**10.1. A Frey curve over a totally real quintic field.** The goal of this section is to prove the following result which settles (almost) Theorem **1.5** in the case $2 | a + b$.

**Theorem 10.1.** For every prime $p \ge 5$, there are no integer solutions $(a, b, c)$ to the equation

$$ (10.2) \qquad x^{11} + y^{11} = z^p $$

such that $abc \ne 0$, $\gcd(a, b, c) = 1$, and $2 | a + b$.

Let $K = \mathbb{Q}(\zeta_{11})^+$ and let $O_K$ denote its ring of integers. Write $\mathfrak{q}_2$ and $\mathfrak{q}_{11}$ for the unique primes in $K$ above 2 and 11, respectively.

Let $(a, b, c)$ be a non-trivial primitive solution to equation (10.2). In particular, $a$ and $b$ are non-zero and coprime.

We consider the Frey curve $F = F_{a,b} := E_{(a,b)}^{(1,2)}$ over $K$ as defined in [45, p. 619] and given by a model of the form

$$ (10.3) \qquad F_{a,b} : Y^2 = X(X - A_{a,b})(X + B_{a,b}), $$

where

$$ A_{a,b} = (\omega_2 - \omega_1)(a + b)^2 $$

$$ B_{a,b} = (2 - \omega_2)(a^2 + \omega_1 ab + b^2) $$

and $\omega_j = \zeta_{11}^j + \zeta_{11}^{-j}$. The standard invariants of model (10.3) are

$$ c_4(F_{a,b}) = 2^4(A_{a,b}^2 + A_{a,b}B_{a,b} + B_{a,b}^2), $$

$$ c_6(F_{a,b}) = 2^5(2A_{a,b}^3 + 3A_{a,b}^2B_{a,b} - 3A_{a,b}B_{a,b}^2 - 2B_{a,b}^3), $$

$$ \Delta(F_{a,b}) = 2^4 (A_{a,b}B_{a,b}C_{a,b})^2, $$

where

$$ C_{a,b} := -(A_{a,b} + B_{a,b}) = (\omega_1 - 2)(a^2 + \omega_2 ab + b^2). $$

Note that $F_{a,b}$ has full 2-torsion defined over $K$.

Write $F = F_{a,b}$ and let

$$ \delta = \begin{cases} -11 & \text{if } 11 + a + b \\ 1 & \text{if } 11 | a + b. \end{cases} $$

We write $F^{(\delta)}$ for the quadratic twist of $F$ by $\delta$ given by the model

$$ y^2 = x(x - \delta A_{a,b})(x + \delta B_{a,b}) $$

with standard invariants

$$ c_4(F^{(\delta)}) = \delta^2 c_4(F), \quad c_6(F^{(\delta)}) = \delta^3 c_6(F), \quad \Delta(F^{(\delta)}) = \delta^6 \Delta(F). $$

For a prime ideal $\mathfrak{q}$ in $K$ and an ideal $\mathfrak{a}$, we denote by $v_{\mathfrak{q}}(\mathfrak{a})$ the valuation at $\mathfrak{q}$ of $\mathfrak{a}$. For an element $\alpha \in O_K \setminus \{0\}$, we write $v_{\mathfrak{q}}(\alpha)$ for $v_{\mathfrak{q}}(\alpha O_K)$.

Let $N(\bar{\rho}_{F^{(\delta)},p})$ be the Serre level of the mod $p$ representation $\bar{\rho}_{F^{(\delta)},p}$ associated with $F^{(\delta)}$ (i.e., the prime-to-$p$ part of its Artin conductor).

**Proposition 10.4.** Suppose $2 | a + b$. Then we have that $N(\bar{\rho}_{F^{(\delta)},p}) = \mathfrak{q}_2\mathfrak{q}_{11}^t$ with

$$ t = \begin{cases} 0 & \text{if } 11 + a + b, \\ 1 & \text{if } 11 | a + b. \end{cases} $$
*Proof.* Recall that $\omega_2 - \omega_1$, $2 - \omega_2$ and $\omega_1 - 2$ all generate the unique prime ideal $\mathfrak{q}_{11}$ above 11 in $K$. Let us show that $A = A_{a,b}$, $B = B_{a,b}$, and $C = C_{a,b}$ are pairwise coprime away from $\mathfrak{q}_{11}$.

Let $\mathfrak{q} \neq \mathfrak{q}_{11}$ be a prime ideal in $K$ above a rational prime $\mathfrak{q}$. Assume that $\mathfrak{q}$ divides $A$. Then, we have $a \equiv -b \pmod{\mathfrak{q}}$ and hence

$$a^2 + \omega_1 ab + b^2 \equiv a^2(2 - \omega_1) \not\equiv 0 \pmod{\mathfrak{q}}, \quad a^2 + \omega_2 ab + b^2 \equiv a^2(2 - \omega_2) \not\equiv 0 \pmod{\mathfrak{q}}$$

Therefore, $\mathfrak{q}$ divides neither $B$ nor $C$.

Assume now that $\mathfrak{q}$ divides $B$. Then, we have $a^2 + b^2 \equiv -\omega_1 ab \pmod{\mathfrak{q}}$ and hence

$$(a+b)^2 \equiv (2-\omega_1)ab \not\equiv 0 \pmod{\mathfrak{q}}, \quad a^2 + \omega_2 ab + b^2 \equiv (2-\omega_2)ab \not\equiv 0 \pmod{\mathfrak{q}}.$$

Therefore, $\mathfrak{q}$ divides neither $A$ nor $C$.

From the formulas above, it follows that $F^{(\delta)}$ is semistable away from $\mathfrak{q}_2$ and $\mathfrak{q}_{11}$ with bad multiplicative reduction precisely at the prime ideals dividing $ABC$.

