Elliptic curves with square-free Δ

Abstract

Under the Riemann Hypothesis for Dirichlet [Formula: see text]-functions, we improve on the error term in a smoothed version of an estimate for the density of elliptic curves with square-free [Formula: see text], where [Formula: see text] is the discriminant, by the author and Browning [Inhomogeneous cubic congruences and rational points on Del Pezzo surfaces, J. Reine Angew. Math. 680 (2013) 69–151]. To achieve this improvement, we elaborate on our methods for counting weighted solutions of inhomogeneous cubic congruences to powerful 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.