20. April, 15:00 -16:00 (st) | Raju Krishnamoorthy | Rank 2 local systems and abelian varieties. Abstract: Inspired by a theorem of Corlette-Simpson on rigid local systems of rank 2, we formulate a conjecture that irreducible rank 2 local systems on smooth varieties over finite fields come from families of abelian varieties. We will explain partial results on this conjecture. Time permitting, we indicate several elements of the proofs. The techniques used include Drinfeld's first work on the Langlands correspondence over finite fields, Katz's notion of a space-filling curve, Poonen's Bertini theorems over finite fields, and, most importantly, the pigeonhole principle. This is joint work with Ambrus Pál |
4. Mai, 15:00 - 16:00 (st) | Fei Ren | Cycle complex and coherent duality. Abstract: Let k be a perfect field of characteristic p>0 and X be a separated scheme of finite type over k. In this talk, we will introduce a complex K_{n,X,log} via Grothendieck's coherent duality theory following Kato and build up a chain map from Bloch's cycle complex mod p^n to K_{n,X,log}. We show that this map is a quasi-isomorphism in the etale topology, and when k is algebraically closed, it is also a quasi-isomorphism in the Zariski topology. This result provides us with a perspective of higher Chow groups of zero cycles with \Z/p^n coefficients from coherent duality theory, and thus introduces the tools from coherent cohomology to the study of cycles. In particular, when k=\bar k, these higher Chow groups are the Cartier invariant part of the hypercohomology of some coherent dualizing complex, so they can be reasonably regarded as an analog of log forms. As corollaries, we deduce certain vanishing, étale descent, Galois descent properties as well as invariance under rational resolutions for higher Chow groups of 0-cycles with \Z/p^n-coefficients. We will also reproduce a finiteness result of Geisser. |
11.Mai, 14:15-15:15 (st) | Souvik Goswami | Higher Arithmetic Chow Groups Abstract: We give a new definition of higher arithmetic Chow groups for smooth projective varieties defined over a number field, which is similar to Gillet and Soulé's definition of arithmetic Chow groups. We also give a compact description of the intersection theory of such groups. An interesting consequence of this theory is the definition of a height pairing between two higher algebraic cycles of `complementary' codimensions, whose real regulator class is zero. This description agrees with Beilinson's height pairing for the classical arithmetic Chow groups. We also give examples of the higher arithmetic intersection pairing in dimension zero that is given by the Bloch-Wigner dilogarithm functions. This is based on a joint work with Jose Ignacio Burgos Gil from ICMAT, Spain. |
18. Mai, 14:15- 15:15 (st) | Holger Kammeyer | L2-invariants, profinite rigidity, and arithmetic groups. Abstract: We give a leisurely introduction to L2-invariants and survey some topics in which they have recently played a roll: fibering of 3-manifolds, profiniteness questions, torsion growth in homology, and algebraic properties of group rings. |
1. Juni 14:15 - 15:15 (st) | Yingying Wang | Cohomology of the structure sheaf on the smooth compactification of Deligne-Lusztig varieties for $GL_n$ . |
8. Juni | Katharina Hübner | Logarithmic differentials on discretely ringed adic spaces . Abstract: The object of interest in this talk is a certain subsheaf $\Omega^+_X$ of the sheaf of differentials $\Omega_X$ of a discretely ringed adic space $X$ over a field $k$. It is expected to play an important role in the tame cohomology theory of~$X$. The first part will be dedicated to an introduction to discretely ringed adic spaces. We will then define $\Omega^+_X$ using K\"ahler seminorms and establish a relation with logarithmic differentials. Finally we study the case where $X = Spa(U,Y)$ for a scheme $Y$ over $k$ and an open subscheme $U$ such that the corresponding log structure on $Y$ is log smooth. It turns out that $\Omega^+_X(X)$ equals $\Omega^{log}_{(U,Y)}(U,Y)$. |
15. Juni | Julian Brough | Towards the Alperin-McKay conjecture for simple groups of type C . Abstract: The local-global counting conjectures aim to understand the deeper connection that exists between a group and it local subgroups, that is centralisers and normalises of to p-subgroups. For a prime p, the Alperin-McKay conjecture relates the height zero characters of a p-block of a finite group and its Brauer correspondent. A reduction theorem reduces this conjecture to validating the inductive Alperin-McKay condition for finite quasi-simple groups. In this talk I will discuss current on going work which contributes to the larger goal of validating this condition in Type C. |
22. Juni | Damiano Rossi | Character Triple Conjecture for Groups of Lie Type . Abstract: Dade’s Conjecture is an important conjecture in representation theory of finite groups. It implies most of the, so called, global-local conjectures. In 2017, Späth introduced a strengthening of Dade’s Conjecture, called the Character Triple Conjecture, which describes the Clifford theory of corresponding characters. Moreover, she proved a reduction theorem, namely that if her conjecture holds for every quasisimple group, then Dade’s Conjecture holds for every finite group. Extending ideas of Broué, Fong and Srinivasan on generalized Harish-Chandra theory, we provide a strategy to prove the Character Triple Conjecture for quasisimple groups of Lie type in the nondefining characteristic. |
6. Juli | Grétar Amazeen | Correspondence actions on Hodge cohomology over a 1-dimensional base . |
13. Juli | Daniel Litt | Arithmetic representations and non-commutative Siegel linearization . Abstract: I'll explain joint work with Borys Kadets, explaining how to prove the following theorem. Let X be a curve over a finitely generated field k, and let \ell be a prime different from the characteristic of k. Then there exists N=N(X,\ell) such that any semisimple arithmetic representation of \pi_1(X_{\bar k}) into GL_n(\overline{\mathbb{Z}_\ell}), which is trivial mod \ell^N, is in fact trivial. This extends previous work of mine from characteristic zero to all characteristics. The main new idea is to introduce techniques from dynamics; in particular a non-commutative version of Siegel's linearization theorem. |