# Institut Camille Jordan

UMR 5208

## Dans les sept prochains jours

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 à venir Séminaire d'arithmétique à Lyon   François Brunault -    jeudi 19 septembre à 11:00 UMPA, ENS de Lyon, Amphi A   Régulateurs explicites et fonctions L de formes modulaires   Résumé : Soutenance d'habilitation Séminaire Combinatoire et Théorie des Nombres ICJ   Maxim Mornev -    jeudi 19 septembre à 14:00 ICJ, Université Lyon 1, Bât. Braconnier, salle séminaire 2   Shtuka cohomology and special values of Goss L-functions (1)   Résumé : Statement of the conjecture, overview of the proof and introduction to shtuka cohomology.   Abstract.  Taelman discovered an analog of BSD conjecture for Drinfeld modules and proved it for the coefficient ring F_q[t]. His methods do not generalize easily to more complicated rings. A different approach to Taelman's BSD conjecture is provided by the theory of shtuka cohomology. This approach allows one to treat all the coefficient rings in a uniform way. It leads to a rather conceptual proof of the conjecture which resonates well with equivariant Tamagawa number conjecture for motives. My aim is to explain the shtuka-theoretic proof of Taelman's analog of the BSD conjecture. The course can be naturally divided into three parts: 1. Statement of the conjecture, overview of the proof and introduction to shtuka cohomology. 2. Shtuka models of Drinfeld modules and their cohomology. 3. Regulator theory and trace formula for elliptic shtukas. Séminaire Combinatoire et Théorie des Nombres ICJ   Maxim Mornev -    vendredi 20 septembre à 10:00 ICJ, Université Lyon 1, Bât. Braconnier, salle séminaire 2   Shtuka cohomology and special values of Goss L-functions (2)   Résumé : Shtuka models of Drinfeld modules and their cohomology.   Abstract.  Taelman discovered an analog of BSD conjecture for Drinfeld modules and proved it for the coefficient ring F_q[t]. His methods do not generalize easily to more complicated rings. A different approach to Taelman's BSD conjecture is provided by the theory of shtuka cohomology. This approach allows one to treat all the coefficient rings in a uniform way. It leads to a rather conceptual proof of the conjecture which resonates well with equivariant Tamagawa number conjecture for motives. My aim is to explain the shtuka-theoretic proof of Taelman's analog of the BSD conjecture. The course can be naturally divided into three parts: 1. Statement of the conjecture, overview of the proof and introduction to shtuka cohomology. 2. Shtuka models of Drinfeld modules and their cohomology. 3. Regulator theory and trace formula for elliptic shtukas. Séminaire Géométries ICJ   Ariyan Javanpeykar -    vendredi 20 septembre à 10:30 ICJ, 112   Arithmetic hyperbolicity   Résumé :  The Green-Griffiths-Lang-Vojta conjectures relate the hyperbolicity of an algebraic variety to the finiteness of sets of "rational points" over number fields. For instance, it suggests a striking answer to the fundamental question "Why do some polynomial equations with integer coefficients have only finitely many solutions in the integers?". Namely, if the zeroes of such a system define a hyperbolic variety, then this system should have only finitely many integer solutions. In this talk I will explain how to verify some of the algebraic, analytic, and arithmetic predictions this conjecture makes. The talk will be a mixture of arithmetic geometry and complex geometry. Séminaire Combinatoire et Théorie des Nombres ICJ   Maxim Mornev -    lundi 23 septembre à 13:00 ICJ, Université Lyon 1, Bât. Braconnier, salle Fokko du Cloux   Shtuka cohomology and special values of Goss L-functions (3)   Résumé :  Regulator theory and trace formula for elliptic shtukas.   Abstract.  Taelman discovered an analog of BSD conjecture for Drinfeld modules and proved it for the coefficient ring F_q[t]. His methods do not generalize easily to more complicated rings. A different approach to Taelman's BSD conjecture is provided by the theory of shtuka cohomology. This approach allows one to treat all the coefficient rings in a uniform way. It leads to a rather conceptual proof of the conjecture which resonates well with equivariant Tamagawa number conjecture for motives. My aim is to explain the shtuka-theoretic proof of Taelman's analog of the BSD conjecture. The course can be naturally divided into three parts: 1. Statement of the conjecture, overview of the proof and introduction to shtuka cohomology. 2. Shtuka models of Drinfeld modules and their cohomology. 3. Regulator theory and trace formula for elliptic shtukas. Séminaire Combinatoire et Théorie des Nombres ICJ   Pooneh Afshari Joo -    mardi 24 septembre à 10:45 ICJ, Université Lyon 1, Bât. Braconnier, salle 112   A la recherche d'une nouvelle version des identités de Gordon   Résumé : Une partition d'un nombre entier positif $n$ est une suite $\Lambda : (\lambda_1 \geq \cdots \geq \lambda_l)$ telle que $\lambda_1 + \cdots + \lambda_l = n$. Les entiers qui apparaissent sont appelés les parties de $\Lambda$. Ma recherche est centrée sur l'étude des partitions des nombres entiers et les identités entre elles. On étudie ce type d'identités en utilisant la relation entre les séries génératrices des partitions et les séries de Hilbert-Poincaré des algèbres graduées associées à un objet important de la géométrie algébrique : l'espace des arcs. Une de ces identités est la suivante : Théorème. (La première identité de Rogers-Ramanujan) Le nombre de partitions d'un nombre naturel $n$ dont les parties sont congruentes à 1 ou 4 modulo 5 est égal au nombre de partitions de $n$ dont les parties ne sont ni égales ni consécutives. En utilisant des idéaux différentiels et des méthodes venant de la théorie d'espace des arcs, nous trouvons une famille d'identités de type Rogers-Ramanujan. Ensuite, nous énonçons une conjecture qui pourrait ajouter un nouveau membre aux identités de Gordon qui sont une généralisation d'identités de Rogers-Ramanujan.