On the distribution of traces of Frobenius for families of elliptic curves and the Lang-Trotter conjecture on average

TL;DR

This paper studies the distribution of Frobenius traces in elliptic curve families, providing new results and bounds related to the Lang-Trotter conjecture, extremal primes, and modular forms, with implications for average behaviors.

Contribution

It extends previous work by generalizing distribution results and establishing bounds for Frobenius traces across various elliptic curve families and related modular forms.

Findings

Distribution results for Frobenius traces in elliptic curve families.

Bounds related to the average Lang-Trotter conjecture.

Results on Frobenius traces for primes in congruence classes.

Abstract

We obtain distribution results for traces of Frobenius for various families of elliptic curves with respect to the Lang-Trotter conjecture, extremal primes, and the central limit theorem. This includes some generalisations and bounds related to the work of Sha-Shparlinski on the average Lang-Trotter conjecture for single-parametric families of elliptic curves and the work of various authors on the trace of Frobenius for primes in congruence classes. Some results are also obtained for modular forms.