Workshop on Local Systems and Monodromy

Title: Malle's conjecture and the Picard rank conjecture via homological stability for Hurwitz spaces

Abstract: We will describe joint work with Ishan Levy, making progress toward Malle's conjecture over function fields and toward the Picard rank conjecture. For G a finite group and K a global field, Malle's conjecture predicts the asymptotic growth of the number of G extensions of K. We compute the asymptotic growth of the number of G extensions of F_q(t), for q sufficiently large and relatively prime to G. We do this by verifying that the homology of Hurwitz spaces associated to an arbitrary group suitably stabilizes. As a more algebro-geometric application of these homological stability results, we also deduce an asymptotic version of the Picard rank conjecture, predicting the rational Picard group of certain Hurwitz spaces vanishes.

Title: Geometric Shafarevich Conjecture for Exceptional Shimura Varieties

Abstract: The Shafarevich conjecture is concerned with finiteness results for families of g-dimensional principally polarized abelian varieties over a base B. Faltings settled the case of B=O_{K,S}. In the case where B is a curve over a finite field, finiteness can never be true as one may always compose with Frobenius. In this setting, to get a theorem one must consider families up to p-power isogenies.

We formulate an analogous statement for Exceptional Shimura varieties S, and describe ongoing work to prove it. This is joint work with Ben Bakker and Jacob Tsimerman.

Title: Integrability of Katzarkov--Zuo foliations

Abstract: Let X be a complex algebraic variety and V a semisimple Q_ell-local system.  Due to work of Gromov and Schoen for X projective and recent work of Brotbek, Daskalopoulos, Deng, and Mese in general, V admits a pluriharmonic norm, which naturally yields a foliation (called a Katzarkov--Zuo foliation) on X along which the monodromy of V preserves a Z_\ell lattice.  Eyssidieux gave a criterion for when this foliation is algebraically integrable in the projective case, and I will describe some joint work with Y. Brunebarbe and J. Tsimerman generalizing this result to the quasiprojective case.  I will also explain how this fits into the proof of the linear Shafarevich conjecture.

Title: Characteristic classes of local systems via the Fargues--Fontaine curve

Abstract: For a prime number p, the category of etale Q_p-local systems on an algebraic variety X over a p-adic field can be "geometrized" in the following way: it admits a fully faithful embedding into the category of vector bundles on the relative Fargues-Fontaine curve of X. For local systems coming from cohomology of a family of varieties over X, the p-adic comparison isomorphisms for that family can be naturally stated in terms of this associated vector bundle. I will discuss how this perspective on local systems can be useful for calculating their additive invariants in etale cohomology. This is joint work with Lue Pan.