Moreover, for a prime ideal $\mathfrak{q} + \mathfrak{q}_2\mathfrak{q}_{11}$, we have $v_{\mathfrak{q}}(\Delta(F^{(\delta)})) = v_{\mathfrak{q}}(\Delta(F)) \equiv 0 \pmod{p}$ by the assumption that $(a, b, c)$ is a solution to (1.6), so the multiplicative primes $\mathfrak{q} \mid \Delta(F^{(\delta)})$ such that $\mathfrak{q} + \mathfrak{q}_2\mathfrak{q}_{11}$ do not divide the Serre level $N(\bar{\rho}_{F^{(\delta)},p})$.

Let us now consider the reduction of $F^{(\delta)}$ at $\mathfrak{q}_{11}$. Assume first that $11 + a + b$. Then, we have $v_{\mathfrak{q}_{11}}(A) = v_{\mathfrak{q}_{11}}(\omega_2 - \omega_1) = 1$. Moreover, we have

$$a^2 + \omega_1 ab + b^2 \equiv a^2 + 2ab + b^2 \equiv (a+b)^2 \not\equiv 0 \pmod{\mathfrak{q}_{11}}.$$

Therefore, we have $v_{\mathfrak{q}_{11}}(B) = 1$ and, similarly, $v_{\mathfrak{q}_{11}}(C) = 1$. It follows that $v_{\mathfrak{q}_{11}}(\Delta(F)) = 6$ and hence $v_{\mathfrak{q}_{11}}(\Delta(F^{(\delta)})) = 12$ since $\delta = -11$ in this case. In particular, a change of variables shows that the curve $F^{(\delta)}$ has good reduction at $\mathfrak{q}_{11}$.

Assume now that $11$ divides $a + b$. Since, $11 \sim \mathfrak{q}_{11}^5$ (the prime 11 is totally ramified in $K$), we have that $v_{\mathfrak{q}_{11}}(A) \geq 11$. Moreover, we have $(a+b)^2 \equiv 0 \pmod{11^2}$ and in particular, it follows that $a^2 + b^2 \equiv -2ab \pmod{\mathfrak{q}_{11}^2}$. We get that

$$B = (2 - \omega_2)(a^2 + \omega_1 ab + b^2) \equiv ab(2 - \omega_2)(\omega_1 - 2) \pmod{\mathfrak{q}_{11}^3}.$$

Therefore, we have $v_{\mathfrak{q}_{11}}(B) = 2$ and, similarly, $v_{\mathfrak{q}_{11}}(C) = 2$. Since $\delta = 1$ in this case, we have

$$v_{\mathfrak{q}_{11}}(c_4(F^{(\delta)})) = v_{\mathfrak{q}_{11}}(c_4(F)) = 4, \quad v_{\mathfrak{q}_{11}}(\Delta(F^{(\delta)})) = v_{\mathfrak{q}_{11}}(\Delta(F)) = 8 + v_{\mathfrak{q}_{11}}(A) \geq 19.$$

It follows that (after a change of variables) the curve $F^{(\delta)}$ has bad multiplicative reduction at $\mathfrak{q}_{11}$.

It remains to deal with the prime ideal $\mathfrak{q}_2$. We have $2 \mid a + b$ by assumption. It then follows that $8 \mid a + b$ and $2 \mid ab$ as $(a, b, c)$ is a solution to (1.6). We have $v_{\mathfrak{q}_2}(A) = 2v_2(a+b) \geq 6$ since 2 is inert in $K$. Moreover, from $a^2 + b^2 \equiv -2ab \pmod{64}$ we get

$$a^2 + \omega_1 ab + b^2 \equiv ab(\omega_1 - 2) \not\equiv 0 \pmod{\mathfrak{q}_2}.$$

Therefore, we have $v_{\mathfrak{q}_2}(B) = 0$ and, similarly, $v_{\mathfrak{q}_2}(C) = 0$. Since $\delta$ is coprime to 2, we have

$$(v_{\mathfrak{q}_2}(c_4(F^{(\delta)})), v_{\mathfrak{q}_2}(c_6(F^{(\delta)})), v_{\mathfrak{q}_2}(\Delta(F^{(\delta)}))) = (4, 6, 4 + 2v_2(A)) = (4, 6, \geq 16).$$

According to [70, Tableau IV], we are either in case 7 in Tate's classification or the equation defining $F^{(\delta)}$ is not minimal at $\mathfrak{q}_2$. We use Proposition 4 in loc. cit. to distinguish between these two possible cases. Write $a_{2,\delta}$ and $b_{8,\delta}$ for the $a_2$- and $b_8$-coefficients of the Weierstrass model of $F^{(\delta)}$, respectively. We have $b_{8,\delta} = -\delta^4(AB)^2$ and hence $b_{8,\delta} \equiv 0 \pmod{\mathfrak{q}_2^5}$. According to this latter proposition, we are in a non-minimal case if and only if $a_{2,\delta} = \delta(B-A)$ is a square modulo $\mathfrak{q}_2^2$.
We have $A \equiv 0 \pmod{q_2^2}$ and

$$
\begin{align*}
B &= (2 - \omega_2)(a^2 + \omega_1 ab + b^2) \\
  &= (4 - \omega_1^2)((a + b)^2 + (\omega_1 - 2)ab) \\
  &= (4 - \omega_1^2)((a + b)^2 + (\omega_6^2 - 4)ab) \\
  &\equiv a^2 \omega_1^2 \omega_6^2 \pmod{q_2^2}.
\end{align*}
$$

Since $\delta \in \{-11, 1\}$ is a square modulo 4, we get that $a_{2,\delta}$ is a square modulo $q_2^2$ and hence we are in the non-minimal case. It follows that the conductor $N_{F^{(\delta)}}$ of $F^{(\delta)}$ is

$$
N_{F^{(\delta)}} = q_2 q_{11}^t \prod_{\substack{q | \Delta(F^{(\delta)}) \\ q + q_2 q_{11}}} q,
$$

where the product runs over the prime ideals in $K$ dividing $\Delta(F^{(\delta)})$ and coprime to $q_2q_{11}$, and $t$ is as in the statement.

