Frobenius sign separation for abelian varieties
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912306057707520 |
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| author | Bucur, Alina Fité, Francesc Kedlaya, Kiran S. |
| author_facet | Bucur, Alina Fité, Francesc Kedlaya, Kiran S. |
| contents | Let A and A' be nonzero abelian varieties defined over a number field k such that Hom(A,A')=0. Under the Generalized Riemann hypothesis for motivic L-functions attached to A and A', we show that there exists a prime p of k of good reduction for A and A' at which the Frobenius traces of A and A' are nonzero and differ by sign, and such that the norm of p is O_{k,g,g'}(log(2NN')^2), where N and N' respectively denote the absolute conductors of A and A'. We also make the dependence of the big-O constant on k and the dimensions g,g' of A,A' explicit up to an effectively computable absolute constant. Our method extends that of Chen, Park, and Swaminathan who considered the case in which A and A' are elliptic curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_10568 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Frobenius sign separation for abelian varieties Bucur, Alina Fité, Francesc Kedlaya, Kiran S. Number Theory 11G10, 11G05, 11R44, 11M41 Let A and A' be nonzero abelian varieties defined over a number field k such that Hom(A,A')=0. Under the Generalized Riemann hypothesis for motivic L-functions attached to A and A', we show that there exists a prime p of k of good reduction for A and A' at which the Frobenius traces of A and A' are nonzero and differ by sign, and such that the norm of p is O_{k,g,g'}(log(2NN')^2), where N and N' respectively denote the absolute conductors of A and A'. We also make the dependence of the big-O constant on k and the dimensions g,g' of A,A' explicit up to an effectively computable absolute constant. Our method extends that of Chen, Park, and Swaminathan who considered the case in which A and A' are elliptic curves. |
| title | Frobenius sign separation for abelian varieties |
| topic | Number Theory 11G10, 11G05, 11R44, 11M41 |
| url | https://arxiv.org/abs/2310.10568 |