Supersingular distribution on average for congruence classes of primes

TL;DR

This paper shows that, on average, supersingular primes for elliptic curves over rationals are not evenly distributed across congruence classes, revealing a bias in their distribution.

Contribution

It establishes the existence of a congruence class bias in supersingular prime distribution using averaging methods and ideas from previous researchers.

Findings

Supersingular primes are twice as common in certain congruence classes.

The bias is demonstrated on average over elliptic curves.

The approach combines Fouvry-Murty averaging with David-Pappalardi's ideas.

Abstract

We demonstrate the existence of a congruence class bias in the distribution of supersingular primes on average for elliptic curves over $\Q$. For example, we show that on average there are twice as many supersingular primes congruent to 2 mod 3 as there are congruent to 1 mod 3. Our result is obtained using the averaging approach of Fouvry-Murty along with ideas of David-Pappalardi.