Elliptic curves in isogeny classes

TL;DR

This paper demonstrates that elliptic curves within isogeny classes distribute nearly uniformly when considering short intervals of Frobenius traces, highlighting a new understanding of their distribution patterns.

Contribution

It establishes that the distribution of elliptic curves in isogeny classes approaches uniformity even over very short Frobenius trace intervals.

Findings

Distribution becomes close to uniform in short intervals

Distribution uniformity holds for various Frobenius trace values

Results improve understanding of elliptic curve distribution patterns

Abstract

We show that the distribution of elliptic curves in isogeny classes of curves with a given value of the Frobenius trace $t$ becomes close to uniform even when $t$ is averaged over very short intervals inside the Hasse-Weil interval.