Finally, we have that $v_q(N(\bar{\rho}_{F^{(\delta)},p})) = v_q(N(\rho_{F^{(\delta)},p})) = v_q(N_{F^{(\delta)}})$ for $\mathfrak{q} = \mathfrak{q}_2$ and $\mathfrak{q} = \mathfrak{q}_{11}$ since the valuation $v_q(\Delta(F^{(\delta)})) \neq 0 \pmod{p}$ for $\mathfrak{q} = \mathfrak{q}_2$ and for $\mathfrak{q} = \mathfrak{q}_{11}$ when $t=1$. $\square$

Let $S_2(N(\bar{\rho}_{F^{(\delta)},p}))$ denote the space of Hilbert cuspforms of level $N(\bar{\rho}_{F^{(\delta)},p})$, parallel weight 2 and trivial character. The curve $F$ is modular by [45, Corollary 6.4] and hence so is $F^{(\delta)}$.

**Proposition 10.5.** *Suppose* $2 \mid a+b$. *The representation* $\bar{\rho}_{F^{(\delta)},p}$ *is irreducible for* $p \neq 2,3$.

*Proof.* Suppose that $\bar{\rho}_{F^{(\delta)},p}$ is reducible, that is,

$$
\bar{\rho}_{F^{(\delta)},p} \sim \begin{pmatrix} \theta & \star \\ 0 & \theta' \end{pmatrix} \quad \text{with } \theta, \theta' : G_K \to \mathbb{F}_p^* \quad \text{satisfying } \theta\theta' = \chi_p.
$$

If $\mathfrak{q} \nmid p$ is a multiplicative prime of $F$ for which $\theta$ ramifies, then $\theta'$ also ramifies at $\mathfrak{q}$ and the conductor exponent at $\mathfrak{q}$ of $\bar{\rho}_{F,p}$ is $\ge 2$, contradicting the fact that the conductor exponent of $F$ at $\mathfrak{q}$ is 1. Therefore, the characters $\theta$ and $\theta'$ ramify only at $p$ and additive primes of $F^{(\delta)}$. We see from Proposition 10.4 that there are no such additive primes, hence $\theta$ and $\theta'$ are unramified away from $p$.

Let us first assume $p \ge 17$. The unit group of $K$ is generated by $\{-1, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4\}$ where

$$
\epsilon_1 = \omega_1^4 - 3\omega_1^2 + 1, \quad \epsilon_2 = \omega_1^2 + \omega_1 - 1, \quad \epsilon_3 = \omega_1, \quad \epsilon_4 = -\omega_1^4 - \omega_1^3 + 3\omega_1^2 + 3\omega_1.
$$

According to Proposition 10.4 and the theory of Tate curves, for every prime ideal $\mathfrak{p}$ above $p$ in $K$, there exists an integer $r_{\mathfrak{p}} \in \{0, 1\}$ such that $\theta|_{I_{\mathfrak{p}}} = (\chi_{\mathfrak{p}}|_{I_{\mathfrak{p}}})^{r_{\mathfrak{p}}}$. Write $s_{\mathfrak{p}} = 12r_{\mathfrak{p}}$. Let $G = \text{Gal}(K/\mathbb{Q})$ be the Galois group of $K$ over $\mathbb{Q}$. Then $G$ acts transitively on $\{\mathfrak{p} \mid p\}$. Fix $\mathfrak{p}_0 \mid p$. For each $\tau \in G$, write $s_{\tau}$ for the number $s_{\mathfrak{p}}$ associated to the ideal $\mathfrak{p} := \tau^{-1}(\mathfrak{p}_0)$ as above. We refer to $\mathfrak{s} := (s_{\tau})_{\tau \in G}$ as the isogeny signature of $F$ at $p$. For an element $\alpha$ of $K$ we define the twisted norm associated to $\mathfrak{s}$ by

$$
\mathcal{N}_{\mathfrak{s}}(\alpha) = \prod_{\tau \in G} \tau(\alpha)^{s_{\tau}}.
$$

According to [47, Corollary 3.2], we have

$$
p \mid A_s := \operatorname{Norm} \left( \operatorname{gcd} \left( (\mathcal{N}_s(\epsilon_1) - 1) \mathcal{O}_K \right) \right)
$$

and hence, in the notation of [47, Theorem 1], $p$ divides the integer $B$ which is defined as the least common multiple of the $A_s$ taken over all signatures $\mathfrak{s} \neq (0)_{\tau \in G}, (12)_{\tau \in G}$. Here, we compute $B=1$.
We conclude that we have $\mathfrak{s} = (0)_{\tau \in G}$ or $(12)_{\tau \in G}$. Therefore, we have either $r_{\mathfrak{p}} = 0$ for every $\mathfrak{p} \mid p$, or $r_{\mathfrak{p}} = 1$ for every $\mathfrak{p} \mid p$. Since $\theta\theta' = \chi_{\mathfrak{p}}$, this means that exactly one of $\theta, \theta'$ ramifies at $p$. So, replacing $F^{(\delta)}$ by a $p$-isogenous curve if necessary, we can assume $\theta$ is unramified at $p$. Thus $\theta$ is unramified at all finite primes. Since $K$ has narrow class number 1 it follows that $\theta = 1$.

