<table><caption>Table 3: Table describing the Artin conductor of the $\lambda$-adic representation $\rho_{(J_r^-(\delta_K), \lambda)}$ in terms of $v, g_r^-$ and $\delta_K$.</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="2">q | 2</td><td>v<sub>2</sub> &le; 0 and v<sub>2</sub> &equiv; 4 mod r</td><td>SQ(&delta;<sub>K</sub>)</td><td>0</td></tr><tr><td>v<sub>2</sub> &le; 0 and v<sub>2</sub> &ne; 4 mod r</td><td>SQ(&delta;<sub>K</sub>)</td><td>2</td></tr><tr><td rowspan="7">q = r</td><td>v<sub>r</sub>(s<sub>0</sub>) &gt; 2</td><td>v<sub>r</sub>(&delta;<sub>K</sub>) = 0</td><td>1</td></tr><tr><td>v<sub>r</sub>(s<sub>0</sub> &minus; 1) &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="2">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 &nabla; 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>) = 0</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>

1.b) If $v_2 \le 0$, $v_2 \ne 4 \mod r$ and $\mathrm{SQ}(\delta_K)$ holds, then $(\mathcal{J}_r^+)^{(\delta_K)}$ has unipotent reduction at $q$, so $\mathbf{n}_{\mathrm{tame}, q}^+ = 2$. However, $(J_r^+)^{(\delta_K)}$ attains good reduction over an extension of $K_q$ with ramification index $r$, which is tame. Lemma 2.27 yields $\mathbf{n}_{\mathrm{wild}, q}^+ = 0$, and thus $\mathbf{n}_q^+ = 2$.

2) Assume that $q = r$.

2.a) If $v_r(s_0) > 2$ and $v_r(\delta_K) = 0$, then $(\mathcal{J}_r^+)^{(\delta_K)}$ has toric reduction at $r$. Similarly, if $v_r(s_0 - 1) > 2$ and $v_r(\delta_K) = 1$, then $(\mathcal{J}_r^+)^{(\delta_K)}$ has toric reduction at $r$. In both cases we have $\mathbf{n}_r^+ = 1$.

2.b) If $v_r \le 2$, then $(\mathcal{J}_r^+)^{(\delta_K)}$ has unipotent reduction at $r$, so $\mathbf{n}_{\mathrm{tame}, r}^+ = 2$. By the discussion at the beginning of the proof, we have $\mathbf{n}_{\mathrm{wild}, r}^+ = \mathbf{n}_{\mathrm{wild}, r}^-$, so the result follows from Theorem 5.10.

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

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

3.b) If $v_q \le 0$, $v_q \equiv 0 \mod r$ and $v_q(\delta_K) = 0$, then $(\mathcal{J}_r^+)^{(\delta_K)}$ has good reduction at $q$, so $\mathbf{n}_q^+ = 0$.

3.c) If $v_q \le 0$, $v_q \ne 0 \mod r$, and $v_q(\delta_K) = 0$, then $(\mathcal{J}_r^+)^{(\delta_K)}$ has unipotent reduction at $q$, so $\mathbf{n}_{\mathrm{tame}, q}^+ = 2$. Just as for $(J_r^+)^{(\delta_K)}$, the Jacobian $(J_r^+)^{(\delta_K)}$ attains good reduction over an extension of degree $r$ or $2r$, which is tame. Therefore $\mathbf{n}_{\mathrm{wild}, q}^+ = 0$ and $\mathbf{n}_q^+ = 2$. $\square$

**Remark 5.12.** In [ACIK$^+$24], the authors compute the Artin conductors at odd places of the $\ell$-adic representations attached Jac$(C_r(a, b, c))/\mathbb{Q}$ and Jac$(C_r^\pm(a, b, c))/\mathbb{Q}$, just as that of their base change to $K$. In loc. cit., the tame conductor was computed exploiting the combinatorial data of the cluster picture, using [BBB$^+$22, Theorem 12.3]. The approach we follow here is different, as we use first describe the reduction types of $(J_r^+)^{(\delta_K)}$ using cluster pictures. Proposition 2.26 automatically gives $\mathbf{n}_{\mathrm{tame}, q}^\pm$, without having to compute the sets $U$ and $V$ from [BBB$^+$22, Theorem 12.3]. The author believes that the strategy in here is more synthetic and beneficial for our Diophantine purposes, as knowing the reduction types of $(\mathcal{J}_r^+)^{(\delta_K)}$ has more applications than just computing Artin conductors (see Proposition 5.3 or Theorem 5.20).

Knowing Theorems 5.10 and 5.11, we can describe the global conductor of the strictly compatible systems of representations $(\rho_{(J_r^\pm(\delta_K), \lambda)})_\lambda$. In order to simplify the notation, we introduce:

**Definition 5.13.** Denote by $\mathbf{n}_2^\pm$ the Artin conductor $\mathbf{n}_q^\pm$ at any even place of $K$. Let $\mathbf{n}_{\mathrm{tor}}$ the square-free product of the primes $q \nmid 2r$ such that $v_q > 0$. Let $\mathbf{n}_{\mathrm{unip}}$ the square-free product of the primes $q \nmid 2r$ such that $v_q < 0$ and $v_q \ne 0 \mod r$.
**Corollary 5.14.** The global conductor of the strictly compatible system $(\rho_{(J_r^{\pm})(\delta_K), \lambda})_\lambda$ is given by

$$ \mathcal{N}(\rho_{(J_r^{\pm})(\delta_K), \lambda}) = 2^{n_2} \mathfrak{r}^{n_r} \mathfrak{n}_{\text{tor}} \mathfrak{n}_{\text{unip}}^2. $$

**Example 5.15.** In the style of Example 4.5, we describe the Artin conductor $\mathfrak{n}_q^{\pm}$ for the specific choice of $s_0, \delta_Q$ done in Proposition 3.18, when $\mathfrak{q} \nmid 2r$. Again, one could deduce in the same way $\mathfrak{n}_q^{\pm}$ when $\mathfrak{q} \mid 2$ or $\mathfrak{q} = \mathfrak{r}$, using Theorems 5.10 and 5.11.

