sub>K</sub>) = 0</td><td>1</td></tr><tr><td>v<sub>q</sub> &le; 0 and v<sub>q</sub> &equiv; 0 mod r</td><td>v<sub>q</sub>(&delta;<sub>K</sub>&delta;<sub>&#x2126;</sub><sup>r</sup>(2 - 4s<sub>0</sub>)) &isin; 2&Zopf;</td><td>0</td></tr><tr><td>v<sub>q</sub> &le; 0 and v<sub>q</sub> &ne; 0 mod r</td><td>&empty;</td><td>2</td></tr></tbody></table>

*Proof.* We use Theorem 4.2 to describe the reduction type of the Néron model $(J_r^{-})^{(\delta_K)}$, and Proposition 2.26 to recover from this the tame part of the conductor. As explained in Corollary 4.49, the choice of $\delta_K$ done in the Figure 1 minimises the semistability defect of $(J_r^{-})^{(\delta_K)}$.

1) Assume that $q$ is even.
1.a) If $\nu_2 > 0$ and $v_q(\delta_Q \delta_K)$ is even, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ has toric reduction at $q$. Proposition 2.26 gives $\mathbf{n}_{\text{tame}, q} = 1$, and $\mathbf{n}_{\text{wild}, q} = 0$ as $(\mathcal{J}_r^{-})^{(\delta_K)}$ is semistable. We deduce that $\mathbf{n}_q = 1$.

1.b) If $\nu_2 \le -4$, $\nu_2 \equiv -4 \pmod r$ and $\text{SQ}(\delta_K \delta_Q^r (2 - 4s_0))$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ has good reduction at $q$, so $\mathbf{n}_q = 0$ by the Néron-Ogg-Shafarevich criterion [ST68].

1.c) If $\nu_2 \le -4$, $\nu_2 \not\equiv -4 \pmod r$ and $\text{SQ}(\delta_K \delta_Q^r (2 - 4s_0))$ holds, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ attains good reduction over a finite extension of $K_q$ with ramification index $r$. Now $r$ being odd, this is a tame extension of $K_q$, so Lemma 2.27 yields $\mathbf{n}_{\text{wild}, q} = 0$, and therefore $\mathbf{n}_q = 2$.

2) Assume that $q = r$.

2.a) If $\nu_r > 2$ and $v_r(\delta_K) = 1$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ has toric reduction at $r$, so $\mathbf{n}_r = \mathbf{n}_{\text{tame}, r} = 1$.

2.b) If $\nu_r \le 2$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ has unipotent reduction at $r$, so $\mathbf{n}_{\text{tame}, r} = 2$.

If $g_r^-$ is reducible over $\mathbb{Q}_r$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ attains good reduction over $\mathbb{Q}_\tau$, which is a quadratic extension of $K_\tau$. Now $\mathbb{Q}_\tau/K_\tau$ is tame, so $\mathbf{n}_{\text{wild}, \tau} = 0$, and thus $\mathbf{n}_r = 2$.

If $g_r^-$ is irreducible over $\mathbb{Q}_r$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ attains good reduction over a wild extension of $K_\tau$. The wild conductor of $\rho_{(\mathcal{J}_r^{-})^{(\delta_K)}, \ell}$ at $r$ is described in Theorem 2.41. In particular, it does not depend on $\delta_K$, but only on the roots of $g_r^-$, so we may choose $\delta_K = 1$. Since $\mathcal{J}_r^-/K_\tau$ is the base change of Jac$(C_r^-(s_0)^{(\delta_Q)})/\mathbb{Q}_r$, and $K_\tau/\mathbb{Q}_r$ is a tame extension, Lemma 2.15 relates the wild conductor of the $\ell$-adic representations attached to the two Jacobians. Combining it with (2.5) gives

$$ \frac{r-1}{2} \mathbf{n}_{\text{wild}} \left( \rho_{\text{Jac}(C_r^-(s_0)^{(\delta_Q)}), \ell |_{G_{\mathbb{Q}_r}}} \right) = \mathbf{n}_{\text{wild}} \left( \rho_{(\mathcal{J}_r^-)^{(\delta_K)}, \ell |_{G_{\mathcal{K}_r}}} \right) = \frac{r-1}{2} \mathbf{n}_{\text{wild}, \mathfrak{r}}^{-} \quad (5.4) $$

To describe the left-most term, we use Theorem 2.41. Since $g_r^-$ is irreducible over $\mathbb{Q}_r$, there is a single orbit in $\mathcal{R}$ under the action of $G_{\mathbb{Q}_r}$, and a representative of this is $\gamma_0$, so $[\mathbb{Q}_r(\gamma_0) : \mathbb{Q}_r] = r$. Theorem 4.46 combined with Remarks 4.39 and 4.40 imply that $\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r$ is totally ramified, so its residue degree is 1. Then (5.4) and Theorem 2.41 give

$$ \mathbf{n}_{\text{wild}, \mathfrak{r}}^{-} = v_r \left( \Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r) - r + 1 \right). $$

The result then follows from the description of $v_r(\Delta(\mathbb{Q}_r(\gamma_0)/\mathbb{Q}_r))$ given in Theorem 4.48.

3) Assume that $q \nmid 2r$.

3.a) If $\nu_q > 0$ and $v_q(\delta_K) = 0$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ has toric reduction at $q$, so $\mathbf{n}_q = \mathbf{n}_{\text{tame}, q} = 1$.

3.b) If $\nu_q \le 0$, $\nu_q \equiv 0 \pmod r$ and $v_q(\delta_K \delta_Q^r (2 - 4s_0)) \in 2\mathbb{Z}$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ has good reduction at $q$, so the Artin conductor is trivial $\mathbf{n}_q = 0$.

3.c) If $\nu_q \le 0$, $\nu_q \not\equiv 0 \pmod r$, and $v_q(\delta_K \delta_Q^r (2 - 4s_0)) \in 2\mathbb{Z}$, then $(\mathcal{J}_r^{-})^{(\delta_K)}$ has unipotent reduction at $q$, so $\mathbf{n}_{\text{tame}, q} = 2$. Moreover, $(\mathcal{J}_r^{-})^{(\delta_K)}$ attains good reduction over an extension of degree $r$ or $2r$, which is tame, as $q \nmid 2r$. Lemma 2.27 yields $\mathbf{n}_{\text{wild}, q} = 0$, so $\mathbf{n}_q = 2$. $\square$

We now proceed to describe the Artin conductor $\mathbf{n}_q^+$.

**Theorem 5.11.** The value of the Artin conductor $\mathbf{n}_q^+$ is described in Table 3.

*Proof.* Recall that, when dealing with the curve $(C_r^+)^{(\delta_K)}$, we assume that $\delta_Q \in \mathbb{Z}$. The defining polynomial of $(C_r^+)^{(\delta_K)}$ is obtained from the one of $(C_r^-)^{(\delta_K)}$ by adjoining the linear factor $(x + 2\delta_Q)$. The extra root $\gamma_r = -2\delta_Q$ is rational, so, whenever $q$ is odd, Theorem 2.41 implies that $\mathbf{n}_{\text{wild}, q}^+ = \mathbf{n}_{\text{wild}, q}^-$ (see [ACIK$^{+24}$, Remark 2.10]). Again, we treat different cases separately.

1) Assume that $q$ is even.

1.a) If $\nu_2 \le 0$, $\nu_2 \equiv 4 \pmod r$ and $\text{SQ}(\delta_K)$, then $(\mathcal{J}_r^+)^{(\delta_K)}$ has good reduction at $q$, so $\mathbf{n}_q^+ = 0$.