• Corpus ID: 233872721

# Chebotarev Density Theorem and Extremal Primes for non-CM elliptic curves

@inproceedings{Malik2020ChebotarevDT,
title={Chebotarev Density Theorem and Extremal Primes for non-CM elliptic curves},
author={Amita Malik and Neha Prabhu},
year={2020}
}
• Published 23 December 2020
• Mathematics
Abstract. For a fixed non-CM elliptic curve E over Q and a prime l, we prove an asymptotic formula on the number of primes p ≤ x for which the Frobenius trace ap(E) satisfies the congruence ap(E) ≡ [2 √ p] (mod l). In order to achieve this, we establish a joint distribution concerning the fractional part of αp for θ ∈ [0, 1], α > 0, and primes p satisfying the Chebotarev condition. As a corollary, we also obtain upper bounds for the number of extremal primes. The results rely on GRH for…

## References

SHOWING 1-10 OF 35 REFERENCES
Extremal primes for elliptic curves without complex multiplication
• Mathematics
Proceedings of the American Mathematical Society
• 2019
Fix an elliptic curve E over Q. An extremal prime for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound.
Bounds for the Lang-Trotter conjectures
For a non-CM elliptic curve $E$ defined over the rationals, Lang and Trotter made very deep conjectures concerning the number of primes $p\leq x$ for which $a_p(E)$ is a fixed integer (and for which
Average Frobenius Distributions for Elliptic Curves: Extremal Primes and Koblitz's Conjecture
Let E/Q be an elliptic curve, and let p be a rational prime of good reduction. Let ap(E) denote the trace of the Frobenius endomorphism of E at the prime p, and let #E(Fp) be the number of
Automorphy for some l-adic lifts of automorphic mod l Galois representations
• Mathematics
• 2008
We extend the methods of Wiles and of Taylor and Wiles from GL2 to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate
Applications of the Sato-Tate conjecture
• Mathematics
• 2020
We use the effective version of the Sato-Tate conjecture for non-CM holomorphic cupsidal newforms $f$ of squarefree level on $\mathrm{GL}_2$ proved by the second author to make some unconditional
A family of Calabi-Yau varieties and potential automorphy
• Mathematics
• 2010
We prove potential modularity theorems for l-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral j-invariant defined
The Theory of the Riemann Zeta-Function
• Mathematics
• 1987
The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects