Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

Abstract

In this paper we prove that for every integer [Formula: see text], there exists an explicit constant [Formula: see text] such that the following holds. Let [Formula: see text] be a number field of degree [Formula: see text], let [Formula: see text] be any rational prime that is totally inert in [Formula: see text] and [Formula: see text] any elliptic curve defined over [Formula: see text] such that [Formula: see text] has potentially multiplicative reduction at the prime [Formula: see text] above [Formula: see text]. Then for every rational prime [Formula: see text], [Formula: see text] has an irreducible mod [Formula: see text] Galois representation. This result has Diophantine applications within the “modular method”. We present one such application in the form of an Asymptotic version of Fermat’s Last Theorem that has not been covered in the existing literature.

On some generalized Fermat equations of the form x2+y2n=zp$x^2+y^{2n} = z^p$

“…By [19,Theorem 8.7], the equation 𝑥 5 + 𝑦 5 = 𝑧 17 has no solutions in non-zero coprime integers 𝑥, 𝑦, and 𝑧. Using this we obtain the following corollary to Theorem 3.…”

“…Proof The prime 3 is inert in K , and by Lemma 4.1, E has multiplicative reduction at $$3\mathcal {O}_K$$. Since $$3 > \deg (K) -1 = 2$$, we can apply [17, Theorem 1.3] to deduce that the representation $$\overline{\rho }_{E,\ell }$$ is irreducible for $$\ell > 65 \cdot 6^6$$.$$\Box$$…”

“…The prime 3 is inert in 𝐾, and by Lemma 4.1, 𝐸 has multiplicative reduction at 3 𝐾 . Since 3 > deg(𝐾) − 1 = 2, we can apply [17,Theorem 1.3] to deduce that the representation 𝜌 𝐸,𝓁 is irreducible for 𝓁 > 65 ⋅ 6 6 . □…”

“…As in Section 3, there exist coprime integers 𝑎 and 𝑏 such that 𝑥 𝓁 + 𝑦 𝑚 𝑖 = (𝑎 + 𝑏𝑖) 17 and 𝑧 = 𝑎 2 + 𝑏 2 .…”

On elliptic curves with $p$-isogenies over quadratic fields

“…The only major barrier to doing this is the lack of a suitable version of Proposition 4.11 for larger number fields. Indeed, such results do exist (see [20,Theorem 1.3] for example), but the lower bounds on p are much larger than 73. Even in the case d = 3, it is necessary to assume that p > 65 • 6 6 = 3032640.…”