Oberseminar Winter term 2023/24
Alle Interessenten sind herzlich eingeladen!
The seminar takes place on Thursday, starting at 4:45pm. The duration of each talk is about 60 minutes.
Everybody who’s interested is welcome to join.
|Sally Gilles (IAS, Princeton)
|On compactly supported p-adic proétale cohomology of analytic varieties (online)
|Michael Rapoport (Univ. Bonn)
|Fundamental lemma and Arithmetic Fundamental lemma for the whole spherical Hecke algebra
|Otto Overkamp (Univ. Düsseldorf)
|A proof of Chai’s conjecture
|Q&A session for the research seminar
|David Schwein (Bonn)
|New supercuspidals at bad primes
|Matei Toma (IECL, Nancy)
|Schur polynomials and the Hodge-Riemann bilinear relations
|Aprameyo Pal (HRI Prayagraj)
|Multivariable (phi,Gamma)-modules and Iwasawa theory
|Wiesława Nizioł (Sorbonne Université, Paris / CNRS)
|Hidden structures on de Rham cohomology of p-adic analytic varieties
|Roman Fedorov (University of Pittsburgh)
| Mixed characteristic case of the conjecture of Grothendieck and Serre
|Stefano Vigni (Università di Genova)
|On the p-part of the Tamagawa number conjecture for motives of modular forms
|Lukas Braun (Univ. Innsbruck)
|Coregularities, Irregularities, and fundamental groups
|Giuseppe Ancona (Université de Strasbourg)
|Lefschetz standard conjecture for some lagrangian fibrations
Sally Giles: On compactly supported p-adic proétale cohomology of analytic varieties
I will define the p-adic proétale cohomology with compact support for rigid analytic varieties and present some properties that it satisfies. In particular, I will discuss comparison theorems between the (compactly supported versions of) proétale and de Rham cohomologies.This is based on a joint work with P. Achinger and W. Niziol.
Meeting ID: 532 517 3167
Password: the first prime >200.
Otto Overkamp: A proof of Chai’s conjecture
The base change conductor is an invariant which measures the failure of a semiabelian variety to have semiabelian reduction. It was conjectured by Chai that this invariant is additive in certain exact sequences. I shall report on recent joint work with Takashi Suzuki which implies this conjecture. Time permitting, I shall also discuss counterexamples to a generalisation of Chai’s conjecture.
Michael Rapoport: Fundamental lemma and Arithmetic Fundamental lemma for the whole spherical Hecke algebra
The FL and the AFL for the unit element in the spherical Hecke algebra of the unitary group are recent theorems (of Z. Yun, W. Zhang, R. Beuzart-Plessis, resp. of W. Zhang, A.Mihatsch/W. Zhang, Z. Zhang). Here FL is a statement in $p$-adic harmonic analysis, whereas AFL is a statement in arithmetic geometry. I will discuss the extension of these statements to an arbitrary element in the spherical Hecke algebra. This is joint work with C. Li and W. Zhang.
David Schwein: New supercuspidals at bad primes
Supercuspidal representations are the elementary particles in the representation theory of reductive p-adic groups and play an important role in number theory as local factors of cuspidal automorphic representations. Constructing such representations explicitly, via (compact) induction, is a longstanding open problem. Although the problem has been solved for large p, a solution remains out of reach in general. I’ll discuss work in progress joint with Jessica Fintzen towards constructing some of these missing supercuspidals when p is (very!) small.
Matei Toma: Schur polynomials and the Hodge-Riemann bilinear relations
The Hard Lefschetz Theorem and the Hodge-Riemann bilinear relations show how special the algebraic topology of complex projective manifolds is and the particular role played herein by ample divisor classes. In this talk we will present recent joint work with Julius Ross, in which we discuss Schur classes of ample vector bundles and show that they enjoy the Hard Lefschetz property and the Hodge-Riemann property in the same way as powers of ample divisor classes. Some applications to positivity in the intersection theory of algebraic cycles will be equally touched upon.