1) Assume that $(a, b, c)$ is a primitive non-trivial solution to the generalised Fermat equation $(\mathcal{E}_{p,p,r})$. If $\delta_K$ is chosen conveniently, then the Artin conductor at $\mathfrak{q}$ of the system of 2-dimensional $\lambda$-adic

$$ \text{representations attached to } \text{Jac}(C_r^{\pm}(a, b, c))^{(\delta_K)} \text{ is } \begin{cases} 1 & \text{if } q \mid ABab, \\ 0 & \text{if } q \mid c \text{ and } q \nmid C, \\ 2 & \text{if } q \mid C. \end{cases} $$

2) Assume that $(a, b, c)$ is a primitive non-trivial solution to the generalised Fermat equation $(\mathcal{E}_{r,r,p})$. If $\delta_K$ is chosen conveniently, then the Artin conductor at $\mathfrak{q}$ of the system of 2-dimensional $\lambda$-adic

$$ \text{representations attached to } \text{Jac}(C_r(a, b, c))^{(\delta_K)} \text{ is } \begin{cases} 1 & \text{if } q \mid Cc, \\ 0 & \text{if } q \mid ab \text{ and } q \nmid AB, \\ 2 & \text{if } q \mid AB. \end{cases} $$

Now that we have computed the Artin conductors of $\rho_{(J_r^{\pm})(\delta_K), \lambda}$, we describe the local inertial type of the associated complex WD-representations. In order to simplify the notation, we introduce:

**Definition 5.16.** Let $\lambda$ and $\mathfrak{q}$ be as above. For any complex embedding $\iota: K_\lambda \hookrightarrow \mathbb{C}$, we denote by $\rho_{\iota, \mathfrak{q}}^{\pm}$ the complex WD-representation

$$ \rho_{\iota, \mathfrak{q}}^{\pm} := W_{\iota} \left( \rho_{(J_r^{\pm})(\delta_K), \lambda} \mid_{W_{\mathfrak{q}}} \right) : W_{\mathfrak{q}} \to \text{GL}_2(\mathbb{C}). $$

For any $x \in \mathbb{Z}$ coprime to $r$, we denote by $\text{ord}^\times(x \text{ mod } r)$ the order of $q$ in $(\mathbb{Z}/r\mathbb{Z})^\times$.

**Proposition 5.17.** The inertial local type of $\rho_{\iota, \mathfrak{q}}^{\pm}$ is described as follows.

1) Assume that $\mathfrak{q} \mid 2$. If $\nu_2 > 0$, then $\rho_{\iota, \mathfrak{q}}^{-}$ is Steinberg. If $\nu_2 \le -4$ and $\nu_2 \ne -4 \bmod r$, then $\rho_{\iota, \mathfrak{q}}^{-}$ is principal series if $\text{ord}^\times(2 \bmod r)$ is odd, and supercuspidal otherwise. If $\nu_2 \le 0$ and $\nu_2 \ne 4 \bmod r$, then $\rho_{\iota, \mathfrak{q}}^{+}$ is principal series if $\text{ord}^\times(2 \bmod r)$ is odd, and supercuspidal otherwise.

2) Assume that $\mathfrak{q} = \mathfrak{r}$. If $\nu_r > 2$, then $\rho_{\iota, \mathfrak{q}}^{\pm}$ is Steinberg, and if $\nu_r \le 2$, then $\rho_{\iota, \mathfrak{q}}^{\pm}$ is principal series.

3) Assume that $\mathfrak{q} \nmid 2r$. If $\nu_q > 0$, then $\rho_{\iota, \mathfrak{q}}^{\pm}$ is Steinberg. If $\nu_q \le 0$ and $\nu_q \ne 0 \bmod r$, then $\rho_{\iota, \mathfrak{q}}^{\pm}$ is principal series if $\text{ord}^\times(q \bmod r)$ is odd, and supercuspidal otherwise.

Moreover, whenever $\rho_{\iota, \mathfrak{q}}^{\pm}$ is supercuspidal, it is non-exceptional and arises as the induction of a character on the unramified quadratic extension of $K_\mathfrak{q}$.

**Proof.** We use Proposition 2.25. If $(J_r^{-})^{(\delta_K)}$ has potential toric reduction at $\mathfrak{q}$, then $\rho_{\iota, \mathfrak{q}}^{\pm}$ is Steinberg. Assume now that $(J_r^{\pm})^{(\delta_K)}$ has potential good reduction at $\mathfrak{q}$. Then $\rho_{\iota, \mathfrak{q}}^{\pm}$ is principal series if and only if the prime to $\mathfrak{q}$ part of the semistability defect $\text{sd}_{(J_r^{\pm})(\delta_K)/K_\mathfrak{q}}$ divides $|\mathbb{F}_\mathfrak{q}^\times|$. One can check (see [DM00, §1]) that the residue degree $[\mathbb{F}_\mathfrak{q} : \mathbb{F}_\mathfrak{q}]$ equals $\text{ord}^\times(q^2 \bmod r) = \frac{\text{ord}^\times(q \bmod r)}{\text{gcd}(2, \text{ord}^\times(q \bmod r))}$. By definition, $r$ divides $q^{\text{ord}^\times(q \bmod r)} - 1$, so we deduce that $r$ divides $|\mathbb{F}_\mathfrak{q}^\times|$ if and only if $\text{ord}^\times(q \bmod r)$ is odd. Knowing this, we treat different cases separately:

1) Assume that $\mathfrak{q} \mid 2$. If $\nu_2 \le -4$ and $\nu_2 \ne -4 \bmod r$, then Corollary 4.25 states that the semistability defect of $(J_r^{-})^{(\delta_K)}$ equals $r$. If $\nu_2 \le 0$ and $\nu_2 \ne 4 \bmod r$, then Corollary 4.25 gives $\text{sd}_{(J_r^{\pm})(\delta_K)/K_\mathfrak{q}} = r$ too. By the discussion above, $\rho_{\iota, \mathfrak{q}}^{\pm}$ is principal series if and only if $\text{ord}^\times(2 \bmod r)$ is odd.
2) Assume that $\mathfrak{q} = \mathfrak{r}$ and $\nu_r \le 2$. The residue field $\mathbb{F}_r$ is isomorphic to $\mathbb{F}_r$. Moreover, Corollary 4.49 implies that the prime to $r$ part of $\mathrm{sd}_{(J_r^{\pm})^{(\delta_K)}/\mathcal{K}_r}$ is 1 or 2. Both options divide $|\mathbb{F}_r^\times| = r-1$, so $\rho_{\ell,\mathfrak{r}}^{\pm}$ is principal series.

3) Assume that $\mathfrak{q} \nmid 2r$, $\nu_q \le 0$ and $\nu_q \not\equiv 0 \pmod r$. Then $(J_r^{\pm})^{(\delta_K)}$ attains good reduction over a (tame) extension of $\mathcal{K}_q$ with ramification index $r$ or $2r$ (Corollary 4.49). Since $\mathfrak{q} \nmid 2$, then $|\mathbb{F}_q^\times|$ is even. Thus, $\rho_{\ell,q}^{-}$ is principal series if and only if $r$ divides $|\mathbb{F}_q^\times|$, if and only if $\mathrm{ord}^\times(q \pmod r)$ is odd.