We conclude that $F^{(\delta)}$ has a $p$-torsion point defined over $K$. From [34, Theorem 1.2], we see that this is impossible for $p > 19$, concluding the proof in this case. Note that $F^{(\delta)}$ has full 2-torsion over $K$ so, for $p = 5, 7, 11, 17, 19$, the curve $F^{(\delta)}$ gives rise to points on the modular curves

$$X = X_0(20), X_0(14), X_0(11), X_0(17), X_0(19),$$

respectively, which are all elliptic curves with finitely many $K$-rational points. Using an explicit map to the $j$-line, we deduce that all $K$-rational points are either cuspidal or have $j$-invariant in the following list:

$$N = 14 : j = 16581375, -3375$$

$$N = 11 : j = -32768, -121, -24729001$$

$$N = 17 : j = -882216989/131072, -297756989/2$$

$$N = 19 : j = -884736.$$

We then check that there are no $a, b \in \mathbb{Z}$ such that the $j$-invariant of $F = F_{a,b}$ lies in the above list.

For $p = 13$, we need to rule out non-cuspidal $K$-rational points on $X_0(26)$. The quotient $X_0(26)/\langle w_2 \rangle$ is isomorphic to an elliptic curve $E$ such that $E(K) = E(\mathbb{Q}) \simeq \mathbb{Z}/3\mathbb{Z}$. Using an explicit map $\psi : X_0(26) \to E$, we deduce that the $K$-rational points of $X_0(26)$ are contained in the inverse image $\psi^{-1}(E(K))$. The $K$-rational points of the inverse image consist exactly of the four cusps of $X_0(26)$. $\square$

*Remark 10.6.* If $11 \mid a + b$, then $\delta = 1$ and the proposition gives the irreducibility of $\bar{\rho}_{F,p}$. Else, if $11 \nmid a + b$, then $\bar{\rho}_{F,p}$ is also irreducible as a twist of $\bar{\rho}_{F(-11),p}$. Therefore, Proposition 10.5 makes [45, Theorem 7.1] explicit and sharp in the case $r = 11$.

We now prove Theorem 10.1. Suppose $2 \mid a+b$. For $p \ge 5$, an application of level lowering theorems for Hilbert modular forms (see [48, 52, 71]), together with the previous result, implies that there is a Hilbert newform $f \in S_2(N(\bar{\rho}_{F(\delta),p}))$ such that for a prime $\mathfrak{p} \mid p$ in $\overline{\mathbb{Q}}$ we have

$$(10.7) \qquad \bar{\rho}_{F(\delta),p} \simeq \bar{\rho}_{f,p}.$$

We have $N(\bar{\rho}_{F(\delta),p}) = \mathfrak{q}_2$ or $\mathfrak{q}_2\mathfrak{q}_{11}$ depending on whether $11 \mid a+b$ or $11 \mid a+b$ by Proposition 10.4.

There is a single Hecke constituent (see Subsection 9.1 for background and notation on Hecke constituents) at level $\mathfrak{q}_2$ and two at level $\mathfrak{q}_2\mathfrak{q}_{11}$. We write $f_1$ for one of the forms belonging to the unique Hecke constituent at level $\mathfrak{q}_2$. Using the auxiliary primes $q = 3, 23$ we discard all the newforms in both levels for all prime exponents $p$ except for the form $f_1$ when $p \in \{5, 31\}$. The field of coefficients of $f_1$ is $\mathbb{Q}_{f_1} = \mathbb{Q}(\sqrt{5})$ in which we have the prime ideal factorizations $(5) = \mathfrak{p}_5^2$ and $(31) = \mathfrak{p}_{31}\mathfrak{p}_{31}'$. Applying refined elimination for $p = 31$ with the auxiliary prime $q = 23$ allows to discard also one of the prime ideals above 31, say $\mathfrak{p}_{31}'$. (We note this calculation in particular shows that $\bar{\rho}_{f_1,p_{31}'}$ is irreducible.) Hence, it remains possible that

$$(10.8) \qquad \bar{\rho}_{F,5} \simeq \bar{\rho}_{f_1,p_5} \quad \text{or} \quad \bar{\rho}_{F,31} \simeq \bar{\rho}_{f_1,p_{31}}.$$

We claim that the representations on the right hand side of the previous isomorphisms are reducible. To prove this claim we apply [65, Theorem 2.1] with $r = 1, d = 5$, $\mathfrak{N} = \mathfrak{N}_1\mathfrak{N}_2$ with $\mathfrak{N}_1 = \mathfrak{q}_2$ and $\mathfrak{N}_2 = 1$. To compute the quantity $m(\mathcal{O})$ we use formula (1.6) in loc. cit. where the value of $\zeta_K(-1)$ is obtained using Magma. We find that $m(\mathcal{O}) = 155/132$ and hence that $p = 5$ and $p = 31$ divide
the numerator of $m(O)$. Therefore, for all primes $\mathfrak{p} \mid p$ in $\mathbb{Q}_{f_1}$, there is at least one newform in the space whose associated mod $\mathfrak{p}$ representation is reducible. The only newforms in the space are $f_1$ and $\sigma f_1$ (obtained by conjugating the Fourier coefficients by the unique non-trivial element $\sigma \in \text{Gal}(\mathbb{Q}_{f_1}/\mathbb{Q})$). Since $\sigma(\mathfrak{p}_{31}) = \mathfrak{p}'_{31}$, it follows from the prime factorizations of 5 and 31 that $\bar{\rho}_{f_1, \mathfrak{p}_5}$, $\bar{\rho}_{\sigma f_1, \mathfrak{p}_5}$, $\bar{\rho}_{f_1, \mathfrak{p}_{31}}$, and $\bar{\rho}_{\sigma f_1, \sigma(\mathfrak{p}_{31})}$ are reducible.

Therefore the isomorphisms (10.8) do not hold as the representations attached to the Frey curve are irreducible by Proposition 10.5. This eliminates the remaining exponents, completing the proof of Theorem 10.1.

