**F. Andreatta** (Università Statale di Milano)

*Title* : Katz type *p*-adic *L*-functions for primes *p* non-split in the CM field

*Abstract* : I will discuss a way to construct anticyclotmic *p*-adic *L*-functions attached to an elliptic eigenform *f* and an imaginary quadratic field *K*, interpolating *p*-adically the central critical values of the Rankin *L*-functions of *f* twisted by anticyclotomic characters of higher infinity type. I will also provide *p*-adic Gross-Zagier formulae for these *p*-adic *L*-functions. Here the prime *p* is assumed to be inert or ramified in *K*. The case that *p* is split is due to Katz (for Eisenstein series) and Bertolini-Darmon-Prasana (for cuspidal eigenforms). This is joint work with Adrian Iovita.

**R. Beuzart-Plessis** (CNRS, Marseille)

*Title* : A new proof of the Jacquet-Rallis fundamental lemma

*Abstract* : The Jacquet-Rallis fundamental lemma is a local identity between (relative) orbital integrals which originates from the relative trace formula approach to the Gan-Gross-Prasad conjecture for unitary groups and is a crucial ingredient in the recent results of W. Zhang on this conjecture. It was established soon after its formulation by Z. Yun in positive characteristic using the same geometric ideas as in Ngô's proof of the endoscopic fundamental lemma and transferred to characteristic 0 by J. Gordon by model-theoretic techniques. In this talk, I will present an alternative proof of this fundamental lemma in characteristic zero which is purely local and based on harmonic analytic tools.

**A. Caraiani** (Imperial College)

*Title* : On the geometry of the Hodge-Tate period morphism

*Abstract* : In this talk, I will describe joint work with Peter Scholze on the geometry
of the Hodge-Tate period morphism for perfectoid Shimura varieties. I will focus on how to relate the fibers of this morphism to perfectoid Igusa varieties, for the open Shimura variety but also for its minimal and toroidal compactifications. I will also mention applications to the cohomology of Shimura varieties and beyond.

**J. Fresán** (École Polytechnique)

*Title* : Irregular Hodge filtration and eigenvalues of Frobenius

*Abstract* : The de Rham cohomology of a connection of exponential type on an algebraic variety carries a filtration, indexed by rational numbers, that generalises the usual Hodge filtration on the cohomology of the trivial connection. I will explain a few results and conjectures relating this filtration to exponential sums over finite fields.

**D. Gaitsgory** (Harvard)

*Title* : Igrushka à la Drinfeld-Lafforgue

*Abstract* : In his ground-breaking work, V. Lafforgue constructed an action of a certain
commutative algebra on the space of automorphic forms, and he showed
that the spectrum of this algebra is (a subset of) the set of semi-simple local
systems for the Langlands dual group. The construction goes through
considering the totality of multi-Galois modules obtained from cohomology of "shtukas".
In this talk we will explain a certain general categorical principle, which makes
this construction automatic, but with one caveat: it literally works in the setting
of constructible sheaves over C, but not for l-adic sheaves over F_q. So in a sense
we are considering a toy model of Lafforgue's situation.
The word "igrushka" means "toy thingy", to contrast with "shtuka" (=thingy).
This is a joint with with D. Kazhdan, N. Rozenblyum and Y. Varshavsky.

**W. T. Gan** (National University of Singapore)

*Title* : Relative character identities and theta correspondence

*Abstract* : In the context of the relative Langlands program, Yiannis Sakellaridis has recently defined a transfer map for certain spaces of test functions on rank 1 spherical varieties. In this talk, we explain how his transfer can be defined from the point of view of theta correspondence, the corresponding fundamental lemmas checked and the desired relative character identities established. This is joint work with Xiaolei Wan.

**Q. Guignard** (ENS & IHÉS)

*Title* : Geometric *l*-adic local factors

*Abstract* : I will explain how to give a cohomological definition of epsilon factors
for *l*-adic sheaves over a henselian trait of positive equicharacteristic
distinct from *l*. The resulting formula is reminiscent of the cohomological
construction by Katz of the *l*-adic Swan representation, and involves Gabber-Katz extensions as well.
These local factors provide a product formula for the determinant of the
cohomology of an *l*-adic sheaf on a curve over a field of positive
characteristic distinct from *l*. When the base field is finite, this
specializes to the classical theory of Dwork, Langlands, Deligne, and Laumon.

**G. Henniart** (Université Paris-Sud)

*Title* : On the conductors of Galois representations of *p*-adic fields

*Abstract* : This is about joint work with Colin J. Bushnell.
Let F be a finite extension of Q_*p*, and G_F the absolute Galois group of F.
A continuous representation r of G_F, of dimension n>0, has a
Swan exponent Sw(r), and a normalized one s(r)=Sw(r)/n. If R is an algebraic
representation of GL_n(C), s(R o r) is at most s(r), when r is irreducible.
But if r is irreducible and minimal in the sense that Sw(xr)>=Sw(r) for all
characters x of G_F, then when R is the adjoint representation, we have
2s(R o r)>=s(r); the proof uses the Langlands correspondence. We conjecture
a similar lower bound when R is irreducible of dimension >1.