To prove the last statement, assume that $\mathfrak{q} \neq \mathfrak{r}$ and that $\rho_{\ell,q}^{\pm}$ is supercuspidal. The semistability defect of $(J_r^{\pm})^{(\delta_K)}$ equals the order of $\rho_{\ell,q}^{\pm}(I_q)$ (see Remark 2.24). Now $\mathrm{sd}_{(J_r^{\pm})^{(\delta_K)}/\mathcal{K}_q}$ is either $r$ or $2r$, so the projective image of $\rho_{\ell,q}^{\pm}$ cannot be $A_4$ nor $S_4$, and the latter WD-representation is therefore non-exceptional. By Remark 2.12, $\rho_{\ell,q}^{\pm}$ is the induction of a character from a quadratic extension of $\mathcal{K}_q$. Since $\mathrm{sd}_{(J_r^{\pm})^{(\delta_K)}/\mathcal{K}_q} = |\rho_{\ell,q}^{\pm}(I_{\mathcal{K}_q})|$ is always coprime to the residue characteristic $q$, we deduce that such quadratic extension is unramified. $\square$

## 5.3 Level lowering

As illustrated in Example 5.15, when choosing the parameters $s_0, \delta_Q$ as in Proposition 3.18, the global conductor $\mathcal{N}(\rho_{(J_r^{\pm})^{(\delta_K)},\lambda})$ depends on the putative solution to the considered equation. Such global conductor equals the level of the newform which gives rise to the system $(\rho_{J_r^{\pm},\lambda})_\lambda$. Its level depending on the solution is problematic for the elimination step, as one has to perform computations with parameters $a, b, c$ that are not meant to exist. In order to make the global conductor (and thus the level of the newform) independent of the solution, we apply level lowering results to $\rho_{(J_r^{\pm})^{(\delta_K)},\lambda}$.

All the content of sections §4 and §5 so far depended only on the prime number $r$, which we fixed since the beginning. Recall that we aim at solving the infinite families of generalised Fermat equations $(\mathcal{E}_{p,p,r})_p$ and $(\mathcal{E}_{r,r,p})_p$. Thus, from now on, we let $p$ be an odd prime number and $\mathfrak{p} \mid p$ a place of $\mathcal{K}$ dividing $p$. We are mostly interested in the 2-dimensional residual representation $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)},\mathfrak{p}} : \mathcal{G}_K \to \mathrm{GL}_2(\mathbb{F}_p)$, which corresponds to the particular case $\lambda = \mathfrak{p}$.

We begin by proving that $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)},\mathfrak{p}}$ is absolutely irreducible. In Proposition 5.3, we proved that $\bar{\rho}_{J_r^{\pm},\mathfrak{r}}$ is absolutely irreducible, so the same holds for its twist $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)},\mathfrak{r}}$. The particular case $\mathfrak{p} = \mathfrak{r}$ being already treated, we focus now on the generic case $\mathfrak{p} \neq \mathfrak{r}$.

**Proposition 5.18.** Assume that there is some odd prime $q \neq r$ such that $\nu_q \le 0$, $\nu_q \not\equiv 0 \pmod r$, and $\mathrm{ord}^\times(q \pmod r)$ is even. Then for any odd place $\mathfrak{p}$ of $\mathcal{K}$, the residual representation $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)},\mathfrak{p}}$ is absolutely irreducible.

*Proof.* As explained above, we focus on the case $\mathfrak{p} \neq \mathfrak{r}$. Since $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)},\mathfrak{p}}$ is odd and $\mathcal{K}$ is totally real, then $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)},\mathfrak{p}}$ is absolutely irreducible if and only if it is irreducible. The assumptions on $\nu_q$ imply that $(J_r^{\pm})^{(\delta_K)}$ has unipotent and potential good reduction at any $\mathfrak{q}$ above $q$. The WD-representation $\rho_{\ell,q}^{\pm}$ is simply given by

$$ \rho_{\ell,q}^{\pm} = \left( \rho_{(J_r^{\pm})^{(\delta_K)}, \mathfrak{p}} |_{W_{\mathcal{K}_q}} \otimes_{\ell} \mathbb{C}, 0 \right) \qquad (5.5) $$

(the tensor product is taken along the embedding $\iota : \mathcal{K}_p \hookrightarrow \mathbb{C}$). The image of $I_{\mathcal{K}_q}$ through $\rho_{\ell,q}^{\pm}$ has order $r$ or $2r$, and by (5.5), the same holds for $\rho_{(J_r^{\pm})^{(\delta_K)},\mathfrak{p}}|_{W_{\mathcal{K}_q}}$. Now Proposition 5.17 implies that $\rho_{\ell,q}^{\pm}$ (and thus $\rho_{(J_r^{\pm})^{(\delta_K)},\mathfrak{p}}|_{W_{\mathcal{K}_q}}$) is an irreducible induction of a character from the quadratic unramified extension of $\mathcal{K}_q$. Since $\mathfrak{p} \nmid 2r$, its reduction $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)},\mathfrak{p}}|_{W_{\mathcal{K}_q}}$ is also an irreducible induction of a character. The latter being a restriction of $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)},\mathfrak{p}}$, we obtain the desired result. $\square$

**Proposition 5.19.** Assume that $q = 2$, $\mathrm{ord}^\times(2 \pmod r)$ is even, and fix some place $\mathfrak{p} \nmid 2r$. If $\nu_2 \le -4$ and $\nu_2 \not\equiv -4 \pmod r$ then $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)},\mathfrak{p}}$ is absolutely irreducible. If $\nu_2 \le 0$ and $\nu_2 \not\equiv 4 \pmod r$, then $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)},\mathfrak{p}}$ is absolutely irreducible.
**Proof.** Corollary 4.25 states that, under the cited conditions, $\rho_{\ell,q}^{\pm}(I_{\mathcal{K}_q})$ has order $r$. Proposition 5.17 implies that $\rho_{\ell,q}^{\pm}$ is an irreducible induction of a character from the quadratic unramified extension of $\mathcal{K}_q$. Since $\mathfrak{p} \nmid 2r$, we conclude just as in the proof of Proposition 5.18. $\square$

In order to apply level lowering results, we need to understand better the local behaviour of the
the residual representation $\bar{\rho}_{(J_r^{\pm})(\delta_K), \mathfrak{p}}$. Namely, we need it to be finite at every $\mathfrak{q} \mid \mathfrak{p}$, meaning that it
arises from a finite flat group scheme over $\mathcal{O}_q$. The next result shows that this is indeed the case.

**Theorem 5.20.** Let $p$ be a rational prime, and $\mathfrak{p} \nmid p$. Let $\mathfrak{q} \neq 2, r$ be a rational prime such that $\nu_q > 0$ and $\nu_q \equiv 0 \bmod p$. Let $\mathfrak{q}$ be a finite place of $\mathcal{K}$ above $q$.

1) If $\mathfrak{q} \nmid p$, then $\bar{\rho}_{(J_r^{\pm})(\delta_K), \mathfrak{p}}$ is unramified at $\mathfrak{q}$.

2) If $\mathfrak{q} \nmid p$, then $\bar{\rho}_{(J_r^{\pm})(\delta_K), \mathfrak{p}}|_{D_q}$ is finite at $\mathfrak{q}$.

