Abstracts

Toby Gee: Moduli stacks of $(\varphi, \Gamma)$-modules

I will review some properties of the moduli stacks of $(\varphi, \Gamma)$-modules which I recently constructed with Matthew Emerton.

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Philipp Habegger: Uniformity in Mordell-Lang for Curves

By Faltings's Theorem, the Mordell Conjecture, a smooth projective curve of genus at least 2 that is defined over the field of rational numbers has at most finitely many rational points. Vojta later gave a new proof which resulted in upper bounds on the number of rational points thanks to work of Bombieri, de Diego, Rémond, and others. I will speak about joint work with Ziyang Gao and Vesselin Dimitrov on a new bound for the number of rational points. The bound depends only on the genus of the curve and the rank of the Mordell-Weil group of the curve's Jacobian.

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Eugen Hellmann: On automorphy lifting for semi-stable representations

Automorphy lifting theorems aim to show that a $p$-adic global Galois representation that is unramified almost everywhere and de Rham at places dividing $p$ is associated to an automorphic representation, provided its reduction modulo $p$ is. In the past years there has been a lot of progress in the case of polarizable representations that are crystalline at $p$. In the semi-stable case much less is known (beyond the 2-dimensional case). I will explain how one can use results about $p$-adic automorphic forms in combination with the Taylor-Wiles method to deduce the semi-stable case from the crystalline case.

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Annette Huber: Exponential periods and o-minimality

(work in progress with Johan Commelin and Philipp Habegger) Exponential periods arise from the period isomorphism for irregular connections established by Hien and Roucairol. The underlying motivic theory was recently developed and studied by Fresan and Jossen. We give an equivalent characterisation as $\int_Ge^{-f}\omega$ with $G\subset \mathbf{C}^n$ semi-algebraic and satisfying additional properties, algebraic $f$ and $\omega$. This can be used to show that the absolute values of real and imaginary part of all effective exponential periods are volumes of sets definable in a certain o-minimal structure. These results are in analogy to a similar statement for classical periods and volumes of semi-algebraic sets. We take this to indicate that there is a deeper connection between the theory of periods and o-minimiality.

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Johannes Nicaise: Stable rationality of complete intersections

I will explain an ongoing project with John Christian Ottem to establish several new classes of stably irrational complete intersections. Our results are based on degeneration techniques and a birational version of the nearby cycles functor that was developed in collaboration with Evgeny Shinder.

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George Pappas: Some symplectic constructions for l-adic local systems over curves

I will discuss certain constructions for l-adic local systems over curves and for their deformations that are inspired by the symplectic theory of character varieties of surface groups and the Chern-Simons theory for 3-manifolds.

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Michael Rapoport: $p$-adic uniformization of unitary Shimura curves

Cherednik's theorem states that a Shimura curve attached to a quaternion algebra over a totally real field $F$ which is split at exactly one archimedean place and is ramified at some $p$-adic place admits $p$-adic uniformization by the Drinfeld $p$-adic upper half plane. This theorem is now almost 50 years old but is still not understood, one of the main difficulties being that outside the case $F=\mathbb Q$ the Shimura curve does not represent a moduli problem of abelian varieties. I will show that when the multiplicative group of the quaternion algebra is replaced by the group of unitary similitudes of a binary hermitian space, things become drastically better. This is joint work with S. Kudla and Th. Zink, and improves on results of P. Scholze (yes, this can be done!).

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Alexander Schmidt: The tame site of a scheme

[Joint work with K. Hübner] For a scheme $X$ of characteristic p>0, the p-part of its étale fundamental $\pi_1^{et}(X)$ is not well-behaved, i.e., it is not $A^1$-homotopy invariant. Therefore the tame fundamental group has been studied in positive and mixed characteristics. It would be helpful to have a tame Grothendieck topology whose associated fundamental group is the tame fundamental group. Such a topology would provide a tame cohomology theory and, in addition, higher tame homotopy groups. The latter seems to be even more desirable because the higher étale homotopy groups of affine varieties vanish in positive characteristic (Achinger, 2017). In this talk we give a definition of a tame site that hopefully will prove useful. Morever, we explain the relation to the tame site of adic spaces as introduced by K. Hübner in 2018.

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Georg Tamme: Excision in algebraic K-theory

The classical excision theorem of Bass, Milnor, and Murthy associates to a Milnor square of rings an exact sequence of algebraic K-groups starting with $K_1$. In this talk I will will explain that this sequence extends naturally to a long exact sequence which involves a new ring spectrum naturally associated with the original Milnor square. In fact, this result holds more generally for any so called localizing invariant as for example topological Hochschild homology and it easily implies and improves excision results of Suslin-Wodzicki, Cortinas, and Geisser-Hesselholt. I will also explain a method that allows us to determine the above new ring spectrum explicitly in several examples. This is joint work with Markus Land.

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Paul Ziegler: Geometric stabilization via $p$-adic integration

The fundamental lemma is an identity of integrals playing an important role in the Langlands program. This identity was reformulated into a statement about the cohomology of moduli spaces of Higgs bundles, called the geometric stabilization theorem, and proved in this form by Ngô. I will give an introduction to these results and explain a new proof of the geometric stabilization theorem, which is joint work with Michael Groechenig and Dimitri Wyss, using the technique of $p$-adic integration.

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