**Remark 10.9.** In principle, the Frey curve $F$ and its twists can also be used to approach the remaining unobstructed congruence class $2 \mid a+b$, $11 \mid a+b$, $11 \mid ab$. Further study would show that this approach requires the computation of Hilbert newforms over $K$ of level $\mathfrak{q}_2^3$. Although this space only has dimension 1,201, a full computation is currently out of reach. The reason is that this level requires the indefinite quaternion algebra algorithm [33], which is slow even for moderate dimensions.

**10.2. Proof of Theorem 1.5.** For $p = 2, 3$, the result follows from [31]. Therefore assume $p \ge 5$.

We will apply a multi-Frey approach which we outline below.

The case $2 \mid a+b$ is proven in Theorem 10.1 using the curve $F$. Therefore, assume that $2 \mid a+b$ and $11 \mid a+b$. For $p = 11$, this is a special case of Fermat's Last Theorem (which is proved using the classical Hellegouarch-Frey curve, different from those in this paper). Hence, we may further suppose that $p \ne 11$.

At the expense of switching $a$ and $b$, and negating both $a$ and $b$, we may assume that $a \equiv 0 \pmod 2$ and $b \equiv 1 \pmod 4$. Let $C = C_{11}(a, b)$ be the hyperelliptic Frey curve constructed in Section 3. To ease notation, we write $J = \text{Jac}(C)/K$ with $K = \mathbb{Q}(\zeta_{11})^+$.

By Theorem 6.1, $\bar{\rho}_{J,p}$ is irreducible and from Corollary 8.7, we have that

$$ (10.10) \qquad \bar{\rho}_{J,p} \otimes \chi_{11}|_{G_K} \simeq \bar{\rho}_{g,\mathfrak{P}} $$

for some Hilbert newform $g$ of parallel weight 2, trivial character, and level $\mathfrak{q}_2^2\mathfrak{q}_{11}$ where $\mathfrak{p} \mid p$ is a prime of $K$ and $\mathfrak{P} \mid p$ is a prime of the field of coefficients $K_g$ of $g$. Moreover, we have $K \subset K_g$.

There are 14 Hilbert newforms at level $\mathfrak{q}_2^2\mathfrak{q}_{11}$, but only 6 of these have coefficient field containing $K$. The indices of these newforms [8] are

$$ \{7, 8, 9, 10, 11, 12\}, $$

and the respective degrees of their coefficient fields are

$$ \{5, 10, 25, 25, 50, 60\}. $$

It takes slightly less than 4 hours (see [8] for the detail on the machine used) to compute the space of Hilbert newforms and around 20 minutes to perform the elimination steps described in Subsection 9.10 using the auxiliary primes $q = 3, 5, 7, 23$. We conclude that the exponent $p$ lies in $\{2, 3, 11\}$, which gives a contradiction.

## 11. REDUCTION TO CM FORMS IN DARMON'S PROGRAM

In this section, we prove Theorem 1.8 and Corollary 1.9. Let $(a, b, c)$ be integers such that

$$ a^5 + b^5 = c^p, \quad abc \neq 0, \quad \text{gcd}(a, b, c) = 1 $$

where $p$ is a prime, $p \notin \{2, 3, 5\}$.
From [10, Theorem 4], we have that $2 + a + b$ and $5 + a + b$. So up to switching the roles of $a$ and $b$ and/or negating both $a$ and $b$, we may assume that $a \equiv 0 \pmod 2$ and $b \equiv 1 \pmod 4$.

Here $K$ denotes the quadratic field $\mathbb{Q}(\zeta_5)^+ = \mathbb{Q}(\sqrt{5})$. Write $\mathfrak{q}_2$ and $\mathfrak{q}_5$ for the unique prime ideals above 2 and 5 in $K$ respectively.

Let $\mathfrak{p}$ be a prime ideal above $p$ in $K$. From Proposition 6.1 and Theorem 8.3 we have

$$ \bar{\rho}_{J_5(a,b),\mathfrak{p}} \simeq \bar{\rho}_{g,\mathfrak{p}}, $$

where $g$ is in the new subspace $S$ of Hilbert newforms of level $\mathfrak{q}_2^2\mathfrak{q}_5^2$, parallel weight 2, and trivial character. Moreover, from conclusion (iv) of that theorem we have $K \subset K_g$ and $K_g$ is the coefficient field of $g$.

The space $S$ has dimension 3 and decomposes into 2 Hecke constituents. The only Hilbert newform with field of coefficients $K$ corresponds to $J_5(0,1)$.

Finally, because switching the roles of $a$ and $b$ and/or negating both $a$ and $b$ changes $\bar{\rho}_{J_5(0,1),\mathfrak{p}}$ by a character of order dividing 2, we obtain the conclusion in the form stated.

We finally prove Corollary 1.9. The variety $J_5(a,b)$ satisfies the hypotheses of Conjecture 1.3. Thus for large enough $p$, the representation $\bar{\rho}_{J_5(a,b),\mathfrak{p}}$ has image not contained in the normalizer of a Cartan subgroup. This contradicts the isomorphism $\bar{\rho}_{J_5(a,b),\mathfrak{p}} \simeq \bar{\rho}_{J_5(0,1),\mathfrak{p}} \otimes \chi$ in Theorem 1.8 because $\bar{\rho}_{J_5(0,1),\mathfrak{p}}$ is contained in the normalizer of a Cartan since $J_5(0,1)$ has CM.