**Proof.** We apply [BCDF23, Theorem 7.5] to the Jacobian Jac$(C_r^{\pm}(s)^{(\delta_Q\delta_K)})$ defined over the function field $\mathcal{K}_q(s)$. Specialising $s = s_0$ yields the well-known $(J_r^{\pm})^{(\delta_K)}$. Theorem 3.5 combined with Lemma 2.18 shows that Jac$(C_r^{\pm}(s)^{(\delta_Q\delta_K)})$ has RM by $\mathcal{K}$. The assumption $\nu_q > 0$ implies that, among $v_q(s_0)$ and $v_q(s_0 - 1)$, exactly one is zero and the other is positive and divisible by $p$. In the notation of [BCDF23, §7], we let $t_1 := s_0$, and then $z_0 := 0$ if $v_q(s_0) > 0$, and $z_0 := 1$ if $v_q(s_0 - 1) > 0$, so that $v_q(\pi_0(t_1)) > 0$ and $v_q(\pi_0(t_1)) \equiv 0 \bmod p$. Moreover, the three other conditions listed in the previous page are satisfied: the first one is due to the fact that $C_r^{\pm}(s)$ are Mumford curves over $\mathcal{K}_q[[s]]$ and $\mathcal{K}_q[[s-1]]$ (see [Dar00, Theorem 1.10]). The second condition follows by considering the discriminants of the models defining $C_r^{\pm}(s)^{(\delta_Q\delta_K)}$ (to obtain them it suffices to replace $s_0$ by $s$ in Corollary 4.13). The third condition follows from $(J_r^{\pm})^{(\delta_K)}$ having toric reduction at $\mathfrak{q}$, as $\nu_q > 0$ (see §4). Theorem 7.5 in [BCDF23] then yields the desired result. $\square$

We are now going to apply level lowering results to the modular compatible system $(\rho_{(J_r^{\pm})(\delta_K), \lambda})_{\lambda}$.
The newform obtained after this will have a suitable level for our Diophantine purposes. As we will
see in Example 5.23, for the specific choice of $s_0$, $\delta_Q$ done in Proposition 3.18, such level will depend
on the considered Diophantine equation, but not on its solution.

Recall the notation $\mathfrak{n}_2^{\pm}$ and $\mathfrak{n}_{\text{unip}}$ introduced in Definition 5.13. For simplicity, we define:

**Definition 5.21.** Denote by $\mathfrak{n}_{\text{tor}, \neq 0(p)}$ the square-free product of the primes $\mathfrak{q} \nmid 2r$ such that $\nu_q > 0$ and $\nu_q \neq 0 \bmod p$.

We now state our level lowering result. We are going to apply a theorem of Breuil-Diamond [BD14], that has the advantage of preserving inertial local types.

**Theorem 5.22.** Assume that $\mathfrak{p} \nmid 2r$ and that $\nu_p \ge 0$. Suppose that $\bar{\rho}_{(J_r^{\pm})(\delta_K), \mathfrak{p}}$ is absolutely irreducible, and when considering $\bar{\rho}_{(J_r^{+})(\delta_K), \mathfrak{p}}$, suppose moreover that $\nu_r > 2$. Then, there is a Hilbert newform $g$ over $\mathcal{K}$ of parallel weight 2, trivial character and level $2^{n_2} \mathfrak{r}^{n_r} \mathfrak{n}_{\text{tor}, \neq 0(p)} \mathfrak{n}_{\text{unip}}^2$, satisfying the following:

1) $\bar{\rho}_{(J_r^{\pm})(\delta_K), \mathfrak{p}} \simeq \bar{\rho}_g, \mathfrak{P}$ for some place $\mathfrak{P} \mid p$ in the field of coefficients $K_g$.

2) We have the field inclusion $\mathcal{K} \subset K_g$.

3) If there is some prime $\mathfrak{q} \nmid 2r$ such that $\nu_q > 0$ and $\nu_q \neq 0 \bmod p$, then $g$ does not have complex multiplication.

**Proof.** By Theorems 5.4 and 5.5, the representation $\bar{\rho}_{(J_r^+)(\delta_K), \mathfrak{p}}$ is modular. The assumption on $\mathfrak{p}$ implies that $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ has good or toric reduction at $\mathfrak{p}$. Moreover, since we assume that Hypothesis 5 holds, there is at least one place $\mathfrak{q}' \nmid 2r$ such that $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ has unipotent reduction at $\mathfrak{q}'$. In particular, $\mathfrak{q}$ and $\mathfrak{p}'$ are coprime. Lemma 6.9 in [BCDF23] implies that $\bar{\rho}_{(J_r^{\pm})(\delta_K), \mathfrak{p}}$ restricted to $G_{\mathcal{K}(\zeta_p)}$ is absolutely irreducible. If $\mathfrak{p} \neq 5$, the result follows from combining Theorem 5.20 with Theorem 3.2.2 in [BD14]. If $\mathfrak{p} = 5$, we claim that $\bar{\rho}_{(J_r^{\pm})(\delta_K), \mathfrak{p}}(G_{\mathcal{K}(\zeta_p)}) \not\equiv \text{PSL}_2(\mathbb{F}_5) \bmod \mathcal{P}$.
implies that $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)}, \mathfrak{p}}(I_{q'})$ has order $r$ or $2r$, as $\rho_{(J_r^{\pm})^{(\delta_K)}, \mathfrak{p}}(I_{q'})$ does not intersect the kernel of the reduction mod $\mathfrak{p}$. Since $[\mathcal{K}(\zeta_5) : \mathcal{K}] = 4$, the claim follows, and we can also apply [BD14, Theorem 3.2.2]. We conclude that $\bar{\rho}_{(J_r^{\pm})^{(\delta_K)}, \mathfrak{p}} \simeq \bar{\rho}_{g, \mathfrak{p}}$, where the level of $g$ is as claimed in the statement.

Next, we prove that $\mathcal{K} \subset K_g$. As explained above, $(\mathcal{J}_r^{\pm})^{(\delta_K)}$ has unipotent reduction at $q'$. Consider the complex WD-representation $\rho_{\ell, q'}^{\pm} : W_{q'} \to \mathrm{GL}_2(\mathbb{C})$ introduced in Definition 5.16, which is isomorphic to $W(\rho_{g, \mathfrak{p}}|_{D_{q'}}) \otimes \mathbb{C}$ (the tensor product is taken along an embedding $(K_g)_{\mathfrak{p}} \hookrightarrow \mathbb{C}$). Proposition 5.17 implies that, whether $\rho_{\ell, q'}^{\pm}$ is principal series or supercuspidal, its restriction to $I_{q'}$ can be written as $\delta \oplus \delta^{-1}$, where $\delta : I_{q'} \to \mathbb{C}^{\times}$ has order $r$ or $2r$. Since $\mathbb{Q}(\zeta_{2r})^{+} = \mathbb{Q}(\zeta_r)^{+}$, [BCDF23, Proposition 8.4] gives the desired inclusion.

