Seminar on Arithmetic Geometry

Jonathan Miles (University of Frankfurt)

We explain how to glue sheaves on the moduli stack of G-bundles on the Fargues-Fontaine curve. In the case of prime-to-p torsion coefficients, the category D_ét(Bun_G) can be thought of as an approximation of the automorphic data appearing in the geometrization of the local Langlands correspondence due to Fargues-Scholze. The stratification of Bun_G arising from the Harder-Narasimhan slope formalism on G-isocrystals yields a semi-orthogonal decomposition of D_ét(Bun_G) into the derived categories of smooth representations of inner forms of Levi subgroups of G. Between such categories there is a (partial) six functor formalism that can be used to compute how sheaves arising on a quasi-compact open substack interact with sheaves on higher strata via nearby cycles functors, which can be interpreted as a derived analogue of Jacquet restriction functors for parabolic subgroups of G (up to inner twisting). We eventually restrict to G=GL_2 and to sufficiently nice coefficients (notably this includes an algebraic closure of F_\ell and Z/\ell^n Z for almost all \ell prime to p), and we will explain how these computations fundamentally reduce to the étale cohomology of p-adic analogues of locally symmetric spaces, such as the Bruhat-Tits building and moduli spaces of mixed p-adic Hodge structures.

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