Effective Chebotarev-Sato-Tate distribution for Abelian varieties with applications
Abstract
We establish a uniform effective Chebotarev-Sato-Tate estimate for abelian varieties, conditional on the Mumford-Tate conjecture and the standard analytic conjectures for L-functions attached to the irreducible representations of the Chebotarev-Sato-Tate group. The proof uses spectral majorants and minorants of indicator functions. For regions whose boundaries have finite upper (d-1)-dimensional Minkowski content, the resulting exponent improves that in Bucur et al. (Theorem 1.1, 2024) whenever the Sato-Tate group has rank greater than one. We establish uniform estimates for regions and finite Galois extensions that vary with the counting parameter, and for prime-dependent regions satisfying an explicit variation condition. As an application, we obtain a conditional upper bound for the number of primes at which a weight-two newform with a quadratic coefficient field has a rational Fourier coefficient. We also formulate a discrepancy conjecture, motivated by a probabilistic model, that implies the asymptotic predicted by Murty (Conjecture 3.4, 1999).
Related
- Spectral Beurling-Selberg Approximation and Applications to Effective Motivic Chebotarev-Sato-Tate Estimates
- Dynamical degrees of automorphisms on abelian varieties
- A holomorphic (2, 2)-Theorem for Abelian varieties of CM-type
- Arithmetic jet spaces and the unramified Zilber--Pink conjecture for subvarieties of abelian varieties
- On the finite transcendence of Frobenius traces for abelian varieties over ℚ
BibTeX
@misc{amri2026effective,
title = {{Effective Chebotarev-Sato-Tate distribution for Abelian varieties with applications}},
author = {Mohammed Amin Amri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36464},
eprint = {2610.36464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36464},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Effective Chebotarev-Sato-Tate distribution for Abelian varieties with applications
AU - Mohammed Amin Amri
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36464
UR - https://arxiv.org/abs/2610.36464
AB - We establish a uniform effective Chebotarev-Sato-Tate estimate for abelian varieties, conditional on the Mumford-Tate conjecture and the standard analytic conjectures for L-functions attached to the irreducible representations of the Chebotarev-Sato-Tate group. The proof uses spectral majorants and minorants of indicator functions. For regions whose boundaries have finite upper (d-1)-dimensional Minkowski content, the resulting exponent improves that in Bucur et al. (Theorem 1.1, 2024) whenever the Sato-Tate group has rank greater than one. We establish uniform estimates for regions and finite Galois extensions that vary with the counting parameter, and for prime-dependent regions satisfying an explicit variation condition. As an application, we obtain a conditional upper bound for the number of primes at which a weight-two newform with a quadratic coefficient field has a rational Fourier coefficient. We also formulate a discrepancy conjecture, motivated by a probabilistic model, that implies the asymptotic predicted by Murty (Conjecture 3.4, 1999).
ER -