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

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.

Preprint of an old paper

References in corpus (2)

• A generalization of the Barban-Davenport-Halberstam Theorem to number fields
• A Barban-Davenport-Halberstam asymptotic for number fields