Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals

TL;DR

This paper investigates the distribution of prime ideals with a specific Frobenius trace in Galois extensions, providing average-case asymptotics that align with heuristic predictions, extending classical conjectures in number theory.

Contribution

It establishes average asymptotic formulas for counting prime ideals with given Frobenius trace in Galois extensions, extending previous results and related to the Lang-Trotter conjecture.

Findings

Asymptotic formulas for prime ideals with fixed Frobenius trace

Results align with standard heuristics for most cases

Extension of classical conjectures to Galois extensions

Abstract

Let $K$ be a fixed number field, assumed to be Galois over $\mathbb Q$. Let $r$ and $f$ be fixed integers with $f$ positive. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree $f$ prime ideals of $K$ with trace of Frobenius equal to $r$. Except in the case $f=2$, we show that "on average," the number of such prime ideals with norm less than or equal to $x$ satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang-Trotter conjecture and extends the work of several authors.