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# Résultats de recherche

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## G. Freixas i Montplet - Automorphic forms and arithmetic intersections (part 2)

In these lectures I will focus on the Riemann-Roch theorem in Arakelov geometry, in the specific context of some simple Shimura varieties. For suitable data, the cohomological part of the theorem affords an interpretation in terms of both holomorphic and non-holomorphic modular forms. The formula relates these to arithmetic intersection numbers, that can sometimes be evaluated through variants of the first Kroenecker limit formula. I will first explain these facts, and then show how the Jacquet-Langlands correspondence allows to relate arithmetic intersection numbers for ... Voir la vidéole (46m14s)

## G.Freixas i Montplet - Automorphic forms and arithmetic intersections (part 3)

In these lectures I will focus on the Riemann-Roch theorem in Arakelov geometry, in the specific context of some simple Shimura varieties. For suitable data, the cohomological part of the theorem affords an interpretation in terms of both holomorphic and non-holomorphic modular forms. The formula relates these to arithmetic intersection numbers, that can sometimes be evaluated through variants of the first Kroenecker limit formula. I will first explain these facts, and then show how the Jacquet-Langlands correspondence allows to relate arithmetic intersection numbers for ... Voir la vidéole (1h29m51s)

## E. Peyre - Slopes and distribution of points (part1)

The distribution of rational points of bounded height on algebraic varieties is far from uniform. Indeed the points tend to accumulate on thin subsets which are images of non-trivial finite morphisms. The problem is to find a way to characterise these thin subsets. The slopes introduced by Jean-Benoît Bost are a useful tool for this problem. These lectures will present several cases in which this approach is fruitful. We shall also describe the notion of locally accumulating subvarieties which arise when one considers rational points of ... Voir la vidéole (1h2m42s)

## E. Peyre - Slopes and distribution of points (part2)

The distribution of rational points of bounded height on algebraic varieties is far from uniform. Indeed the points tend to accumulate on thin subsets which are images of non-trivial finite morphisms. The problem is to find a way to characterise these thin subsets. The slopes introduced by Jean-Benoît Bost are a useful tool for this problem. These lectures will present several cases in which this approach is fruitful. We shall also describe the notion of locally accumulating subvarieties which arise when one considers rational points of ... Voir la vidéole (55m57s)

## E. Peyre - Slopes and distribution of points (part3)

The distribution of rational points of bounded height on algebraic varieties is far from uniform. Indeed the points tend to accumulate on thin subsets which are images of non-trivial finite morphisms. The problem is to find a way to characterise these thin subsets. The slopes introduced by Jean-Benoît Bost are a useful tool for this problem. These lectures will present several cases in which this approach is fruitful. We shall also describe the notion of locally accumulating subvarieties which arise when one considers rational points of ... Voir la vidéole (1h28m48s)

## E. Peyre - Slopes and distribution of points (part4)

The distribution of rational points of bounded height on algebraic varieties is far from uniform. Indeed the points tend to accumulate on thin subsets which are images of non-trivial finite morphisms. The problem is to find a way to characterise these thin subsets. The slopes introduced by Jean-Benoît Bost are a useful tool for this problem. These lectures will present several cases in which this approach is fruitful. We shall also describe the notion of locally accumulating subvarieties which arise when one considers rational points of ... Voir la vidéole (1h1m14s)

## A. von Pippich - An analytic class number type formula for PSL2(Z)

For any Fuchsian subgroup Γ⊂PSL2(R) of the first kind, Selberg introduced the Selberg zeta function in analogy to the Riemann zeta function using the lengths of simple closed geodesics on Γ∖H instead of prime numbers. In this talk, we report on a formula that determines the special value at s=1 of the derivative of the Selberg zeta function for Γ=PSL2(Z). This formula is obtained as an application of a generalized Riemann-Roch isometry for the trivial sheaf on ¯¯¯¯¯¯¯¯¯¯¯Γ∖H, equipped with the Poincaré metric. This is joint ... Voir la vidéole (1h3m1s)

## Y. Tang - Exceptional splitting of reductions of abelian surfaces with real multiplication

Chavdarov and Zywina showed that after passing to a suitable field extension, every abelian surface A with real multiplication over some number field has geometrically simple reduction modulo p for a density one set of primes p. One may ask whether its complement, the density zero set of primes p such that the reduction of A modulo p is not geometrically simple, is infinite. Such question is analogous to the study of exceptional mod p isogeny between two elliptic curves in the recent work of Charles. ... Voir la vidéole (50m46s)