Parity conjectures for elliptic curves over global fields of positive characteristic

TL;DR

This paper proves the p-parity conjecture for elliptic curves over global fields of characteristic p > 3 and offers partial results for the 1-parity conjecture for other primes, advancing understanding of elliptic curve parity properties.

Contribution

The paper establishes the p-parity conjecture in positive characteristic and provides initial results for the 1-parity conjecture for primes different from p, filling gaps in the theory.

Findings

Proved the p-parity conjecture for elliptic curves over global fields of characteristic p > 3.

Presented partial results on the 1-parity conjecture for primes 1 1 p.

Enhanced understanding of parity conjectures in positive characteristic settings.

Abstract

We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\ell$-parity conjecture for primes $\ell \neq p$.