Caractère d'isogénie et critères d'irréductibilité

Abstract

This article deals with the Galois representation attached to the torsion points of an elliptic curve defined over a number field. We first determine explicit uniform criteria for the irreducibility of this Galois representation for elliptic curves varying in some infinite families, characterised by their reduction type at some fixed places of the base field. Then, we deduce from these criteria an explicit form for a bound that appear in a theorem of Momose. Finally, we use these results to precise a previous theorem of the author about the homotheties contained in the image of the Galois representation.

Shifted powers in Lucas–Lehmer sequences

“…We would like to apply level-lowering to the mod p representation ρ En,p , and for this we need to show that it is irreducible. We shall make use of the following result due to Freitas and Siksek [12], which is based on the work of David [8] and Momose [18]. Proposition 4.2.…”

A result on the equation $x^p + y^p = z^r$ using Frey abelian varieties

“…The aim of this section is to provide a uniform bound on the residual characteristic of prime ideals p for which the corresponding representations ρ A,p is reducible when A runs through certain families of abelian varieties of GL 2 -type. For elliptic curves over totally real fields, such irreducibility criteria were previously known and different variants (for various families of curves) can be found in the work of Serre [11], Kraus [7,8], Billerey [1], David [3], Dieulefait-Freitas [4] and Freitas-Siksek [5].…”

Criteria for Irreducibility of mod p Representations of Frey Curves

“…The following is closely related to a result of David [5,Theorem 2], but formulated in a way that is more suitable for attacking specific examples.…”

“…For a general number field K, it is expected that there is some B K , such that for all elliptic curves E/K without complex multiplication, and all p > B K , the mod p representation ρ E,p is irreducible. Several papers, including those by Momose [17], Kraus [13], [14] and David [5], establish a bound B K depending on the field K, under some restrictive assumptions on E, such as semistability. The Frey elliptic curves one deals with in the modular approach are close to being semistable [21,Section 15.2.4].…”