Abstracts

Johannes Anschütz: $\mathbb C_p$-semilinear Galois representations of $p$-adic fields via Cartier-Witt stacks

This talk will report on joint work in progress with Arthur-César Le Bras and Ben Heuer. The aim of this project is a possible description of the category of $\mathbb{C}_p$-semilinear Galois representations of a p-adic field $K$ via vector bundles on Cartier-Witt stacks of rings of integers of finite extensions of $K$.

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Ana Caraiani: On the modularity of elliptic curves over imaginary quadratic fields

I will survey how to prove the modularity of elliptic curves defined over the rational numbers, as pioneered by Wiles and Taylor-Wiles and completed by Breuil, Conrad, Diamond and Taylor. I will also mention the case of elliptic curves defined over real quadratic fields, more recently completed by Freitas, Le Hung and Siksek. I will then explain why the case of imaginary quadratic fields is qualitatively different from the previous ones. Finally, I will discuss joint work in progress with James Newton, where we prove a local-global compatibility result in the crystalline case for Galois representations attached to torsion classes occurring in the cohomology of locally symmetric spaces. This has an application to the modularity of elliptic curves over imaginary quadratic fields, which also builds on recent work of Allen, Khare and Thorne.

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Laurent Fargues: Extension du domaine de la lutte

I will explain how to define an extended Kottwitz set that contains the usual one and whose basic elements allows us to reach all inner forms of a given reductive $p$-adic group, even if its center is not connected. This relies on the work of Kaletha. In particular I give a geometric interpretation of its gerb in terms of the curve. One can then define an extended stack of $G$-bundles that contains the one we defined and studied with Scholze as a closed / open substack and whose points are the extended Kottwitz set. At the end this allows us to formulate a geometrization conjecture for any reductive group even if it's not an extended pure inner form of its quasisplit inner form.

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Toby Gee: An introduction to the categorical $p$-adic Langlands program

I will attempt to explain what the categorical $p$-adic Langlands program is, and how it connects to more classical questions about congruences between modular forms.

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Eugen Hellmann: On the K-theory of the stack of L-parameters for GL_n

The categorical version of the local Langlands conjecture of Fargues-Scholze asserts, in the case of $GL_n$, that there is an equivalence between a certain category of $\ell$-adic sheaves on the stack of rank $n$ vector bundles on the Fargues-Fontaine curve and a certain category of coherent sheaves on the stack of $n$-dimensional Weil-Deligne representations. We will give some evidence for this conjecture by showing that there is an isomorphism between the K-groups (in particular: on Grothendieck groups) of these categories. In particular we will (rather explicitly) compute the K-theory of the stack of $n$-dimensional Weil-Deligne representations.

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Moritz Kerz: The Picard-Lefschetz formula for normal crossings

In the study of semi-stable degeneration of Lefschetz pencils one is led to a generalization of the classical Picard-Lefschetz formula for certain perverse sheaves on normal crossing spaces. In the talk I will recall the formalism of nearby cycle and vanishing cycle functors and I will explain how Hodge theory allows one to obtain the normal crossing Picard-Lefschetz formula. Joint work with A. Beilinson and H. Esnault.

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Sabrina Pauli: Arithmetic enrichments in enumerative geometry

Results from motivic homotopy theory allow to answer questions in enumerative geometry over an arbitrary base field. In the talk I will explain how this works and give some examples.

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

Cherednik discovered that a Shimura curve attached to a quaternion algebra over a totally real field F admits p-adic uniformization by the Drinfeld p-adic upper half plane if is ramified at some $p$-adic place. This observation is now almost 50 years old and there is finally the hope for a published proof. One of the main difficulties is that outside the case $F=\mathbb Q$ the Shimura curve does not represent a moduli problem of abelian varieties. This difficulty disappears when the multiplicative group of the quaternion algebra is replaced by the group of unitary similitudes of a binary hermitian space, but then other interesting problems arise. I will discuss work of Boutot-Zink, of Kudla-Rapoport-Zink, and of Scholze-Weinstein.

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Michael Temkin: Functorial resolution of singularities: comparison of the methods

Until very recently we knew different descriptions of essentially the same method to canonically resolve singularities (in characteristic zero), but in the past few years a few new methods were found. Perhaps, this is already a good time to compare the methods, point out principles which are common to all of them, and pin down what was the main ingredient which changes from one method to another and essentially determines the method. In my talk I will suggest such a systematization and will give a panoramic view of the known desingularization algorithms.

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Xinwen Zhu: Isogenies of mod $p$ CM abelian varieties via main theorem of complex multiplication

The main theorem of complex multiplication describes how automorphisms of $\mathbb C$ act on CM abelian varieties and their torsion points. I will explain an extension of this theory to describe the action of $p$-quasi-isogenies between mod $p$ reductions of CM abelian varieties on prime-to-$p$ torsion points. Joint work with Liang Xiao.

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