On endomorphism algebras of $\text{GL}_2$-type abelian varieties and Diophantine applications
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| Format: | Preprint |
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2022
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| _version_ | 1866908287536988160 |
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| author | Madriaga, Franco Golfieri Pacetti, Ariel Torcomian, Lucas Villagra |
| author_facet | Madriaga, Franco Golfieri Pacetti, Ariel Torcomian, Lucas Villagra |
| contents | 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 $x^2+dy^6=z^p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_16334 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On endomorphism algebras of $\text{GL}_2$-type abelian varieties and Diophantine applications Madriaga, Franco Golfieri Pacetti, Ariel Torcomian, Lucas Villagra Number Theory 11D41, 11F80 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 $x^2+dy^6=z^p$. |
| title | On endomorphism algebras of $\text{GL}_2$-type abelian varieties and Diophantine applications |
| topic | Number Theory 11D41, 11F80 |
| url | https://arxiv.org/abs/2211.16334 |