## REFERENCES

[1] Groupes de monodromie en géométrie algébrique. I. Lecture Notes in Mathematics, Vol. 288. 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. 5.2, c

[2] Martin Azon, Mar Curcó-Iranzo, Maleeha Khawaja, Céline Maistret, and Diana Mocanu. Conductor exponents for families of hyperelliptic curves. ArXiv preprint, https://arxiv.org/abs/2410.21134, 2024. 1.4

[3] Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor. Potential automorphy and change of weight. Ann. of Math. (2), 179(2):501-609, 2014. 2

[4] Michael Bennett, Preda Mihăilescu, and Samir Siksek. The Generalized Fermat Equation. In Open problems in mathematics, pages 173-205. Springer, [Cham], 2016. 1.1

[5] Michael A. Bennett and Sander R. Dahmen. Klein forms and the generalized superelliptic equation. Ann. of Math. (2), 177(1):171-239, 2013. 3.8

[6] Nicolas Billerey. Formes homogènes de degré 3 et puissances $p$-ièmes. J. Number Theory, 128(5):1272-1294, 2008. 3.8

[7] Nicolas Billerey, Imin Chen, Lassina Dembélé, Luis Dieulefait, and Nuno Freitas. Some extensions of the modular method and Fermat equations of signature $(13,13,n)$. Publ. Mat., 67(2):715-741, 2023. 5.17, 9.6

[8] Nicolas Billerey, Imin Chen, Luis Dieulefait, and Nuno Freitas. Supporting files for this paper, https://github.com/NicolasBillerey/xhyper. 1.6, 10.2, 57

[9] Nicolas Billerey, Imin Chen, Luis Dieulefait, and Nuno Freitas. A result on the equation $x^p + y^p = z^r$ using Frey abelian varieties. Proc. Amer. Math. Soc., 145(10):4111-4117, 2017. 6.1

[10] Nicolas Billerey, Imin Chen, Luis Dieulefait, and Nuno Freitas. A multi-Frey approach to Fermat equations of signature $(r,r,p)$. Trans. Amer. Math. Soc., 371(12):8651-8677, 2019. 1.5, 5.17, 9.11, 11

[11] Nicolas Billerey, Imin Chen, Luis Dieulefait, and Nuno Freitas. On Darmon's program for the generalized Fermat equation, II. Math. Comp., 94(354):1977-2003, 2025. 1, 1.5, 3, 6.1, 6.1, 9

[12] Nicolas Billerey, Imin Chen, Luis Dieulefait, and Nuno Freitas. On Darmon's program for the generalized Fermat equation, I. Journal für die reine und angewandte Mathematik (Crelles Journal), 2025. 1

[13] Yuri Bilu and Pierre Parent. Serre's uniformity problem in the split Cartan case. Ann. of Math. (2), 173(1):569-584, 2011. 4.7

[14] 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. 5

[15] Richard Brauer and Cecil Nesbitt. On the modular characters of groups. Ann. of Math. (2), 42:556-590, 1941. 2.2
[16] 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. 1.5, 8, 8.2

[17] Henri Carayol. Sur les représentations *l*-adiques associées aux formes modulaires de Hilbert. *Ann. Sci. École Norm. Sup.* (4), 19(3):409-468, 1986. 2

[18] Imin Chen and Angelos Koutsianas. A modular approach to Fermat Equations of signature (*p*,*p*,5) using Frey hyperelliptic curves. ArXiv preprint, https://arxiv.org/abs/2210.02316, 2025. 1.4

[19] Imin Chen and Samir Siksek. Perfect powers expressible as sums of two cubes. *J. Algebra*, 322(3):638-656, 2009. 1.7, 3

[20] Wen-Chen Chi. *l*-adic and *λ*-adic representations associated to abelian varieties defined over number fields. *Amer. J. Math.*, 114(2):315-353, 1992. 2.6

[21] Henri Cohen. *Advanced topics in computational number theory*, volume 193 of *Graduate Texts in Mathematics*. Springer-Verlag, New York, 2000. 6.1, 6.1

[22] Brian Conrad, Fred Diamond, and Richard Taylor. Modularity of certain potentially Barsotti-Tate Galois representations. *J. Amer. Math. Soc.*, 12(2):521-567, 1999. 4.1

[23] Charles W. Curtis and Irving Reiner. *Representation theory of finite groups and associative algebras*. AMS Chelsea Publishing, Providence, RI, 2006. Reprint of the 1962 original. 2.2

[24] Sander R. Dahmen. *Classical and modular methods applied to Diophantine equations*. PhD thesis, University of Utrecht, 2008. 1.5

[25] Henri Darmon. The equation $x^4 - y^4 = z^p$. *C. R. Math. Rep. Acad. Sci. Canada*, 15(6):286-290, 1993. 1.2

[26] Henri Darmon. The equations $x^n + y^n = z^2$ and $x^n + y^n = z^3$. *Internat. Math. Res. Notices*, (10):263-274, 1993. 1.2

[27] Henri Darmon. Faltings plus epsilon, Wiles plus epsilon, and the generalized Fermat equation. *C. R. Math. Rep. Acad. Sci. Canada*, 19(1):3-14, 1997. 1.3, 2.1

[28] Henri Darmon. Modularity of fibres in rigid local systems. *Ann. of Math. (2)*, 149(3):1079-1086, 1999. 4.2

[29] Henri Darmon. Rigid local systems, Hilbert modular forms, and Fermat's Last Theorem. *Duke Math. J.*, 102(3):413-449, 2000. 1.3, 1.4, 1.4, 1.4, 1.8, 2, 2.1, 2.1, 2.4, 2.2, 2.4, 2.4, 2.44, 4, 4.1, 4.1, 4.2, 4.2, 4.12, 7

[30] 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. 1.1

[31] Henri Darmon and Loïc Merel. Winding quotients and some variants of Fermat's Last Theorem. *J. Reine Angew. Math.*, 490:81-100, 1997. 10.2

[32] Pierre Deligne and David Mumford. The irreducibility of the space of curves of given genus. *Inst. Hautes Études Sci. Publ. Math.*, (36):75-109, 1969. 2.3