Finally, if the condition on the third item is satisfied, then $\rho_{g, \mathfrak{p}}$ has a Steinberg prime, so $g$ cannot have complex multiplication. $\square$

**Example 5.23.** Once again, we specialise Theorem 5.22 to the particular Frey objects obtained by setting $s_0, \delta_{\mathbb{Q}}$ as in Proposition 3.18. For any $x \in \mathbb{Z}$, define $\mathrm{rad}^*$ to be the product of all prime ideals of $\mathcal{O}_{\mathcal{K}}$ dividing $x$ that are coprime to $2r$.

1) Assume that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{p, p, r}) : Ax^p + By^p = Cz^r$, the generalised Fermat equation of signature $(p, p, r)$. Let $s_0, \delta_{\mathbb{Q}}$ be as in Proposition 3.18. The level of the newform $g$ given by Theorem 5.22 is

$$ \mathcal{N}_g = 2^{n_2^{\pm}} \mathfrak{r}^{n_r^{\pm}} \mathrm{rad}^*(AB) \mathrm{rad}^*(C)^2. $$

2) Assume that $(a, b, c)$ is a primitive non-trivial solution to $(\mathcal{E}_{r, r, p}) : Ax^r + By^r = Cz^p$, the generalised Fermat equation of signature $(r, r, p)$. Let $s_0, \delta_{\mathbb{Q}}$ be as in Proposition 3.18. The level of the newform $g$ given by Theorem 5.22 is

$$ \mathcal{N}_g = 2^{n_2^{\pm}} \mathfrak{r}^{n_r^{\pm}} \mathrm{rad}^*(C) \mathrm{rad}^*(AB)^2. $$

# 6 Solving families of GFEs of signatures $(p, p, r)$ and $(r, r, p)$

In this section we explain how to effectively perform the elimination step to solve infinite families of generalised Fermat equations. We begin by summarising the discussion above to solve specific families of equations. We then explain how to discard isomorphisms of Galois representations by comparing traces of Frobenius. To conclude, we discuss some computational aspects of our Magma implementation.

## 6.1 Specialising $s_0$ and $\delta_{\mathbb{Q}}$ for solving specific families of equations

The whole content of sections §4 and §5 was done for generic values of $s_0$ and $\delta_{\mathbb{Q}}$, although we left a trail of examples after every substantial result. We are now going to fix the values of these parameters as displayed in Table 1. We begin by summarising the strategy of the modular method to solve an instance of a family of GFEs. We focus on the signature $(p, p, r)$, the discussion for $(r, r, p)$ would follow the same lines.

### The modular method in practice

Let $r, A, B, C$ be three fixed integers as in Definition 2.1, and let $p > r$ be any prime number. Assume that there exists a primitive non-trivial solution $(a, b, c)$ to the generalised Fermat equation

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

We proceed in various steps.

1) To $(a, b, c)$ correspond two Frey hyperelliptic curves $C_r^{\pm}(a, b, c)/\mathbb{Q}$. By Theorem 3.5, the base-changed Jacobians $J_r^{\pm}(a, b, c) := \mathrm{Jac}(C_r^{\pm}(a, b, c) \times \mathcal{K})$ have RM by $\mathcal{K}$. To them, we associate 2-dimensional representations $\rho_{J_r^{\pm}(a, b, c), \lambda} : G_{\mathcal{K}} \to \mathrm{GL}_2(\mathcal{K}_{\lambda})$, for every $\lambda \in \mathrm{Spec}(\mathcal{O}_{\mathcal{K}})$.
2) Theorem 5.4 implies that the compatible system $(\rho_{J_r^{-}(a,b,c),\lambda})_{\lambda}$ is modular. If $r | ab$, or $v_r(AB) > 2$, then Theorem 5.5 also gives the modularity of $(\rho_{J_r^{+}(a,b,c),\lambda})_{\lambda}$. Consider a finite place $p$ dividing $p$. By Theorem 5.22, the residual representation $\bar{\rho}_{J_r^{\pm}(a,b,c),p}$ arises from a Hilbert newform over $\mathcal{K}$ of parallel weight 2 and level $\mathcal{N} := 2^{\frac{n_p}{2}} \mathfrak{r}^{\frac{n_p}{2}} \mathrm{rad}^*(AB) \mathrm{rad}^*(C)^2$.

3) We numerically compute the space of Hilbert newforms of parallel weight 2 and level $\mathcal{N}$, which we denote by $S_2(\mathcal{N})$. Following the content of subsection 6.2, we prove that, for any $g \in S_2(\mathcal{N})$, and any place $\mathfrak{P} | p$ in the field of coefficients $K_g$, we have $\bar{\rho}_{J_r^{\pm}(a,b,c),p} \not\cong \bar{\rho}_{g,\mathfrak{P}}$. This contradicts the previous point, so we conclude that $(a, b, c)$ does not exist.

## 6.2 Discarding isomorphisms by comparing traces of Frobenius

We now explain how to effectively discard isomorphisms of Galois representations as above, both from a theoretical and algorithmic point of view. For simplicity, we write $J := J_r^{\pm}(a, b, c)$, and we keep using the notation from the previous subsection.

**Definition 6.1.** Let $g$ be a newform in $S_2(\mathcal{N})$. We say that we have eliminated the pair $(p, g)$ if we show that, for any place $\mathfrak{P} | p$ in the coefficient field $K_g$, we have

$$ \bar{\rho}_{J, p} \not\cong \bar{\rho}_{g, \mathfrak{P}}. \qquad (6.1) $$

To prove the non-existence of primitive non-trivial solutions to $(\mathcal{E}_{p,p,r})$, one has to eliminate all pairs $(p, g)$ (at least for $p$ greater than a certain bound). Let us fix a newform $g \in S_2(\mathcal{N})$, and a place $\mathfrak{P} | p$ in its field of coefficients. For any finite place $\mathfrak{q}$ of $\mathcal{K}$, we consider the traces of Frobenius

$$ a_{\mathfrak{q}}(J) := \mathrm{Tr}(\bar{\rho}_{J, \mathfrak{p}}(\mathrm{Frob}_{\mathfrak{q}})), \quad \text{and} \quad a_{\mathfrak{q}}(g) := \mathrm{Tr}(\bar{\rho}_{g, \mathfrak{P}}(\mathrm{Frob}_{\mathfrak{q}})) $$

assuming that the Néron model of $J$ has good reduction at $\mathfrak{q}$ in the former case, and that $\mathfrak{q} \nmid \mathcal{N}$ in the latter case. If the isomorphism $\bar{\rho}_{J, p} \simeq \bar{\rho}_{g, \mathfrak{P}}$ holds then $a_{\mathfrak{q}}(J)$ and $a_{\mathfrak{q}}(g)$ are equal in $\mathbb{F}_p$. To test if such an equality holds, we would need to fix embeddings $\mathbb{F}_p \hookrightarrow \bar{\mathbb{F}}_p$ and $\mathbb{F}_{\mathfrak{P}} \hookrightarrow \bar{\mathbb{F}}_p$, which are not canonical. In order to get rid of this ambiguity, we consider the divisibility relationship:

$$ p \mid \gcd_{\sigma \in \mathrm{Gal}(\mathcal{K}/\mathbb{Q})} N_{K_g/\mathbb{Q}}(a_{\sigma(\mathfrak{q})}(J) - a_{\sigma(\mathfrak{q})}(g)). \qquad (6.2) $$

