An introduction to the algebraic geometry of the Putman–Wieland 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 Marković.

Generating the homology of covers of surfaces

“…Let $$\pi \colon \widetilde{\Sigma }\rightarrow \Sigma$$ be a finite branched cover between closed oriented surfaces. The homology of $$\widetilde{\Sigma }$$ encodes subtle information about the mapping class group of $$\Sigma$$, and over the last decade has been intensely studied [5, 9–14, 16–18, 20]. Much of this is motivated by a conjecture of Putman–Wieland [20] we discuss below.…”

“…Conjecture 1.1 has been proved in a variety of cases; see, for example, [5, 11–14]. However, we know very little about when $$\operatorname{H}^{\operatorname{scc}}_1(\widetilde{\Sigma };\mathbb {Q}) = \operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$$.…”