[33] Lassina Dembélé and John Voight. Explicit methods for Hilbert modular forms. In *Elliptic curves, Hilbert modular forms and Galois deformations*, Adv. Courses Math. CRM Barcelona, pages 135-198. Birkhäuser/Springer, Basel, 2013. 10.9

[34] Maarten Derickx, Sheldon Kamienny, William Stein, and Michael Stoll. Torsion points on elliptic curves over number fields of small degree. *Algebra Number Theory*, 17(2):267-308, 2023. 10.1

[35] Fred Diamond and Jerry Shurman. *A first course in modular forms*, volume 228 of *Graduate Texts in Mathematics*. Springer-Verlag, New York, 2005. 2.4

[36] Luis Dieulefait and Nuno Freitas. Fermat-type equations of signature (13,13,*p*) via Hilbert cuspforms. *Math. Ann.*, 357(3):987-1004, 2013. 1.2, 1.5

[37] Luis Dieulefait and Nuno Freitas. The Fermat-type equations $x^5 + y^5 = 2z^p$ or $3z^p$ solved through $\mathbb{Q}$-curves. *Math. Comp.*, 83(286):917-933, 2014. 1.5

[38] Luis Dieulefait, Ariel Pacetti, and Panagiotis Tsaknias. On the number of Galois orbits of newforms. *J. Eur. Math. Soc. (JEMS)*, 23(8):2833-2860, 2021. 5.1

[39] Luis Dieulefait and Ariel Martín Pacetti. A simplified proof of Serre's conjecture. *Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM*, 117(4):Paper No. 153, 17, 2023. 8.2

[40] Tim Dokchitser, Vladimir Dokchitser, Céline Maistret, and Adam Morgan. Semistable types of hyperelliptic curves. In *Algebraic curves and their applications*, volume 724 of *Contemp. Math.*, pages 73-135. Amer. Math. Soc., [Providence], RI, [2019] ©2019. 5

[41] Tim Dokchitser, Vladimir Dokchitser, Céline Maistret, and Adam Morgan. Arithmetic of hyperelliptic curves over local fields. *Math. Ann.*, 385(3-4):1213-1322, 2023. 5

[42] Bas Edixhoven. The weight in Serre's conjectures on modular forms. *Invent. Math.*, 109(3):563-594, 1992. 7.1

[43] Jordan S. Ellenberg. Finite flatness of torsion subschemes of Hilbert-Blumenthal abelian varieties. *J. Reine Angew. Math.*, 532:1-32, 2001. 1.4, 7
[44] Walter Feit. *The representation theory of finite groups*, volume 25 of *North-Holland Mathematical Library*. North-Holland Publishing Co., Amsterdam-New York, 1982. 6.2

[45] Nuno Freitas. Recipes to Fermat-type equations of the form $x^r + y^r = Cz^p$. *Math. Z.*, 279(3-4):605–639, 2015. 1.3, 1.5, 2.1, 3.8, 10.1, 10.1, 10.6

[46] Nuno Freitas. On the Fermat-type equation $x^3 + y^3 = z^p$. *Comment. Math. Helv.*, 91(2):295–304, 2016. 1.7, 3

[47] Nuno Freitas and Samir Siksek. Criteria for irreducibility of mod $p$ representations of Frey curves. *J. Théor. Nombres Bordeaux*, 27(1):67–76, 2015. 10.1

[48] Kazuhiro Fujiwara. Level optimization in the totally real case. *ArXiv preprint*, https://arxiv.org/abs/math/0602586, 2006. 8, 10.1

[49] Stephen Gelbart. Three lectures on the modularity of $\bar{\rho}_{E,3}$ and the Langlands reciprocity conjecture. In *Modular forms and Fermat's last theorem (Boston, MA, 1995)*, pages 155–207. Springer, New York, 1997. 9.6

[50] David Geraghty. Modularity lifting theorems for ordinary Galois representations. *Math. Ann.*, 373(3-4):1341–1427, 2019. 6.1

[51] Xavier Guitart. Abelian varieties with many endomorphisms and their absolutely simple factors. *Rev. Mat. Iberoam.*, 28(2):591–601, 2012. 2.6

[52] Frazer Jarvis. Correspondences on Shimura curves and Mazur's principle at *p*. *Pacific J. Math.*, 213(2):267–280, 2004. 8, 10.1

[53] Monsur A. Kenku. On the number of $\mathbb{Q}$-isomorphism classes of elliptic curves in each $\mathbb{Q}$-isogeny class. *J. Number Theory*, 15(2):199–202, 1982. 4.7

[54] Chandrashekhar Khare and Jack A. Thorne. Automorphy of some residually $S_5$ Galois representations. *Math. Z.*, 286(1-2):399–429, 2017. 4.1

[55] Chandrashekhar Khare and Jean-Pierre Wintenberger. Serre's modularity conjecture. I. *Invent. Math.*, 178(3):485–504, 2009. 4.1

[56] Chandrashekhar Khare and Jean-Pierre Wintenberger. Serre's modularity conjecture. II. *Invent. Math.*, 178(3):505–586, 2009. 4.1

[57] Alain Kraus. On the equation $x^r + y^r = z^p$. Notes for a talk given at IEM, Universität Duisburg-Essen, 1998. A copy has been included in [8]. 1.4, 1.4, 1.8, 3

[58] Alain Kraus. Sur l'équation $a^3 + b^3 = c^p$. *Experiment. Math.*, 7:1–13, 1998. 1.5, 3, 4.1

[59] Alain Kraus. Courbes elliptiques semi-stables sur les corps de nombres. *Int. J. Number Theory*, 3(4):611–633, 2007. 6.1

[60] Robert P. Langlands. *Base change for GL(2)*, volume 96 of *Annals of Mathematics Studies*. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. 4.1

