Abelian varieties not isogenous to Jacobians over global fields

TL;DR

This paper demonstrates the existence of certain abelian varieties not isogenous to Jacobians over various global fields, using Hecke operators and Serre Tate coordinates, extending results across different types of fields.

Contribution

It introduces new methods involving Hecke operators and Serre Tate coordinates to prove the existence of non-isogenous abelian varieties to Jacobians over multiple global fields.

Findings

Existence of abelian varieties not isogenous to Jacobians over characteristic p function fields.

Extension of results to number fields and finite fields.

Development of new proof techniques involving Hecke operators and formal neighborhoods.

Abstract

We prove the existence of abelian varieties not isogenous to Jacobians over characterstic $p$ function fields. Our methods involve studying the action of degree $p$ Hecke operators on hypersymmetric points, as well as their effect on the formal neighborhoods using Serre Tate co-ordinates. We moreover use our methods to provide another proof over number fields, as well as proving a version of this result over finite fields.