Distribution of Farey Fractions in Residue Classes and Lang--Trotter Conjectures on Average

TL;DR

This paper proves uniform distribution of Farey fractions in residue classes modulo a prime under certain conditions and applies this to derive average bounds for Lang--Trotter conjectures on elliptic curves.

Contribution

It establishes uniform distribution results for Farey fractions in residue classes and uses these to obtain average bounds for Lang--Trotter conjectures.

Findings

Farey fractions are uniformly distributed modulo p for T ≥ p^{1/2 + ε}.

Derived upper bounds for Lang--Trotter conjectures on average.

Applied distribution results to elliptic curve Frobenius traces and fields.

Abstract

We prove that the set of Farey fractions of order $T$, that is, the set $\{\alpha/\beta \in \Q : \gcd(\alpha, \beta) = 1, 1 \le \alpha, \beta \le T\}$, is uniformly distributed in residue classes modulo a prime $p$ provided $T \ge p^{1/2 +\eps}$ for any fixed $\eps>0$. We apply this to obtain upper bounds for the Lang--Trotter conjectures on Frobenius traces and Frobenius fields ``on average'' over a one-parametric family of elliptic curves.