[61] Pedro Lemos. Serre's uniformity conjecture for elliptic curves with rational cyclic isogenies. *Trans. Amer. Math. Soc.*, 371(1):137–146, 2019. 4.7

[62] 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. 2.3, 2.3, 2.3

[63] Paul Lockhart. On the discriminant of a hyperelliptic curve. *Trans. Amer. Math. Soc.*, 342(2):729–752, 1994. 2.3, 2.3

[64] Werner Lütkebohmert. From Tate's elliptic curve to abeloid varieties. *Pure Appl. Math. Q.*, 5(4, Special Issue: In honor of John Tate. Part 1):1385–1427, 2009. 7.1

[65] Kimball Martin. The Jacquet-Langlands correspondence, Eisenstein congruences, and integral $L$-values in weight 2. *Math. Res. Lett.*, 24(6):1775–1795, 2017. 10.1

[66] John C. Mason and David C. Handscomb. *Chebyshev polynomials*. Chapman & Hall/CRC, Boca Raton, FL, 2003. 2.4, iii

[67] R. Daniel Mauldin. A generalization of Fermat's Last Theorem: the Beal conjecture and prize problem. *Notices Amer. Math. Soc.*, 44(11):1436–1437, 1997. 1.1

[68] David Mumford. An analytic construction of degenerating abelian varieties over complete rings. *Compositio Math.*, 24:239–272, 1972. 7.1, 7.1

[69] Filip Najman. On $r$-isogenies over $\mathbb{Q}(\zeta_r)$ 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. 4.1

[70] Ioannis Papadopoulos. Sur la classification de Néron des courbes elliptiques en caractéristique résiduelle 2 et 3. *J. Number Theory*, 44(2):119–152, 1993. 10.1

[71] Ali Rajaei. On the levels of mod $\ell$ Hilbert modular forms. *J. Reine Angew. Math.*, 537:33–65, 2001. 8, 10.1

[72] Michel Raynaud. Schémas en groupes de type $(p, \dots, p)$. *Bull. Soc. Math. France*, 102:241–280, 1974. 6.1, 6.2

[73] Kenneth A. Ribet. Galois action on division points of Abelian varieties with real multiplications. *Amer. J. Math.*, 98(3):751–804, 1976. 2.2, 2.2, 2.2
[74] Kenneth A. Ribet. Abelian varieties over $\mathbb{Q}$ and modular forms. In *Algebra and topology 1992 (Taejön)*, pages 53–79. Korea Adv. Inst. Sci. Tech., Taejön, 1992. 2.6, 2.2

[75] Kenneth A. Ribet. Images of semistable Galois representations. Number Special Issue, pages 277–297. 1997. Olga Taussky-Todd: in memoriam. 6.2, 6.2

[76] Matthieu Romagny. Models of curves. In *Arithmetic and geometry around Galois theory*, volume 304 of *Progr. Math.*, pages 149–170. Birkhäuser/Springer, Basel, 2013. 5.2, 5.2

[77] Jean-Pierre Serre. Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. *Invent. Math.*, 15(4):259–331, 1972. 4.7

[78] Jean-Pierre Serre. Sur les représentations modulaires de degré 2 de $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. *Duke Math. J.*, 54:179–230, 1987. 1.2

[79] Goro Shimura. The special values of the zeta functions associated with Hilbert modular forms. *Duke Math. J.*, 45(3):637–679, 1978. 9.1

[80] Misja F.A. Steinmetz. *Explicit Serre weights for two-dimensional Galois representations*. PhD thesis, King's College London, 2020. 7.1

[81] Walter Tautz, Jaap Top, and Alain Verberkmoes. Explicit hyperelliptic curves with real multiplication and permutation polynomials. *Canad. J. Math.*, 43(5):1055–1064, 1991. 2.4

[82] Douglas Ulmer. Conductors of $\ell$-adic representations. *Proc. Amer. Math. Soc.*, 144(6):2291–2299, 2016. 5

[83] Andrew Wiles. Modular elliptic curves and Fermat's Last Theorem. *Annals of Mathematics*, 144:443–551, 1995. 1.1

[84] Chenyan Wu. *F-virtual Abelian Varieties of $GL_2$-type and Rallis Inner Product Formula*. PhD thesis, Columbia University, 2011. Available at https://academiccommons.columbia.edu/doi/10.7916/D8RF621W. 2.2

[85] David Zywina. On the possible images of the mod $\ell$ representations associated to elliptic curves over $\mathbb{Q}$. *ArXiv preprint*, https://arxiv.org/abs/1508.07660, 215. 4.1

LABORATOIRE DE MATHÉMATIQUES BLAISE PASCAL, UNIVERSITÉ CLERMONT AUVERGNE ET CNRS, CAMPUS UNIVERSITAIRE DES CÉZEAUX, 3, PLACE VASARELY, 63178 AUBIÈRE CEDEX, FRANCE

Email address: nicolas.billerey@uca.fr

DEPARTMENT OF MATHEMATICS, SIMON FRASER UNIVERSITY, BURNABY, BC V5A 1S6, CANADA

Email address: ichen@sfu.ca

DEPARTAMENT DE MATEMÀTIQUES I INFORMÀTICA, UNIVERSITAT DE BARCELONA (UB), GRAN VIA DE LES CORTS CATALANES 585, 08007 BARCELONA, SPAIN; CENTRE DE RECERCA MATEMÀTICA (CRM), EDIFICI C, CAMPUS BELLATERRA, 08193 BELLATERRA, SPAIN

Email address: ldieulefait@ub.edu

INSTITUTO DE CIENCIAS MATEMÁTICAS, CSIC, CALLE NICOLÁS CABRERA 13-15, 28049 MADRID, SPAIN

Email address: nuno.freitas@icmat.es