On congruences between normalized eigenforms with different sign at a Steinberg prime

2015/05/28 by Dieulefait, Luis, Soto, Eduardo
#(Secondary) #11F33 (Primary) #11F80 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1505.07883

Abstract

Let f be a newform of weight 2 on Γ₀(N) with Fourier q-expansion f(q)=q+∑_{n≥ 2} a_n qⁿ, where Γ₀(N) denotes the group of invertible matrices with integer coefficients, upper triangular mod N. Let p be a prime dividing N once, p∥ N, a Steinberg prime. Then, it is well known that a_p∈\1,-1\. We denote by K_f the field of coefficients of f. Let λ be a finite place in K_f not dividing 2p and assume that the mod λ Galois representation attached to f is irreducible. In this paper we will give necessary and sufficient conditions for the existence of another Hecke eigenform f'(q)=q+∑_{n≥ 2} a'_n qⁿ p-new of weight 2 on Γ₀(N) and a finite place λ' of K_f' such that a_p=-a'_p and the Galois representations ρ_f,λ and ρ_f',λ' are isomorphic.

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