Criteria for Irreducibility of mod p Representations of Frey Curves

Abstract

Soit K un corps galoisien totalement réel, et soit E un ensemble de courbes elliptiques sur K. Nous donnons des conditions suffisantes pour l'existence d'un ensemble calculable de premiers rationnels P tels que, pour p / ∈ P et E ∈ E, la représentation Gal(K/K) → Aut(E[p]) soit irréductible. Nos conditions sont en général satisfaites par les courbes de Frey associées à des solutions d'équations diophantiennes; dans ce contexte, l'irréductibilité de la mod p représentation est une hypothèse requise pour l'application des théorèmes d'abaissement du niveau. Comme illustration de notre approche, nous avons amélioré le résultat de [6] pour les équations de Fermat de signature (13, 13, p).

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp

“…Alas, there is not yet a uniform boundedness theorem for isogenies. The papers of Kraus (2007), David (2012), Freitas and Siksek (2015b) do give effective bounds C K such that for ℓ > C K the representation ρ E,ℓ is irreducible, however these bounds are too large for our present purpose. We will refine the arguments in those papers making use of the fact that the curve E is semistable, and the number fields K = Q(ζ + ζ −1 ) all have narrow class number 1.…”

A multi-Frey approach to Fermat equations of signature $(r,r,p)$

“…The equation (1.2) with r = 5 and d = 2, 3 has already been subject of the papers [4], [6] and [23], where it was resolved for 3 4 of prime exponents p. For r = 13 and d = 3, it has been resolved in the papers [22], [28] under the assumption 13 ∤ z.…”

“…The following illustrates a fundamental difference between the Frey elliptic curves E and F . Note that irreducibility of ρ E,p followed by an application of [28,Theorem 3] which makes crucial use of the presence of explicit primes of good reduction of E. This was guaranteed by the fact that all the primes not dividing 2 ⋅ 13 of bad reduction of E must have residual characteristic congruent to 1 mod 13 (see Lemma 5). This is no longer the case for F due to the factor a + b in ∆(F ).…”

“…By [29,Lemma 6.3] for instance, we have that θ and θ ′ are unramified outside p and the additive primes of C. Furthermore, θ and θ ′ have the same conductor away from p. The unit group of K is generated by {−1, ǫ} where ǫ 2 − ǫ − 1 = 0. In the notation of [28], we compute B = −2 6 ⋅ 5. From the first paragraph of the proof of [28, Theorem 1] we thus conclude that exactly one of θ, θ ′ ramifies at (the primes above) p. Let us therefore assume that θ is unramified at p.…”

“…This is no longer the case for F due to the factor a + b in ∆(F ). Therefore, we can only apply [28,Theorem 2] which guarantees that ρ F,p is irreducible when p > (1 + 3 18 ) 2 . This bound is insufficient for our purposes.…”