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

TL;DR

The paper advances the Putman-Wieland conjecture by combining algebraic and geometric viewpoints, showing that when the cover group satisfies $|H|<g^2$, the cohomology $H^1$ cannot have finite-orbit vectors under the induced mapping class action, and providing an accessible geometric route via canonical maps and quadrics to the same conclusion. It further constructs families of origami curves with arbitrarily large isotrivial isogeny factors, arguing through degeneracy loci and Clifford-type bounds that certain representations must satisfy $\,\dim\rho\ge g$, which underpins asymptotic PW results. The work also documents explicit low-genus counterexamples, notably genus $0$ and $1$ origami families, and extends to hyperelliptic cases, where PW fails for all $g\ge 2$ (via Marković and Bogomolov–Tschinkel constructions). Overall, the authors connect Hodge theory, monodromy, Chevalley–Weil theory, and the geometry of canonical embeddings to frame PW, propose streamlined proofs, and chart directions toward Ivanov-type conjectures in higher genus. The results sharpen the understanding of when PW can hold and illuminate the geometry of families with nontrivial isogeny factors in their Jacobians.

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.

Figures (2)

• Figure 1: A picture of the variety $V$ swept out by colored planes, which is a component of the variety $Q_1 \cap Q_2$ cut out by certain quadrics containing the canonical image of $X$ in the case that a $2$-dimensional representation lies in the kernel of $\theta$. Each colored plane corresponds to a point $x \in \mathbb P \rho$ and is the span $x$ together with the corresponding hyperplanes in each copy of $\mathbb P \rho^\vee$ on which $x$ vanishes.
• Figure 2: A picture of the variety $V$ swept out by colored planes, which is contained in the intersection of certain quadrics containing the canonical image of $X$ when a $3$-dimensional representation $\rho$ lies in the kernel of $\theta$. Each colored plane corresponds to a point $x \in \mathbb P \rho$ and is the span $x$ together with the corresponding hyperplanes in each copy of $\mathbb P \rho^\vee$ on which $x$ vanishes.

Theorems & Definitions (44)

• Conjecture 1.1: Putman-Wieland, putmanW:abelian-quotients
• Theorem 1.2: landesmanL:canonical-representations
• Remark 1.3
• Remark 1.4
• Theorem 1.5: putmanW:abelian-quotients
• Definition 2.2
• Conjecture 2.3: Putman-Wieland, putmanW:abelian-quotients
• Remark 2.4
• Remark 2.5
• Theorem 2.6: landesmanL:canonical-representations