Elliptic curves with square-free $\Delta$

TL;DR

This paper improves the error term in estimating the density of elliptic curves with square-free discriminant under the Riemann Hypothesis by refining methods for counting solutions to cubic congruences and evaluating exponential sums.

Contribution

It introduces a novel approach to explicitly evaluate exponential sums and averages over moduli, enhancing previous density estimates for elliptic curves.

Findings

Improved error bounds under the Riemann Hypothesis

Refined counting methods for cubic congruences

Enhanced evaluation of exponential sums

Abstract

Under the Riemann Hypothesis for Dirichlet L-functions, we improve on the error term in a smoothed version of an estimate for the density of elliptic curves with square-free $\Delta=D/16$, where D is the discriminant, by T.D. Browning and the author. To achieve this improvement, we elaborate on our methods for counting weighted solutions of inhomogeneous cubic congruences to power-ful moduli. The novelty lies in going a step further in the explicit evaluation of complete exponential sums and saving a factor by averaging over the moduli.