# PAS Seminar

Date: every Wednesday, 10:30-11:30 (unless stated otherwise)

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45

Organizer: Dennis Eriksson

## Wednesday, January 22

10:30 - 11:30 Francois Charles: Frobenius distribution for pairs of elliptic curves

Abstract: We will discuss a proof of the following result: Let E and E' be two elliptic curves over a number field. Then there exist infinitely many primes p such that the reductions mod p of E and E' are geometrically isogenous. This result was previously known in the function field case, and can be seen as a partial analog of a density criterion for Hodge loci due to M. Green.

## Tuesday, February 18

10:30 - 11:30 Giuseppe Ancona: Shimura varieties as moduli spaces of motives, after Milne

Abstract: This is an expository and informal talk. We will start by recalling the different notions of motives available and also explain the variant using Deligne's absolute hodge cycles (and give some applications to periods of the latter). Then, following Milne, we will explain how "many" Shimura varieties can be described as moduli spaces of motives.

This class of Shimura varieties is larger than the more familiar PEL type, which are described as moduli of abelian varieties with some given extra endomorphisms

## Thursday, February 20

10:30 - 11:30 Giuseppe Ancona: On the motive of a commutative algebraic group

Abstract: We will show that the motive of a commutative algebraic group (for instance a semiabelian variety) has a canonical Kunneth decomposition in the category of Voevodsky's motives. This generalizes the results of Deninger, Murre and Kunnemann on the motive of an abelian variety. Our techniques uses Kimura finiteness. This is a joint work with Steven Enright-Ward and Annette Huber. (The talk is independent with the one on Tuesday, February 18.)

## Wednesday, February 26

exceptionally at Raum 1.016 (Lipschitz-Saal), Mathematik-Zentrum, Endenicher Allee 60

10:30 - 11:30 Hélène Esnault: Étale fundamental groups and stratifications

Abstract: Over the field of complex numbers, the étale fundamental group controls the local systems, or equivalently the regular singular flat connections, as the topological fundamental group is finitely generated (Malčev-Grothendieck). We describe analogies in characteristic p>0, for stratifications (or equivalently O-coherent D-modules, or equivalently Frobenius divided sheaves, or equivalently crystals in the infinite site). In particular, we list some non-trivial examples of smooth non-proper varieties which are simply connected in characteristic p>0.

## Wednesday, March 12

exceptionally at basement lecture hall

10:30 - 11:30 Haoran Wang: On the geometry and cohomology of the tame cover of Drinfeld's upper half space

Abstract: I will discuss the geometry and cohomology of the tamer cover of the Drinfeld upper-half space in which some Deligne-Lusztig variety naturally appears. This gives in particular a purely local study of the local Langlands correspondence and the Jacquet-Langlands correspondence for depth 0 supercuspidal representations.

## Tuesday, April 8

13:30 - 14:30 Jan Bruinier: Modularity of special cycles and formal Fourier Jacobi series

Abstract: A famous theorem of Gross, Kohnen, and Zagier states that Heegner divisors on modular curves are the coefficients of weight 3/2 modular forms with values in the first Chow group. A far reaching conjecture of Kudla predicts more generally that special cycles of codimension r on Shimura varieties associated to orthogonal groups of signature of signature (n,2) are the coefficients of Siegel modular forms of genus r with values in the r-th Chow group. We report on older results on the subject, and on more recent work by W. Zhang, M. Raum and myself. These results involve modularity theorems for vector valued formal Fourier-Jacobi series.

## Wednesday, April 9

10:30 - 11:30 Jens Funke: Singular theta lifts and the construction of Green currents for the unitary group

Abstract: In this talk I will explain how one can introduce a singular theta lift of Borcherds type for the unitary group U(p,q) which is adjoint to the geometric Kudla-Millson theta lift for this group. This is analogous to the dual pair SL_{2} x O(n,2) previously considered by Bruinier and myself. In particular, this leads to the construction of Green currents codimension q cycles.

15:00 - 16:00 Frédéric Déglise: Mixed motives following Beilinson and Voevodsky

16:30 - 17:30 Frédéric Déglise: Etale motives and the rigidity theorem

## Wednesday, April 16

10:30 - 11:30 Stephan Ehlen: Lattices with many Borcherds products

Abstract: We prove that there are only ﬁnitely many isometry classes of even lattices L of signature (2,n) for which the space of cusp forms of weight 1+n/2 for the Weil representation of the discriminant group of L is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of L can be realized as the divisor of a Borcherds product. We obtain similar classiﬁcation results in greater generality for ﬁnite quadratic modules.

16:30 - 17:30 Luis Garcia: Periods of automorphic forms and singular theta lifts

Abstract: Consider two different holomorphic Hecke eigenforms f_{i} ∈ π_{i}, i=1,2 of weight 2 on a Shimura curve X/Q. We will first discuss Beilinson's conjecture relating the image of the complex regulator map from a higher Chow group with the special value of L(π_{1}×π_{2},s) at s=0. Then we will review Borcherds' construction of meromorphic functions on X with divisors supported on CM points. Finally we will show how to use the theta correspondence to compute, assuming that the f_{i} have full level and up to an archimedean zeta integral, the period integrals arising as regulators of higher Chow cycles constructed using Borcherds' functions.

## Wednesday, April 23

16:30 - 17:30 Geordie Williamson: Decomposition theorem - perverse sheaves