If we had $\bar{\rho}_{J, p} \simeq \bar{\rho}_{g, \mathfrak{P}}$, then (6.2) would hold for every $\mathfrak{q}$. Therefore, as soon as we find some $\mathfrak{q}$ such that $p$ does not divide the RHS of (6.2), then the pair $(p, g)$ is eliminated. But the Jacobian $J = J_r^{\pm}(a, b, c)$ is not meant to exist, so how to compute the RHS of (6.2)? We use Remark 3.13, which explains that the curves $C_r^{\pm}(a, b, c)$ admit models that depend only on $a$ and $c$. Moreover, the curve $C_r^{\pm}(a, b, c)$ is the base change of a curve defined over $\mathbb{Q}$, so the $a_{\mathfrak{q}}(J)$'s depend only on the congruence classes of $a, c \bmod q$. Therefore, it suffices to compute the RHS of (6.2) for $a$ and $c$ ranging through all possible values of $\mathbb{F}_q$ that do not yield singular curves over $\mathbb{F}_q$.

Recall that $G_{\mathbb{Q}}$ acts on $S_2(\mathcal{N})$ via its action on Fourier coefficients. An orbit under this action is called a Hecke constituent. For any $g \in S_2(\mathcal{N})$, we denote by $[g]$ its Hecke constituent. Let $q$ be a prime number and $\mathfrak{q} \subset \mathcal{K}$ a place above $q$. Assume first that the Néron model of $J$ has good reduction at $\mathfrak{q}$. Following [BCDF23, §9.4], we introduce

$$ N_{\mathrm{good}}(\mathfrak{q}, g) := \prod_{\mathfrak{q}'|q} \gcd_{\sigma \in \mathrm{Gal}(\mathcal{K}/\mathbb{Q})} N_{K_g/\mathbb{Q}}(\sigma(a_{\mathfrak{q}'}(J)) - a_{\sigma(\mathfrak{q})}(g)). \qquad (6.3) $$

As explained in loc. cit., this quantity depends only on the Hecke constituent $[g]$, and not on the choice of a representative. If $p \nmid N_{\mathrm{good}}(\mathfrak{q}, g)$, then every pair $(p, g')$ is eliminated, for $g' \in [g]$.

Suppose now that the Néron model of $J$ has toric reduction at $\mathfrak{q}$. Assuming that we had $\bar{\rho}_{J, p} \not\cong \bar{\rho}_{g, \mathfrak{P}}$, then $p$ would divide

$$ M_{\mathrm{toric}}(\mathfrak{q}, g) := \gcd_{\sigma \in \mathrm{Gal}(\mathcal{K}/\mathbb{Q})} N_{K_g/\mathbb{Q}}(a_{\sigma(\mathfrak{q})}(g)^2 - (N_{\mathcal{K}/\mathbb{Q}}(\mathfrak{q}) + 1)^2). $$
In practice, to eliminate a pair $(p, g)$ we first range through the different $g$'s, and for each of these, we bound the set of $p$'s that are not yet eliminated. When implementing this discussion, we proceed as follows. First, we initialise $S_2(\mathcal{N})$, and compute a basis of eigenforms: Magma returns a list of the corresponding Hecke constituents. For each of these, we check if the field $\mathcal{K}$ is included in $K_g$ (cf. Theorem 5.22). If not, we can already eliminate the form $g$ for every prime $p$. If $\mathcal{K} \subset K_g$, we initialise the set of $p$'s to eliminate as an infinite set of prime numbers. We run a loops on ascending $q$'s, and on appropriate pairs $(a, c) \in \mathbb{F}_q^2$. At every stage, we compute the set of prime divisors of $N_{\text{good}}(q, g) M_{\text{toric}}(q, g)$, and we intersect this set with the one obtained for the previous $q$. The remaining $p$'s are those for which the pair $(p, g)$ is not yet eliminated.

This procedure is completely effective, and the reader can find a Magma implementation in [Azo25]. To illustrate its explicitness, we specialise the parameter $r = 5$ and solve families of equations of signature $(p, p, 5)$ and $(5, 5, p)$. In this case, the field $\mathcal{K}$ equals $\mathbb{Q}(\sqrt{5})$. We denote by $\mathfrak{r}_5$ the unique prime ideal in $\mathcal{O}_{\mathcal{K}}$ above 5. More precisely, we prove the following asymptotic results:

**Theorem 6.2.** Let $p > 71$ be any prime number. 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 $10 \mid ab$.

*Proof.* We fix $r = 5, A = 7, B = 1$ and $C = 3$. If $2 \mid ab$, the Artin conductor of $\rho_{J_r^{-}(a,b,c), p}$ at any even place is 1. If $5 \mid ab$ and the twisting parameter $\delta_{\mathcal{K}}$ has $\mathfrak{r}_5$-adic valuation 1, the Artin conductor at $\mathfrak{r}_5$ is also 1. Assuming that $10 \mid ab$, we get $\mathcal{N} = 2^1 3^2 \mathfrak{r}_5^1 7^1$: for this level, $S_2(\mathcal{N})$ has dimension 680, and there are 101 Hecke constituents. The output of the elimination process can be found in the file "ex1_pp5.txt" in [Azo25]. □

**Theorem 6.3.** Let $p > 41$ be any prime number. 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 $10 \mid c$.

*Proof.* This time we set $r = 5, A = 1, B = 7$ and $C = 1$. If $10 \mid c$, the level given by Theorem 5.22 is $\mathcal{N} = 2^1 \mathfrak{r}_5^1 7^2$. For this level, $S_2(\mathcal{N})$ has dimension 471, and there are 71 Hecke constituents. The output of the elimination process can be found in the file "ex1_55p.txt" in [Azo25]. □

**Remark 6.4.** If one drops the assumption $10 \mid c$ in Theorem 6.3, there exist primitive non-trivial solutions to the equation. For $p = 2$, there is $(2, -1, 5)$, and for $p = 3$, there is $\pm(1, 1, 2)$. This is problematic, as to these solutions correspond well-defined Jacobians, which correspond to certain Hilbert newforms of the level given by Theorem 5.22. In this case, in the elimination step, there is some $g$ for which we cannot bound the set of $p$'s such that $(p, g)$ is eliminated.

Nevertheless, in our particular case, we manage to overcome this difficulty. Indeed, to the mentioned solutions correspond newforms whose level is greater than $\mathcal{N} = 2^1 \mathfrak{r}_5^1 7^2$. Therefore, these newforms do not live in the same space as the one where we perform the elimination.

