Average Frobenius distribution for the degree two primes of a number field

TL;DR

This paper studies the distribution of degree two primes in a number field with a specific Frobenius trace, providing average asymptotic results that align with heuristic predictions and extending prior research in the area.

Contribution

It establishes average asymptotic formulas for degree two primes with a given Frobenius trace in certain number fields, extending previous work and relating to the Lang-Trotter conjecture.

Findings

Average count of degree two primes matches heuristic predictions

Asymptotic identities are established under specific restrictions

Extends classical results on Frobenius distributions in number fields

Abstract

Let $K$ be a number field and $r$ an integer. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree two prime ideals of $K$ with trace of Frobenius equal to $r$. Under certain restrictions on $K$, 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.