Frobenius sign separation for abelian varieties
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.
Keywords
Cite
@article{arxiv.2310.10568,
title = {Frobenius sign separation for abelian varieties},
author = {Alina Bucur and Francesc Fité and Kiran S. Kedlaya},
journal= {arXiv preprint arXiv:2310.10568},
year = {2025}
}
Comments
10 pages. Accepted in Proc. Amer. Math. Soc. Several changes were made in order to turn the implied constant in the O-notation into a universal constant independent of the field of definition and the dimensions. Includes material formerly appearing in arXiv:2002.08807