As explained in Remark 3.13, the Frey curves depend only on two parameters among $a, b, c$. On the other hand, the level provided by Theorem 5.22 depends on the radical of the coefficients, but not on the valuations of the latter. For instance, for the signature $(p, p, r)$, the LHS in isomorphism 6.1 is independent of $Bb^p$, whereas the RHS depends only on the primes dividing $B$.

This allows for a very interesting phenomenon, that we call "Pay for 1 equation, get n for free." As a direct consequence of Theorems 6.2 and 6.3, we obtain:
**Theorem 6.5.** Let $p > 71$ be any prime number. For any $i \in [1..4]$ and $j \in \{3,4\}$, there are no primitive non-trivial solutions to the generalised Fermat equation

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

*Proof.* For this choice of coefficients, the level given by Theorem 5.22 is $\mathcal{N} = 2^1 3^2 \mathbf{r}_5^1 7^1$, just as in the proof of Theorem 6.2. Since the Jacobian $J_r^{-}(a, b, c)$ does not depend on $b$, the elimination process from "ex1_pp5.txt" in [Azo25] shows the non-existence of solutions for any of the considered equations. $\square$

**Theorem 6.6.** Let $p > 411$ be any prime number. For any $i \in [1..4]$ and $j \in [2..4]$, there are no primitive non-trivial solutions to the generalised Fermat equation

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

*Proof.* Just as in the proof of Theorem 6.3, the level here is $\mathcal{N} = 2^1 \mathbf{r}_5^1 7^2$. This time, the Jacobian depends only on $a, b$, so the elimination process from "ex1_55p.txt" in [Azo25] allows to conclude. $\square$

## 6.3 Computational aspects of the elimination step

The method above presents some computational limitations that make it hard to apply in full generality. The execution time of our scripts grows very quickly with the size of the parameters $r, A, B, C$ (the reader can find details about these execution times in [Azo25]). In particular, computing a basis of eigenforms of the space $S_2(\mathcal{N})$ presents a huge computational cost. Indeed, the time of execution for this task grows very quickly with the level. The examples treated in Theorems 6.2 and 6.3 present a suitable compromise between the interest of the Diophantine equation, and the length of the execution time of our scripts.

Most of the newforms are eliminated because their field of coefficients does not contain the field $\mathcal{K}$. For the other ones, the execution time for bounding the set of primes grows with the degree of the field of coefficients of the form. For some given forms, we observe that certain primes are difficult to eliminate. To eliminate such pairs $(p, g)$, one could use refined versions of the elimination step as in [BCDF23, §9.11]. The author is currently working on implementing this.

Finally, we stress out the fact that our scripts are written using functional programming. We aim at writing functions and methods that may be reused as much as possible. For instance, the files "ex1_pp5.txt" and "ex1_55p.txt" both use the same elimination functions.

## References

[ACIK$^{+24}$] Martin Azon, Mar Curcó-Iranzo, Maleeha Khawaja, Céline Maistret, and Diana Mocanu. Conductor exponents for families of hyperelliptic curves, 2024. https://arxiv.org/abs/2410.21134.

[Azo25] Martin Azon. Electronic resources and supporting files for this paper, 2025. https://github.com/martin-azon/effective_Darmon.

[BBB$^{+22}$] Alex J. Best, L. Alexander Betts, Matthew Bisatt, Raymond van Bommel, Vladimir Dokchitser, Omri Faraggi, Sabrina Kunzweiler, Céline Maistret, Adam Morgan, Simone Muselli, and Sarah Nowell. A user's guide to the local arithmetic of hyperelliptic curves. *Bull. Lond. Math. Soc.*, 54(3):825–867, 2022.

[BCDF23] Nicolas Billerey, Imin Chen, Luis Dieulefait, and Nuno Freitas. On Darmon's program for the generalized Fermat equation, i, 2023. https://arxiv.org/pdf/2205.15861.

[BCP97] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. *J. Symbolic Comput.*, 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993).

[BD13] Michael A. Bennett and Sander R. Dahmen. Klein forms and the generalized superelliptic equation. *Ann. of Math.* (2), 177(1):171–239, 2013.

[BD14] Christophe Breuil and Fred Diamond. Formes modulaires de Hilbert modulo $p$ et valeurs d'extensions entre caractères galoisiens. *Ann. Sci. Éc. Norm. Supér.* (4), 47(5):905–974, 2014.

[Bis22] Matthew Bisatt. Clusters, inertia, and root numbers. *Funct. Approx. Comment. Math.*, 66(2):203–243, 2022.
[BLR90] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud. *Néron models*, volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [*Results in Mathematics and Related Areas (3)*]. Springer-Verlag, Berlin, 1990.

[BVY04] Michael A. Bennett, Vinayak Vatsal, and Soroosh Yazdani. Ternary Diophantine equations of signature $(p, p, 3)$. *Compos. Math.*, 140(6):1399–1416, 2004.

[Bö13] Gebhard Böckle. Galois representations. In *Travaux mathématiques. Vol. XXIII*, volume 23 of *Trav. Math.*, pages 5–35. Fac. Sci. Technol. Commun. Univ. Luxemb., Luxembourg, 2013.

[Car86] Henri Carayol. Sur les représentations *l*-adiques associées aux formes modulaires de Hilbert. *Ann. Sci. École Norm. Sup. (4)*, 19(3):409–468, 1986.

[CK22] Imin Chen and Angelos Koutsianas. A modular approach to Fermat equations of signature $(p, p, 5)$ using Frey hyperelliptic curves, 2022. https://arxiv.org/abs/2210.02316.

[CK25] Imin Chen and Angelos Koutsianas. Darmon's program: A survey, 2025. available at https://www.imonchen.org/publications.

[Dar00] Henri Darmon. Rigid local systems, Hilbert modular forms, and Fermat's last theorem. *Duke Math. J.*, 102(3):413–449, 2000.

[DDMM23] Tim Dokchitser, Vladimir Dokchitser, Céline Maistret, and Adam Morgan. Arithmetic of hyperelliptic curves over local fields. *Math. Ann.*, 385(3-4):1213–1322, 2023.

[Del73] P. Deligne. Les constantes des équations fonctionnelles des fonctions *L*. In *Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972)*, volume Vol. 349 of Lecture Notes in Math., pages 501–597. Springer, Berlin-New York, 1973.

[DFV24] Lassina Dembélé, Nuno Freitas, and John Voight. On galois inertial types of elliptic curve over $\mathbb{Q}_p$, 2024. https://arxiv.org/abs/2203.07787.

[DG95] Henri Darmon and Andrew Granville. On the equations $z^m = F(x, y)$ and $Ax^p + By^q = Cz^r$. *Bull. London Math. Soc.*, 27(6):513–543, 1995.

[DM69] P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus. *Inst. Hautes Études Sci. Publ. Math.*, (36):75–109, 1969.

