Institut Camille Jordan

Soutenance d'habilitation

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.

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.

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.

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.