**A. Ichino** (Kyoto University)

*Title* : Hodge classes and the Jacquet-Langlands correspondence

*Abstract* : I will report a joint work with Kartik Prasanna on the relation
between Langlands functoriality and the theory of algebraic cycles. In
particular, I will explain our approach to constructing a Hodge class
in the case of GL(2) over a totally real field and its inner forms.

**T. Koshikawa** (RIMS, Kyoto University)

*Title* : Remarks on the cohomology of compact unitary Shimura varieties

*Abstract* : Caraiani and Scholze proved that the generic part of the
cohomology of certain compact unitary Shimura varieties vanishes outside
the middle degree. Prior to their result, Boyer obtained a stronger result
about the vanishing range in the Harris-Taylor case, which also treats
non-generic classes. I will explain Boyer's result and its consequences,
and an attempt to generalize Boyer's result to more general Shimura
varieties using the approach of Caraiani-Scholze.

**Timo Richarz** (TU Darmstadt)

*Title* : Cohen-Macaulayness of parahoric local models

*Abstract* : The singularities arising in the mod-*p*-reduction of Shimura varieties with parahoric level structure
can be described in terms of linear algebra via so called local models. By the work of G. Pappas and X. Zhu,
these models are known to be normal with reduced special fibre in many cases. Further, they conjecture
that the local models and hence the corresponding integral models of Shimura varieties are Cohen-Macaulay
as well. In the special case of unramified groups with Iwahori level, this conjecture was proven by X. He. In my talk,
I explain a proof of this conjecture in general if *p* > 2. This is joint work with T. Haines.

**B. Schraen** (Université Paris-Sud)

*Title* : Infinitesimal characters and application to canonical dimension

*Abstract* : I will prove some local-global compatibility for the action of the Harish-Chandra center on some Hecke eigenspaces in *p*-adic completed cohomology spaces. I will explain how this leads to a non trivial upper bound for the canonical dimension of these spaces as representations of a *p*-adic Lie group. This is a joint work with Gabriel Dospinescu and Vytautas Paskunas.

**K. Shimizu** (UC Berkeley)

*Title* : Constancy of generalized Hodge-Tate weights of a *p*-adic local system

*Abstract* : A *p*-adic local system on a rigid analytic variety can be regarded as a family of local Galois representations parametrized by the variety. In this talk, we will discuss the constancy of generalized Hodge-Tate weights of a *p*-adic local system. This is one instance of rigidity phenomena of geometric families of Galois representations in contrast to arithmetic families in the Galois deformation theory.

**Y. Tian** (Université de Strasbourg)

*Title* : On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives

*Abstract* : In this talk, I will report some recent progress on the rank 0 zero case of Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives attached to certain cuspidal automorphic representations of GL_n*GL_{n+1} over a CM field. More precisely, we show that, under some technical assumptions, the non-vanishing of certain Rankin-Selberg L-function implies the vanishing of the corresponding Selmer group. The main ingredients for the proof include Gan-Gross-Prasad conjecture for unitary groups, the geometry of unitary Shimura varieties and Kolyvagin?s machinery of Euler systems. This is a joint work in progress with Yifeng Liu, Liang Xiao, Wei Zhang and Xinwen Zhu.

**T. Tsuji** (University of Tokyo)

*Title* : Coefficients in integral *p*-adic Hodge theory

*Abstract* : This is joint work with Matthew Morrow. Some more details will be discussed in
the lectures by Morrow in the summer school. In the integral *p*-adic Hodge theory
by Bhatt, Morrow, and Scholze, a new cohomology theory: the Ainf cohomology
intertwines the integral p-adic cohomology groups: de Rham, crystalline, and *p*-adic étale
via its specializations. In this talk, we introduce a theory of coefficients,
which may be regarded as a relative theory of Breuil-Kisin-Fargues modules, and
study its A_inf cohomology and specializations. Admissibility with respect to relative
A_crys and relation to *p*-divisible groups will also be discussed.

**M.-F. Vigneras** (Sorbonne Université)

*Title* : Representations over non-algebraically closed fields

*Abstract* : Fintzen proved that any irreducible smooth cuspidal admissible representation of a tame reductive *p*-adic group over any algebraically closed field of characteristic *l* different from *p* is compactly induced if *p* does not divide the order of the Weyl group of G. As an application of a general theorem with Henniart on scalar extensions, we prove that Fintzen's result is still true when the field is finite, and with Herzig and Koziol that any reductive *p*-adic group admits an irreducible smooth cuspidal admissible representation over any field of characteristic *p*.

**S. Zhang** (Princeton)

*Title* : Admissible pairing of algebraic cycles

*Abstract* : For a smooth and projective variety X over a global field of dimension n with an adelic polarization, I will propose some canonical local and global height pairings for two cycles Y , Z of pure codimension p, q satisfying p + q = n + 1. I will also discuss some applications to abelian varieties and Shimura varieties.

**X. Zhu** (Caltech)

*Title* : Kloosterman crystals for reductive groups

*Abstract* : I will first review the relationship between the classical Bessel equation and the Kloosterman sum. Then I will discuss the generalizations of this story for arbitrary reductive groups using ideals from the geometric Langlands program, based on the works by Frenkel-Gross, Heinloth-Ngo-Yun, myself, and the recent joint work in progress with Daxin Xu.