On endomorphism algebras of $\text{GL}_2$-type abelian varieties and Diophantine applications

TL;DR

This paper establishes that congruences between Galois representations of certain modular forms imply isomorphisms of their associated endomorphism algebras, leading to applications in solving specific exponential Diophantine equations.

Contribution

It proves a new link between Galois representation congruences and endomorphism algebra isomorphisms for abelian varieties of GL2-type, with implications for Diophantine equations.

Findings

Congruences imply isomorphism of endomorphism algebras for associated abelian varieties.

No non-trivial primitive solutions for certain exponential equations when prime p is large.

Results connect modular form properties with solutions to specific Diophantine equations.

Abstract

Let $f$ and $g$ be two different newforms without complex multiplication having the same coefficient field. The main result of the present article proves that a congruence between the Galois representations attached to $f$ and to $g$ for a large prime $p$ implies an isomorphism between the endomorphism algebras of the abelian varieties $A_f$ and $A_g$ attached to $f$ and $g$ by the Eichler-Shimura construction. This implies important relations between their building blocks. A non-trivial application of our result is that for all prime numbers $d$ congruent to $3$ modulo $8$ satisfying that the class number of $\mathbb{Q}(\sqrt{-d})$ is prime to $3$, the equation $x^4+dy^2 =z^p$ has no non-trivial primitive solutions when $p$ is large enough. We prove a similar result for the equation $x^2+dy^6=z^p$.