[DM00] Henri Darmon and Jean-François Mestre. Courbes hyperelliptiques à multiplications réelles et une construction de Shih. *Canad. Math. Bull.*, 43(3):304–311, 2000.

[DS21] Luis Dieulefait and Eduardo Soto. Solving $ax^p + by^p = cz^p$ with *abc* containing an arbitrary number of prime factors. *Mediterr. J. Math.*, 18(2):Paper No. 48, 24, 2021.

[Fon94] Jean-Marc Fontaine. Représentations *l*-adiques potentiellement semi-stables. Number 223, pages 321–347. 1994. *Périodes *p*-adiques* (Bures-sur-Yvette, 1988).

[Fre90] Eberhard Freitag. *Hilbert modular forms*. Springer-Verlag, Berlin, 1990.

[Gra03] Georges Gras. *Class field theory*. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. From theory to practice, Translated from the French manuscript by Henri Cohen.

[Kra97] Alain Kraus. Majorations effectives pour l'équation de Fermat généralisée. *Canad. J. Math.*, 49(6):1139–1161, 1997.

[KT17] Chandrashekhar B. Khare and Jack A. Thorne. Automorphy of some residually $S_5$ Galois representations. *Math. Z.*, 286(1-2):399–429, 2017.

[KW09a] Chandrashekhar Khare and Jean-Pierre Wintenberger. Serre's modularity conjecture. I. *Invent. Math.*, 178(3):485–504, 2009.

[KW09b] Chandrashekhar Khare and Jean-Pierre Wintenberger. Serre's modularity conjecture. II. *Invent. Math.*, 178(3):505–586, 2009.

[Lan80] Robert P. Langlands. *Base change for GL(2)*, volume No. 96 of *Annals of Mathematics Studies*. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1980.

[Lan02] Serge Lang. *Algebra*, volume 211 of *Graduate Texts in Mathematics*. Springer-Verlag, New York, third edition, 2002.

[Liu96] Qing Liu. Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète. *Trans. Amer. Math. Soc.*, 348(11):4577–4610, 1996.

[Liu02] Qing Liu. *Algebraic geometry and arithmetic curves*, volume 6 of *Oxford Graduate Texts in Mathematics*. Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné, Oxford Science Publications.

[Liu23] Qing Liu. Computing minimal Weierstrass equations of hyperelliptic curves. *Res. Number Theory*, 9(4):Paper No. 76, 22, 2023.
[LMF25] The LMFDB Collaboration. The L-functions and modular forms database. https://www.lmfdb.org, 2025. [Online; accessed 16 March 2025].

[Loc94] P. Lockhart. On the discriminant of a hyperelliptic curve. Trans. Amer. Math. Soc., 342(2):729-752, 1994.

[Mel22] Lukas Melninkas. On the root numbers of abelian varieties with real multiplication. Acta Arith., 203(2):137-163, 2022.

[Mil17] J. S. Milne. *Algebraic groups*, volume 170 of *Cambridge Studies in Advanced Mathematics*. Cambridge University Press, Cambridge, 2017. The theory of group schemes of finite type over a field.

[MP24] Franco Golfieri Madraga and Ariel Pacetti. Hypergeometric motives and the generalized fermat equation, 2024. https://arxiv.org/abs/2412.08804.

[Naj24] Filip Najman. On *r*-isogenies over $Q(ζ_r)$ of elliptic curves with rational *j*-invariants. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 118(3):Paper No. 134, 9, 2024.

[Neu99] Jürgen Neukirch. *Algebraic number theory*, volume 322 of *Grundlehren der mathematischen Wissenschaften* [*Fundamental Principles of Mathematical Sciences*]. Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder.

[Pac25] Ariel Pacetti. On transformation properties of hypergeometric motives and diophantine equations, 2025. https://arxiv.org/abs/2502.02776.

[Pan22] Lue Pan. The Fontaine-Mazur conjecture in the residually reducible case. J. Amer. Math. Soc., 35(4):1031-1169, 2022.

[Rib76] Kenneth A. Ribet. Galois action on division points of Abelian varieties with real multiplications. Amer. J. Math., 98(3):751-804, 1976.

[Roh94] David E. Rohrlich. Elliptic curves and the Weil-Deligne group. In *Elliptic curves and related topics*, volume 4 of CRM Proc. Lecture Notes, pages 125-157. Amer. Math. Soc., Providence, RI, 1994.

[Ser79] Jean-Pierre Serre. *Local fields*, volume 67 of *Graduate Texts in Mathematics*. Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg.

[Ser08] Jean-Pierre Serre. *Topics in Galois theory*, volume 1 of *Research Notes in Mathematics*. A K Peters, Ltd., Wellesley, MA, second edition, 2008. With notes by Henri Darmon.

[SGA72] *Groupes de monodromie en géométrie algébrique. I*, volume Vol. 288 of *Lecture Notes in Mathematics*. Springer-Verlag, Berlin-New York, 1972. Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7 I), Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim.

[Sil86] Joseph H. Silverman. *The arithmetic of elliptic curves*, volume 106 of *Graduate Texts in Mathematics*. Springer-Verlag, New York, 1986.

[Sil94] Joseph H. Silverman. *Advanced topics in the arithmetic of elliptic curves*, volume 151 of *Graduate Texts in Mathematics*. Springer-Verlag, New York, 1994.

[ST68] Jean-Pierre Serre and John Tate. Good reduction of abelian varieties. Ann. of Math. (2), 88:492-517, 1968.

[SW99] C. M. Skinner and A. J. Wiles. Residually reducible representations and modular forms. *Inst. Hautes Études Sci. Publ. Math.*, (89):5-126, 1999.

[Tat79] J. Tate. Number theoretic background. In *Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2*, volume XXXIII of Proc. Sympos. Pure Math., pages 3-26. Amer. Math. Soc., Providence, RI, 1979.

[Tay04] Richard Taylor. Galois representations. Ann. Fac. Sci. Toulouse Math. (6), 13(1):73-119, 2004.

[TTV91] Walter Tautz, Jaap Top, and Alain Verberkmoes. Explicit hyperelliptic curves with real multiplication and permutation polynomials. *Canad. J. Math.*, 43(5):1055-1064, 1991.

[Ulm16] Douglas Ulmer. Conductors of $\ell$-adic representations. Proc. Amer. Math. Soc., 144(6):2291-2299, 2016.

[WG24] Ashleigh Wilcox and Bogdan Grechuk. Generalised fermat equation: a survey of solved cases, 2024. https://arxiv.org/pdf/2412.11933.

[Wil95] Andrew Wiles. Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2), 141(3):443-551, 1995.

<footnote>MARTIN AZON (martin.azon@uca.fr) - LABORATOIRE DE MATHÉMATIQUES B. PASCAL, UNIVERSITÉ CLERMONT AUVERGNE, Campus des Cézeaux. 3 place Vasarely, TSA 60026 CS 60026, 63178 Aubière Cedex (France).</footnote>