Date/Time : 26 Oct 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Arnaud Vanhaeke, Morningside Center

On the Galois side, Fontaine’s theory characterizes indecomposable semistable non-crystabelian p-adic representations in terms of a filtration, parametrized by the so-called Fontaine-Mazur higher L-invariant. In recent work, Z. Qian computed extension groups between locally analytic Steinberg representations for GL_d(Q_p) and introduced the corresponding Breuil-Schraen higher L-invariant. These invariants naturally give rise to complexes of locally analytic representations involving Steinberg representations. In ongoing joint work with Zicheng Qian and Benchao Su, motivated by the case of GL_2(Q_p), we show that the pro-etale complex of Drinfeld’s symmetric space, with coefficients in a p-adic local system, matches both the Fontaine-Mazur and the Breuil-Schraen higher L-invariants. This provides a geometric realization of the locally analytic special series for GL_d(Q_p). In the talk, I will state the main expected theorems and outline the proof, which proceeds through several reduction steps.

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Date/Time : 19 Oct 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : David Hansen, National University of Singapore

Motivated by the recent amazing discovery of a complex structure on the six-sphere, I will revisit some basic questions around the classification of smooth proper rigid analytic spaces, primarily in dimensions two and three. One theme of this exploration is that in the presence of one (or more) vanishing ell-adic (resp. p-adic) Betti numbers, the Grothendieck-Riemann-Roch theorem (resp. the Hirzebruch-Riemann-Roch theorem) puts strong constraints on the possible Hodge numbers.

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Date/Time : 12 Oct 2026 13:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Yap Jit Wu, National University of Singapore

Recently, Junyi Xie and Ziquan Yang gave a proof of Faltings’ isogeny theorem by using non-Archimedean equidistribution to reduce it to Tate’s isogeny theorem over finite fields. In this talk, I will present a proof, obtained by GPT and based on their work, that each isogeny class over a fixed number field contains only finitely many isomorphism classes. The argument again uses non-Archimedean equidistribution, this time reducing the problem directly to the corresponding finiteness statement over finite fields, where it is immediate. This leads to a new proof of the Shafarevich conjecture and, in turn, the Mordell conjecture. I will begin with a historical overview of the various proofs of the Mordell conjecture before explaining the new argument.

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Date/Time : 05 Oct 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Finn Wiersig, National University of Singapore

Regular holonomic D-modules form a particularly well-behaved class of systems of algebraic differential equations. Over the complex numbers, they are distinguished by two remarkable properties: the Riemann–Hilbert correspondence gives them a topological description in terms of perverse sheaves, and their derived category admits a six-functor formalism. In this talk, I will report on joint work in progress with Andreas Bode towards a p-adic analogue of this theory. Building on Diao–Lan–Liu–Zhu’s work on the p-adic Riemann–Hilbert correspondence, we introduce regular holonomic complexes of D-cap-modules on smooth rigid analytic varieties. We develop a six-functor formalism for this category, establishing stability under tensor products, internal Hom, duality, and inverse images, and discussing progress on direct images under suitable compactifiability conditions.

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Date/Time : 28 Sep 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Jack Thorne, University of Cambridge

The Langlands correspondence predicts the existence of Galois representations associated to certain automorphic representations of GL(n, AF). These are known to exist in the case that F is a CM field and the automorphic representation in question is regular algebraic (this includes, for example, the automorphic representations that should correspond to elliptic curves over imaginary quadratic fields). For arithmetic applications, it is essential to know that this automorphic to Galois dictionary satisfies compatibility with the local Langlands correspondence: in particular, when considering the restriction of the p-adic Galois representation to a p-adic decomposition group, the representation should be potentially semistable, in the sense of Fontaine, and its (semisimple) Weil—Deligne representation should be the one predicted by the local Langlands correspondence. This local-global compatibility has been proved in many cases when the p-adic Galois representation satisfies a global hypothesis (such as being `decomposed generic’). I will discuss a proof of the general case, and describe an application to the Bloch—Kato conjecture. This is joint work with A’Campo, Hevesi, and Whitmore.

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Date/Time : 14 Sep 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Jack Sempliner, University of California (Los Angeles)

We will discuss a new result on the p-adic geometry of Igusa stacks, which allows one to relate the p-adic geometry of Shimura varieties for a pair of groups $G, G’$ which are global extended pure inner forms. Using this geometric input and the spectral action of Fargues-Scholze, we will discuss how this input may be used to prove the existence of Hecke-equivariant cohomological “Jacquet-Langlands” maps between the cohomology (even with integral or torsion coefficients) of the two Shimura varieties of abelian type associated to $G, G’$.

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Date/Time : 07 Sep 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : David Hansen, National University of Singapore

I will formulate a collection of vanishing conjectures in the Fargues-Scholze setting, which describe the maps on suitable Grothendieck groups induced by Eisenstein functors, constant term functors, and the gluing functors between strata of Bun_G. I will also discuss the relationships among these conjectures, and explain how they imply a very general form of the Bertoloni Meli-Harris-Viehmann conjecture (which I will also recall in detail). I will then sketch a proof of the vanishing conjectures (and hence a proof of the generalized BMHV conjecture). Joint work in progress with Linus Hamann and Peter Scholze.

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Date/Time : 31 Aug 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : David Hansen, National University of Singapore

I will formulate a collection of vanishing conjectures in the Fargues-Scholze setting, which describe the maps on suitable Grothendieck groups induced by Eisenstein functors, constant term functors, and the gluing functors between strata of Bun_G. I will also discuss the relationships among these conjectures, and explain how they imply a very general form of the Bertoloni Meli-Harris-Viehmann conjecture (which I will also recall in detail). I will then sketch a proof of the vanishing conjectures (and hence a proof of the generalized BMHV conjecture). Joint work in progress with Linus Hamann and Peter Scholze.

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Date/Time : 24 Aug 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Alberto Vezzani, University of Milan

We present how the framework of rigid analytic motivic homotopy theory provides streamlined definitions of relative Hyodo-Kato cohomologies, constructibility results and comparison isomorphisms to de Rham cohomology, while also proving generalizations of ell-adic local monodromy theorems. This is work in progress with Fabrizio Andreatta and Federico Binda.

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