**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)

**Organizers **

A. Abbes (CNRS & IHÉS),
C. Breuil (CNRS, Orsay),
M. Harris (Columbia University),

A. Mézard (Sorbonne Université),
T. Saito (University of Tokyo)

**Organized in partnership with **

** Registration on the web is mandatory** both for the Summer School (17-21 June 2019) and the Conference (24-28 June 2019).

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