arXiv subjects
Publications and source records attributed to Eduardo Soto.
In this paper we prove new cases of the asymptotic Fermat equation with coefficients. This is done by solving remarkable $S$-units equations and applying a method of Frey-Mazur.
In this paper we prove a level raising theorem for some weight $2$ trivial character newforms at almost every prime $p$. This is done by ignoring the residue characteristic at which the level raising appears.
Let $f$ be a newform of weight $2$ on $Γ_0(N)$ with Fourier $q$-expansion $f(q)=q+\sum_{n\geq 2} a_n q^n$, where $Γ_0(N)$ 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 $λ$ 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 $f'(q)=q+\sum_{n\geq 2} a'_n q^n$ $p$-new of weight $2$ on $Γ_0(N)$ and a finite place $λ'$ of $K_{f'}$ such that $a_p=-a'_p$ and the Galois representations $\barρ_{f,λ}$ and $\barρ_{f',λ'}$ are isomorphic.