Galois theory of the canonical theta structure

TL;DR

This paper provides a Galois-theoretic framework for understanding the canonical theta structure, linking it to p-adic theta relations and enabling algebraic proofs of theta identities in characteristic 2.

Contribution

It introduces a Galois-theoretic characterization of the canonical theta structure and derives algebraic proofs of 2-adic theta identities for canonical lifts.

Findings

Galois property translates into p-adic theta relations

Algebraic proof of 2-adic theta identities in characteristic 2

Explicit description of theta null points of canonical lifts

Abstract

In this article we give a Galois-theoretic characterization of the canonical theta structure. The Galois property of the canonical theta structure translates into certain $p$-adic theta relations which are satisfied by the canonical theta null point of the canonical lift. As an application we give a purely algebraic proof of some 2-adic theta identities which describe the set of theta null points of the canonical lifts of ordinary abelian varieties in characteristic 2. The latter theta relations are suitable for explicit canonical lifting.