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

TL;DR

The paper proves that congruences between Galois representations attached to GL$_2$-type newforms with identical coefficient fields force injections between the rational endomorphism algebras of their Eichler–Shimura abelian varieties for large primes, provided the reference form has no CM and a square-free level quotient. Central to the method are Ribet’s inner twists and their endomorphisms, which yield a concrete link between the endomorphism algebras through the group of inner twists $\Gamma_f$. This framework is then applied to Fermat-type Diophantine equations over imaginary quadratic fields: the associated GL$_2$-type abelian varieties decompose into 1-dimensional building blocks (elliptic curves) over suitable fields, enabling a descent to eliminate nontrivial primitive solutions for large primes $p$ under certain arithmetic conditions on $d$. Consequently, the paper delivers two asymptotic nonexistence results for the equations $x^4+dy^2=z^p$ (with $d\equiv 3\pmod{8}$ and class number conditions) and $x^2+dy^6=z^p$ (with $d\equiv 19\pmod{24}$ and similar class-number constraints), illustrating the power of endomorphism-algebra techniques in Diophantine problems. The results hinge on a careful analysis of inner twists, fields of definition for building blocks, and the modular method's lift to quadratic base fields via $\mathbb{Q}$-curves and $K$-rational torsion data.

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 $x^2+dy^6=z^p$.

Theorems & Definitions (54)

• Definition
• Definition
• Remark 1
• Example
• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Lemma 1.3
• proof