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

Abstract

Let f be a newform of weight 2 on \Gamma_0(N) with Fourier q -expansion f(q)=q+\sum_{n\geq 2} a_n q^n , where \Gamma_0(N) denotes the group of invertible matrices with integer coefficients, upper triangular mod N . Let p be a prime dividing N once, p\parallel N , a Steinberg prime. Then, it is well known that a_p\in\{1,-1\} . We denote by K_f the field of coefficients of f . Let \lambda be a finite place in K_f not dividing 2p and assume that the mod \lambda 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+\sum_{n\geq 2} a'_n q^n p -new of weight 2 on \Gamma_0(N) and a finite place \lambda' of K_{f'} such that a_p=-a'_p and the Galois representations \bar\rho_{f,\lambda} and \bar\rho_{f',\lambda'} are isomorphic.

Locations

• Revista Matemática Iberoamericana - View
• arXiv (Cornell University) - View - PDF
• Dipòsit Digital de la Universitat de Barcelona (Universitat de Barcelona) - View - PDF