Abstracts for Conference on "Arithmetic Geometry", July 22 - 26, 2024

Formal-analytic arithmetic surfaces are arithmetic analogues of germs of formal surfaces along a proper curve over some base field, in the same way as arithmetic surfaces are analogues of projective algebraic surfaces over some base field. It is possible to develop a  calculus  on formal-analytic arithmetic surfaces, involving Hermitian vector bundles and suitably defined arithmetic intersection numbers à la Arakelov. This calculus admits applications to finiteness results concerning the étale fundamental group of arithmetic surfaces, notably of integral models of modular curves.

This is a report on work in progress with Beuzart-Plessis and Thorne. Let G be a quasi-split group over a local field K of positive characteristic p. We show, conditionally on the existence of a version of the twisted trace formula adequate for stable cyclic base change, that every irreducible parameter of the Weil group of K with values in the L-group of G is the image of a (necessarily) supercuspidal representation of G(K) under the local parametrization constructed by Genestier-Lafforgue  and Fargues-Scholze.

There are two different ways to construct families of ordinary p-adic Siegel modular forms. One is by p-adically interpolating classes in Betti cohomology, first introduced by Hida and then given a more representation-theoretic interpretation by Emerton. The other is by p-adically interpolating classes in coherent cohomology, once again pioneered by Hida and generalised in recent years by Boxer and Pilloni. I will explain these two constructions and then discuss joint work in progress with James Newton and Juan Esteban Rodríguez Camargo that aims to compare them.

I will describe the objects one obtains by descent from $G$-bundles on a tame Galois cover of a given curve, answering a question raised by Grothendieck in his Bourbaki talk in 1956 on Weil's "memoire paper" from 1938. Joint work with George Pappas.