The spin of prime ideals and level-raising of even Galois representations

TL;DR

This paper extends the concept of spin of prime ideals to connect a character sum conjecture with the density of primes affecting the level of even Galois representations, confirming a longstanding conjecture.

Contribution

It introduces an extended notion of spin for prime ideals and links a character sum conjecture to the density of primes that raise the level of Galois representations.

Findings

Primes raising the level have density 2/3 under the conjecture

Extension of spin notion to prime ideals

Supports Ramakrishna's 1998 conjecture

Abstract

By extending the notion of spin of prime ideals, we show that a short character sum conjecture implies that the set of primes raising the level of a certain even Galois representation has density 2/3, as conjectured by Ramakrishna in 1998.