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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.
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Abstract We give two applications of our prior work toward the Putman–Wieland conjecture. First, we deduce a strengthening of a result of Marković–Tošić on virtual mapping class group actions on the homology of covers. Second, let and let be a finite ‐cover of topological surfaces. We show the virtual action of the mapping class group of on an ‐isotypic component of has nonunitary image.more » « less
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Abstract Fix a positive integernand a finite field$${\mathbb {F}}_q$$ . We study the joint distribution of the rank$${{\,\mathrm{rk}\,}}(E)$$ , then-Selmer group$$\text {Sel}_n(E)$$ , and then-torsion in the Tate–Shafarevich group Equation missing<#comment/>asEvaries over elliptic curves of fixed height$$d \ge 2$$ over$${\mathbb {F}}_q(t)$$ . We compute this joint distribution in the largeqlimit. We also show that the “largeq, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains.more » « less
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For$$2 \leq d \leq 5$$, we show that the class of the Hurwitz space of smooth degree$$d$$, genus$$g$$covers of$$\mathbb {P}^1$$stabilizes in the Grothendieck ring of stacks as$$g \to \infty$$, and we give a formula for the limit. We also verify this stabilization when one imposes ramification conditions on the covers, and obtain a particularly simple answer for this limit when one restricts to simply branched covers.more » « less
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We show that the minimum rank of a non-isotrivial local system of geometric origin on a suitably general -pointed curve of genus is at least . We apply this result to resolve conjectures of Esnault-Kerz and Budur-Wang. The main input is an analysis of stability properties of flat vector bundles under isomonodromic deformations, which additionally answers questions of Biswas, Heu, and Hurtubise.more » « less
