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arXiv · 2310.10568

Frobenius sign separation for abelian varieties

Alina Bucur·Francesc Fité·Kiran S. Kedlaya
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Abstract

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.

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Alina Bucur, Francesc Fité, Kiran S. Kedlaya. 2023-10-16. Frobenius sign separation for abelian varieties. https://arxiv.org/abs/2310.10568

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