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  1. ; ( , Proceedings of the London Mathematical Society)
    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.

     
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  2. ; ; ( , Mathematische Annalen)
    Abstract

    Fix a positive integernand a finite field$${\mathbb {F}}_q$$Fq. We study the joint distribution of the rank$${{\,\mathrm{rk}\,}}(E)$$rk(E), then-Selmer group$$\text {Sel}_n(E)$$Seln(E), and then-torsion in the Tate–Shafarevich group Equation missing<#comment/>asEvaries over elliptic curves of fixed height$$d \ge 2$$d≥2over$${\mathbb {F}}_q(t)$$Fq(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.

     
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  3. ; ; ( , Bulletin of the London Mathematical Society)