Humboldt Arithmetic Geometry Seminar

Humboldt University Berlin, Summer 2022

Time: Tuesdays 13:15 - 14:45

Venue: Room 3.006, Rudower Chaussee 25, 12489 Berlin

— See also the Algebraic Geometry Seminars at HU and FU

03.05.2022 Dougal Davis (U Edinburgh)
Title: Mixed Hodge modules on flag varieties and representations of real reductive groups
Abstract: In this talk, I will give a gentle introduction to applications of mixed Hodge modules in the representation theory of real reductive groups. Since the work of Beilinson-Bernstein, Kazhdan-Lusztig and Lusztig-Vogan, it has been understood how to realise representations of real groups in terms of twisted D-modules on flag varieties, and how weight filtrations coming from the corresponding mixed sheaves over a finite field can be used to gain strong control over their structure. Much more recently, it was suggested by Schmid and Vilonen that passing instead to mixed Hodge modules reveals more information about representations than is accessible over a finite field, through Hodge filtrations and polarisations. I will sketch this broad story, including the main conjecture of Schmid-Vilonen relating Hodge structures to unitarity, and, as time permits, explain some concrete results appearing in recent joint work of myself and Vilonen (arXiv:2202.08797) on the computation of the Hodge structures in Kazhdan-Lusztig-Vogan theory and their connection with the unitarity algorithm of Adams-van Leeuwen-Trapa-Vogan.
10.05.2022 Marco Flores (HU Berlin)
Title: A cohomological approach to formal Fourier-Jacobi series
Abstract: Siegel modular forms admit various expansions, one of the most important being the Fourier-Jacobi expansion. Algebraically, these expansions take the form of a series whose coefficients are Jacobi forms satisfying a certain symmetry condition. One then poses the following modularity question: does every formal series of that shape come from a Siegel modular form? Bruinier and Raum answered the question affirmatively, over the complex numbers, in 2014. In this talk I will consider this question over the ring of integers, and reformulate it as a matter of cohomological vanishing. I will present a weaker version of the desired cohomological vanishing, and a result highlighting how special the case of genus g=2 potentially is.
24.05.2022 Sarah Zerbes (ETH Zürich)
Title: Euler systems and the Birch—Swinnerton-Dyer conjecture for abelian surfaces
Abstract: Euler systems are one of the most powerful tools for proving cases of the Bloch-Kato conjecture, and other related problems such as the Birch and Swinnerton-Dyer conjecture. I will recall a series of recent works (variously joint with Loeffler, Pilloni, Skinner) giving rise to an Euler system in the cohomology of Shimura varieties for \( \mathrm{GSp}(4)\), and an explicit reciprocity law relating the Euler system to values of \( L\)-functions. I will then explain recent work with Loeffler, where we use this Euler system to prove new cases of the BSD conjecture for modular abelian surfaces over \( \mathbb{Q} \), and for modular elliptic curves over imaginary quadratic fields.
07.06.2022 Morten Lüders (U Hannover)
Title: Milnor K-theory of p-adic rings and motivic cohomology
Abstract: We explain a joint work with Matthew Morrow on \(p\)-adic Milnor K-theory. Our main theorem is a comparison of mod \(p^r\) Milnor K-groups of \(p\)-henselian local rings with the Milnor range of a newly defined syntomic cohomology theory by Bhatt, Morrow and Scholze. We begin by putting our result into context. Then we sketch the proof which builds on an analysis of a filtration on Milnor K-groups and a new technique called the left Kan extension from smooth algebras.
14.06.2022 Christopher Deninger (U Münster)
Title: Dynamical systems for arithmetic schemes
Abstract: For any arithmetic scheme \(X\) we construct a continuous time dynamical system whose periodic orbits come in compact packets that are in bijection with the closed points of \( X \). All periodic orbits in a given packet have the same length equal to the logarithm of the order of the residue field of the corresponding closed point. For \( X = \mathrm{Spec}\, \mathbb{Z}\) we get a dynamical system whose periodic orbits are related to the prime numbers. The construction uses new ringed spaces which are constructed from rational Witt vector rings. In the zero-dimensional case we recover a construction of Kucharczyk and Scholze who realized certain Galois groups as étale fundamental groups of ordinary topological spaces. A p-adic version of our construction turns out to be closely related to the Fargues-Fontaine curve of p-adic Hodge theory.
21.06.2022 José Ignacio Burgos Gil (ICMAT)
Title: Chern-Weil theory and Hilbert-Samuel theorem for semi-positive singular toroidal metrics on line bundles
Abstract: In this talk I will report on joint work with A. Botero, D. Holmes and R. de Jong. Using the theory of b-divisors and non-pluripolar products we show that Chern-Weil theory and a Hilbert Samuel theorem can be extended to a wide class of singular semi-positive metrics. We apply the techniques relating semipositive metrics on line bundles to b-divisors to study the line bundle of Siegel-Jacobi forms with the Peterson metric. On the one hand we prove that the ring of Siegel-Jacobi forms of constant positive relative index is never finitely generated, and we recover a formula of Tai giving the asymptotic growth of the dimension of the spaces of Siegel-Jacobi modular forms.
28.06.2022 Fabrizio Barroero (U Roma Tre)
Title: On the Zilber-Pink conjecture for complex abelian varieties and distinguished categories
Abstract: The Zilber-Pink conjecture is a very general statement that implies many well-known results in diophantine geometry, e.g., Manin-Mumford, Mordell-Lang and André-Oort. I will report on recent joint work with Gabriel Dill in which we proved that the Zilber-Pink conjecture for a complex abelian variety A can be deduced from the same statement for its trace, i.e., the largest abelian subvariety of A that can be defined over the algebraic numbers. This gives some unconditional results, e.g., the conjecture for curves in complex abelian varieties (over the algebraic numbers this is due to Habegger and Pila) and the conjecture for arbitrary subvarieties of powers of elliptic curves that have transcendental j-invariant. While working on this project we realised that many definitions, statements and proofs were formal in nature and we came up with a categorical setting that contains most known examples and in which (weakly) special subvarieties can be defined and a Zilber-Pink statement can be formulated. We obtained some conditional as well as some unconditional results.
12.07.2022 No Seminar (ICM sectional workshop Number Theory and Algebraic Geometry at ETH)
19.07.2022 Ya Deng (U Lorraine)

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