Summer School: June 17-21, 2019

F. Andreatta (Università Statale di Milano)
Title : Faltings heights of abelian varieties with complex multiplication
Abstract : After introducing Shimura varieties of orthogonal type, their Heegner divisors and the so called CM (complex Multiplication) points, I will review a conjecture of Buinier-Kudla-Yang which provides explicit formulas for their arithmetic intersection. I will show that they imply an averaged version of a conjecture of Colmez on the height of CM abelian varieties.

R. Beuzart-Plessis (CNRS, Marseille)
Title : Introduction to the Gan-Gross-Prasad and Ichino-Ikeda conjectures
Abstract : The aim of this mini-course will be to present the so-called global Gan-Gross-Prasad and Ichino-Ikeda conjectures which, in broad terms, relate certain explicit integrals of automorphic forms (called 'automorphic periods') to special values of (automorphic) L-functions. Once properly introduced, we will survey recent progress on these conjectures, most notably by W. Zhang, as well as remaining open problems.

M. Morrow (CNRS, Paris)
Title : Recent developments in integral p-adic cohomology
Abstract : This mini-course will introduce the audience to some recent developments in integral p-adic Hodge theory, originating from the cohomology theory A\Omega introduced in the 2016 article "Integral p-adic Hodge theory". This provides a natural interpolation between de Rham, crystalline, and p-adic étale cohomology. It may be constructed in several different fashions: either as a modification of Galois/étale cohomology, or via topological cyclic homology, or most recently (and most generally) in its guise as a canonical q-deformation of de Rham cohomology via prisms (work in progress by Bhatt-Scholze). Depending on developments before the workshop, the course will most likely focus on some aspects of the relative theory, for example how modules with q-connection correspond to older notions in p-adic Hodge theory such as Faltings? generalised representations and relative Fontaine modules (joint with T. Tsuji).

T. Saito (University of Tokyo)
Title : Characteristic cycle of a constructible sheaf
Abstract : We discuss basic ingredients in the definition of the singular support and the characteristic cycle of a constructible sheaf on a smooth scheme over a field of positive characteristic. We also discuss their main properties including the index formula, functoriality for pull-back and push-forward etc.

B. Schraen (CNRS, Paris)
Title : Companion forms for p-adic automorphic forms
Abstract : Let f be a eigenform on a definite unitary group. A companion form of f is a p-adic eigenform which has same prime to p Hecke eigenvalues than f. Companion forms can be non classical and can be of weight different from the weight of f. In the lectures, I will explain how companion forms of f can be predicted by the Galois representation associated to f when f has a level prime to p and give a sketch of the proof in some particular cases, following a work in collaboration with Breuil and Hellmann. The proof is linked to a local analogue for trianguline Galois representations. Here is a provisional program for the three hours :
Lecture 1 : statement of the result on companion forms.
Lecture 2 : trianguline local representations and trianguline variety.
Lecture 3 : a sketch of proof of the global result.

Lectures (1 hour each)

A.-C. Le Bras (Universität Bonn), H. Hu (MPIM, Bonn) J. Lin (Universität Duisburg-Essen),
L. Mocz (Princeton), D. Xu (Caltech), E. Yang (Peking University)

Conference : June 24-28, 2019

Speakers

A. Beilinson (University of Chicago), A. Caraiani (Imperial College), L. Fargues (CNRS, Paris),
J. Fresán (École Polytechnique), D. Gaitsgory (Harvard), W. T. Gan (National University of Singapore),
Q. Guignard (ENS & IHÉS), G. Henniart (Orsay), A. Ichino (Kyoto University), M. Kim (Oxford),
T. Koshikawa (RIMS, Kyoto University), M. Rapoport (Universität Bonn), K. Shimizu (UC Berkeley),
Y. Tian (Universität Bonn), T. Tsuji (University of Tokyo), M.-F. Vigneras (Sorbonne Université),
S. Zhang (Princeton), X. Zhu (Caltech)