An Asymptotic for the Number of Solutions to Linear Equations in Prime Numbers from Specified Chebotarev Classes

TL;DR

This paper extends asymptotic formulas for solutions to linear equations in primes within specified Chebotarev classes, incorporating advanced number theory tools, and applies these results to identify elliptic curves with particular discriminant properties.

Contribution

It provides new asymptotic results for prime solutions in Chebotarev classes using class field theory, with applications to elliptic curve discriminants.

Findings

Asymptotic formulas for prime solutions in Chebotarev classes derived.

Incorporation of Chebotarev Density Theorem and Class Field Theory corrections.

Application to elliptic curves with specified discriminant splitting.

Abstract

We extend known results on the number of solutions to a linear equation in at least three prime numbers when the primes involved are required to lie in specified Chebotarev classes. We prove asymptotic results similar to previous ones only now taking into account corrections coming form the Chebotarev Density Theorem and Global Class Field Theory. We then apply these results to find elliptic curves whose discriminants split completely of a given number field.