On the almost-product structure on the moduli of bounded global $G$-shtuka
Jack Sempliner (Imperial College London)
Abstract: Let $X$ be an algebraic curve over $\mathbb{F}_q$ and $G$ be a reductive algebraic group over $\mathbb{F}_q(X)$. Under mild technical hypotheses we construct families of stacks over the moduli $\text{Sht}_{G, X, I}^{\mu_*}$ of bounded global $G$-shtuka (a small generalization of the stacks studied by Lafforgue and Varshavsky) which provide natural analogues of Igusa varieties in the function field setting. Our main result is an isomorphism between certain Igusa varieties associated to moduli of shtuka for reductive groups $G, G'$ which are related by an inner twist. Along the way we prove an almost-product formula computing the compactly supported cohomology of the special fibers of $\text{Sht}_{G, X, I}^{\mu_*}$ with trivial coefficients in terms of the cohomology of our Igusa stacks and a function-field analogue of Rapoport-Zink spaces constructed in previous work of Hartl and Arasteh Rad.
number theory
Audience: researchers in the topic
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