Aprameyo Pal: Multivariable (phi, Gamma)-modules and Iwasawa theory
In the first half of the talk, I recall the motivation and construction for multivariable (Phi, Gamma)-modules. In the second half of the talk (joint work, partly in progress, with Gergely Zabradi), I show how to pass to Robba-style versions via overconvergence. The group cohomology can also be computed from the generalized Herr complex over the multivariate Robba ring. If time permits, I will indicate how the analytic Iwasawa cohomology (computed also from a generalized Herr-complex) will hopefully be useful for the (re)formulation of Bloch-Kato exponential maps in this setting.
Wiesława Nizioł: Hidden structures on de Rham cohomology of p-adic analytic varieties
I will survey what we know about extra structures (Hodge filtration, Frobenius, monodromy) appearing on de Rham cohomology of analytic varieties over local fields of mixed characteristic.
Roman Fedorov: Mixed characteristic case of the conjecture of Grothendieck and Serre on torsors.
A conjecture of Grothendieck and Serre predicts that for an integral
regular scheme S and a reductive S-group scheme G, a G-torsor is
Zariski locally trivial if it is rationally trivial. The conjecture is
known for schemes over fields. Recently, K. Cesnavicius and the
speaker proved the conjecture in the case when S is unramified over
Spec Z and the group scheme satisfies a certain isotropy condition.
The talk will be devoted to these results.
After carefully formulating the conjecture and explaining its
importance and consequences, I will outline the general proof
strategy. Then I will describe the difficulties arising in the mixed
characteristic case and the ways to go around them. I will assume only
very basic knowledge of algebraic geometry.
Georg Tamme: On the vanishing of negative K-groups
The negative algebraic K-groups of a scheme encode some information about its geometry. For example, they vanish when the scheme is regular. In this talk, I will discuss the following vanishing result for negative K-groups: For any quasi-compact, quasi-separated scheme X, the K-groups K_i(X) vanish for i less than the negative valuative dimension of the scheme. The valuative dimension is a variant of the Krull dimension introduced by Jaffard in 1960 which still behaves well for non-Noetherian schemes. The main new ingredient in the proof is a descent result for algebraic K-theory under abstract blowups in the non-Noetherian setting. This is joint work with Shane Kelly and Shuji Saito and also based on earlier joint work with Markus Land, and with Moritz Kerz and Florian Strunk.
Stefano Vigni: On the p-part of the Tamagawa number conjecture for motives of modular forms
The Tamagawa number conjecture of Bloch—Kato and Fontaine—Perrin-Riou predicts formulas for special values of L-functions of motives and can be seen as a vast generalisation of the Birch—Swinnerton-Dyer conjecture for abelian varieties. In this talk, I will describe results on the p-part (for a prime number p) of the Tamagawa number conjecture for motives of modular forms of higher (even) weight in analytic rank 1. This is joint work with Matteo Longo.
Lukas Braun: Coregularities, Irregularities, and fundamental groups
In this talk, based on joint work with Fernando Figueroa, I will investigate the connections between a rather new invariant, the coregularity c, and almost-abelianity of fundamental groups of varieties of klt type.
We conjecture a precise correspondence between the rank of an abelian subgroup of finite index in the fundamental group, and the coregularity, similar to the correspondence between the (augmented) irregularity and the Albanese variety in the case of trivial canonical class. We prove the conjecture in the cases c=0,1,2 (and in arbitrary dimension) by revealing the connection to varieties with trivial canonical class of dimension c.
Giuseppe Ancona: Lefschetz standard conjecture for some lagrangian fibrations
The Lefschetz standard conjecture predicts the existence of some specific algebraic classes on the square of an algebraic variety, namely the inverse of the Lefschetz operator should be induced by an algebraic correspondence. We will show this conjecture for the hyper-kähler varieties constructed by Laza-Saccà-Voisin. This will be a special case of a general criterion which will tell that hyper-kähler varieties admitting lagrangian fibrations « of Ngô type » satisfy this conjecture.
I will start by recalling the conjecture and the known results.Then I will discuss how the conjecture behaves under fibration and explain why several difficulties appear. Finally I will explain why in the setting of lagrangian fibrations these difficulties can be treated. A crucial input is Ngô’s support theorem, which I will recall as well.
This is a joint work with Mattia Cavicchi, Robert Laterveer and Giulia Saccà.