HKR theorem and residue sequences for logarithmic Hochschild homology

• Federico Binda (Università di Milano Statale)

Room: Aula Magna Using a geometric definition of logarithmic Hochschild homology of derived pre-log rings, we construct an André-Quillen type spectral sequence and show a logarithmic version of the Hochschild-Kostant-Rosenberg theorem. We use this to show that (log) Hochschild homology is representable in the category of log motives. Among the applications, we deduce a residue sequence for Hochschild homology involving blow-ups of log schemes, generalising results of Rognes-Sagave-Schlichtkrull. This is a joint work with Tommy Lundemo, Doosung Park and Paul Arne Østvær.

Homotopy theory of adic spaces and applications

• Alberto Vezzani (Università di Milano Statale)

Room: Aula Magna In this talk, we will discuss some recent advances in the theory of motivic rigid analytic geometry. In particular, we show how to define and study a relative de Rham cohomology for adic spaces in mixed characteristic and how to extend this construction to the equi-characteristic p case (taking values on the relative Fargues-Fontaine curve). Finiteness results, and the relation to "tilting" and to classical cohomology theories are also discussed. As an application, we give a proof of the p-adic weight-monodromy conjecture for smooth projective hypersurfaces. These results are part of joint works with J. Ayoub and M. Gallauer, with A.-C. Le Bras and with F. Binda and H. Kato.

Endoscopy for $\mathsf{GSp}(4)$ and rational points on elliptic curves

• Rodolfo Venerucci (Università di Milano Statale)

Room: Aula Magna I will report on a joint work in progress with F. Andreatta, M. Bertolini, and M. A. Seveso, aimed at studying the equivariant Birch and Swinnerton-Dyer conjecture for rational elliptic curves twisted by certain 4-dimensional modular Artin representations. Our setting is rich enough to embrace the study of the BSD conjecture for rational elliptic curves over abelian extensions of quadratic fields. Our ultimate goal is to construct non-trivial Selmer classes (or more ambitiously rational points) when their existence is predicted by the BSD conjecture. These arise as p-adic limits of distinguished (motivic) classes in the arithmetic p-adic étale cohomology of the product of a Siegel threefold and two modular curves. Our construction uses the endoscopy for $\mathsf{GSp}(4)$ in a crucial way.

Differential operators on spaces of p-adic modular forms

• Leonardo Fiore (Università di Milano Statale)

Room: Aula Magna We discuss the possibility of constructing differential operators on spaces of p-adic Siegel modular forms by deriving the action of suitable infinitesimal groups on the Igusa tower. In particular, we present the construction in the case of modular curves, which was first proposed by S. Howe in 2020, and discuss the possibility of extending it to the Siegel threefold, by working at Klingen level on the locus of p-rank greater or equal to 1.