A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for the Lang-Trotter conjectures

Abstract

We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and zero repulsion phenomenon for Hecke L-functions that were recently proved by the authors. Our result produces an improvement for the best unconditional bounds toward two conjectures of Lang and Trotter regarding the distribution of traces of Frobenius for elliptic curves and holomorphic cuspidal modular forms. We also obtain new results on the distribution of primes represented by positive-definite integral binary quadratic forms.

26 pages; v2 several typographical and minor errors are fixed, some of which had a small impact on the quality of the exponents in Thm 1.1, Thm 1.2, and Cor 1.3

References in corpus (1)

• An explicit bound for the least prime ideal in the Chebotarev density theorem

Cited by in corpus (13)

• A unified and improved Chebotarev density theorem
• Effective forms of the Sato--Tate conjecture
• Exceptional splitting of reductions of abelian surfaces
• An unconditional $\mathrm{GL}(n)$ large sieve
• Primes represented by positive definite binary quadratic forms
• Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves
• Pointwise Bound for $\ell$-torsion in Class Groups II: Nilpotent Extensions
• Composite values of shifted exponentials
• Lang--Trotter Conjecture for CM Elliptic Curves
• Pointwise Bound for $\ell$-torsion in Class Groups: Elementary Abelian Extensions
• The Lang-Trotter Conjecture for the elliptic curve $y^2=x^3+Dx$
• Elliptic curves with missing Frobenius traces
• On the distribution of traces of Frobenius for families of elliptic curves and the Lang-Trotter conjecture on average