The spin of prime ideals

TL;DR

This paper introduces a new invariant called 'spin' for prime ideals in a number field, enabling analysis of their distribution without relying on traditional L-function Euler products, with applications to elliptic curve Selmer groups.

Contribution

It defines the 'spin' invariant for prime ideals and develops techniques to study its distribution, surpassing traditional L-function methods.

Findings

Established distribution results for the spin of prime ideals

Demonstrated applications to Selmer groups of elliptic curves

Developed bilinear forms techniques for non-L-function-based analysis

Abstract

Fixing a nontrivial automorphism of a number field K, we associate to ideals in K an invariant (with values in {0,1,-1}) that we call the "spin" and for which the associated L-function does not possess Euler products. We are nevertheless able, using the techniques of bilinear forms, to handle spin value distribution over primes, obtaining stronger results than the analogous ones which follow from the technology of L-functions in its current state. The initial application of our theorem is to the arithmetic statistics of Selmer groups of elliptic curves.