We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Effective Sato--Tate conjecture for abelian varieties and applications.
- Authors
Bucur, Alina; Fité, Francesc; Kedlaya, Kiran S.
- Abstract
From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato-Tate measure that only depends on certain invariants of A. We discuss three applications of this conditional result. First, for an abelian variety defined over k, we consider a variant of Linnik's problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius traces attain the integral part of the Hasse--Weil bound. Third, for a pair of abelian varieties A and A0 defined over k with no common factors up to k-isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces of A and A0 have opposite sign.
- Subjects
ABELIAN groups; RIEMANN hypothesis; INVARIANTS (Mathematics); FROBENIUS manifolds; GROUP theory
- Publication
Journal of the European Mathematical Society (EMS Publishing), 2024, Vol 26, Issue 5, p1713
- ISSN
1435-9855
- Publication type
Article
- DOI
10.4171/jems/1443