Motivic Fundamental Group of CM Elliptic Curves and Geometry of Bianchi Hyperbolic Threefolds

• Alexander Goncharov (Yale University & IHES)

Room: Centre de conférences Marilyn et James Simons Let N be an ideal in the ring O of gaussian integers. We consider the action of the motivic Galois group on the motivic fundamental group of the elliptic curve with CM by the ring O, punctured at the N-torsion points, and relate it to the geometry of the Bianchi threefold obtained by taking the quotient of the hyperbolic space by a congruence subgroup of GL(2,O) determined by the ideal N.

Purity for Flat Cohomology

• Kęstutis Česnavičius (Université Paris-Sud)

Room: Centre de conférences Marilyn et James Simons The absolute cohomological purity conjecture of Grothendieck proved by Gabber ensures that on regular schemes étale cohomology classes of fixed cohomological degree extend uniquely over closed subschemes of large codimension. I will discuss the corresponding phenomenon for flat cohomology. The talk is based on joint work with Peter Scholze.

Application of Functional Transcendence to Counting Rational Points on Curves

• Ziyang Gao (IMJ-PRG)

Room: Centre de conférences Marilyn et James Simons With Philipp Habegger we recently proved a height inequality, using which one can bound the number of rational points on 1-parameter families of curves in terms of the genus, the degree of the number field and the Mordell-Weil rank (but no dependence on the Faltings height). This gives an affirmative answer to a conjecture of Mazur for pencils of curves. In this talk I will give a blueprint to generalize this method to an arbitrary family of curves. In particular I will focus on: (1) how establishing a criterion for the Betti map to be immersive leads to the desired bound; (2) how to apply mixed Ax-Schanuel to establish such a criterion. This is work in progress, partly joint with Vesselin Dimitrov and Philipp Habegger.

Sur le programme de Langlands p-adique

• Pierre Colmez (IMJ-PRG)

Room: Centre de conférences Marilyn et James Simons Le programme de Langlands p-adique a pour origine les travaux de Serre et de Hida sur les familles p-adiques de formes modulaires et les représentations galoisiennes qui leur sont associées. Mazur, en collaboration avec Gouvéa et avec Coleman, a joué un grand rôle dans la maturation de ce programme, mais celui-ci n'a toujours pas de forme vraiment définitive. Je présenterai des travaux récents en lien avec ce programme.

New Rational Points of Algebraic Curves over Extension Fields

• Barry Mazur (Harvard University)

Room: Centre de conférences Marilyn et James Simons For L/K an extension of fields and V an algebraic variety over K say that V is Diophantine Stable for the extension L/K if V(L) = V(K). That is, if `V acquires no new rational points’ when one makes the field extension from K to L. I will describe some recent results joint with Karl Rubin regarding Diophantine Stability and give a survey of related recent statistics, heuristics, and conjectures.