论文标题
关于$ \ text {gl} _2 $ -Type Abelian品种和Diophantine应用的内态代数
On endomorphism algebras of $\text{GL}_2$-type abelian varieties and Diophantine applications
论文作者
论文摘要
令$ f $和$ g $为两个不同的新形式，而没有复杂的乘法具有相同的系数字段。本文的主要结果证明，Galois表示与$ f $的代表与$ f $的$ g $之间的一致性表示，大质量$ p $ to to abelian品种$ a_f $和$ a_g $ a_g $ a_g $ f $ f $ f $ f $ f $ f $ f $ g $ g $ G $ by Eichimler construction the eichimera construction的内态代数之间的同构。这意味着它们的构建基础之间的重要关系。 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.我们证明了方程$ x^2+dy^6 = z^p $的结果类似。
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 $x^2+dy^6=z^p$.
