An introduction to the algebraic geometry of the Putman-Wieland conjecture

TL;DR

This paper explores algebraic and geometric approaches to the Putman-Wieland conjecture, introduces new origami curve families with special Jacobian properties, and discusses a hyperelliptic analogue failure.

Contribution

It provides new algebraic and geometric insights into the conjecture, constructs novel origami curves, and analyzes the failure of a hyperelliptic analogue.

Findings

Construction of families of origami curves with high-dimensional isotrivial Jacobian factors

Algebraic and geometric perspectives on the Putman-Wieland conjecture

Failure of the hyperelliptic analogue of the conjecture

Abstract

We give algebraic and geometric perspectives on our prior results toward the Putman-Wieland conjecture. This leads to interesting new constructions of families of "origami" curves whose Jacobians have high-dimensional isotrivial isogeny factors. We also explain how a hyperelliptic analogue of the Putman-Wieland conjecture fails, following work of Markovi\'{c}.