## Seminario di Geometria Aritmetica, Motivi e Teoria dei numeri

This year advanced undergraduates, PhD students and Post-Docs will give a talk. There will be a short course given by Bruno Kahn.

Short Course by Bruno Kahn "Finite generation theorems in arithmetic and algebraic geometry"
Abstract: I will give an overview of the main known finite generation theorems in arithmetic geometry: K-groups of rings of integers, and for higher-dimensional schemes, units and Picard group. Then a sketch (or detailed, time permitting) proof of the Mordell-Weil and Néron-Severi theorems. Finally I shall explain how to deduce the Lang-Néron theorem from the latter in a simple way.

The course will take place on April 12, 13, 14 and 15 in Aula dottorato starting from 14.30 to 16.30.

April 22, 14.30-16.30, Aula C, Cristiana Bertolin Deligne's conjecture on extensions of 1-motives
Abstract: We introduce the notion of extension of 1-motives. Using the dictionary between strictly commutative Picard stacks and complexes of abelian sheaves concentrated in degrees -1 and 0, we check that an extension of 1-motives induces an extension of the corresponding strictly commutative Picard stacks.We prove then the following conjecture of Deligne: if M and N are two 1-motives, then any mixed realization, which is an extension of the mixed realization of M by the mixed realization of N, and which comes from geometry, is isomorphic to the mixed realization of a 1-motive extension of M by N

April 29, 14.30-16.30, Aula C, Alberto Vezzani Geometry over $\F_1$
Abstract: We introduce the notion of schemes over F_1. In particular, we provide a proof of an equivalence between Deitmar's and Toën-Vaquié's notions of schemes over F_1, establishing a symmetry with the classical case of schemes.

May 6, 14.30-16.30, Aula C, Marco Seveso L-invariants, monodromy modules and Darmon cycles attached to modular forms (joint with Victor Rotger)
Abstract: Let f be a modular form on the Shimura curve X attached to an indefinite quaternion Q-algebra of discriminant D and to an Eichler order of level N. Suppose that the weight of f is even and >=2. We explain how to attach to a modular form which is new at p an L-invariant defined in the spirit of Darmon-Greenberg-Orton. We then explain how to attach to the modular form f a monodromy module D_f. When the L-invariant equals the Mazur L-invariant the above monodromy module is isomorphic to the one attached to the Deligne p-adic representation of f. In this case a construction of so called Darmon cycles generalizing the Stark-Heegner points is available.

May 13 No seminar

May 20, 14.30-16.30, Aula C, Riccardo Brasca P-adic modular forms of non-integral weight over Shimura curves
Abstract: After a review of the theory of Shimura curves over a totally real field and their interpretation as moduli spaces, we will introduce the space of classical and p-adic modular forms, of integral weight, over such curves. Then we will give an equivalent, but less natural, definition of these spaces, that allows us to consider also the case of non-integral weights. Hecke operators will also be introduced.

May 27, 14.30-16.30, Aula C, Somayeh Habibi The Chow motive of Hilbert scheme of points on a surface
Abstract: we shall first introduce the Hilbert scheme of points and punctual Hilbert scheme, we also define Hilbert chow morphism. Then we state some general properties of them. We introduce the natural stratification of Hilb^n(X). And then after giving the definition of refined Gysin map, we compute the Chow group of Hilbert scheme of point on a surface, sketchy. Finally we will briefly speak about the motives of Hilbert scheme of points on a surface

June 10, 14.30-16.30, Aula C Rodolfo Venerucci p-adic regulators in Hida theory
Abstract: Let E be an elliptic curve over Q ordinary at a prime p and let f_{\infty} be the p-adic Hida family of modular forms attached to E. We recall the construction of a p-adic analytic Hida L-function L_p(k) which interpolates the central complex L values of the elements in f_{\infty}. Recently Bertolini and Darmon proved a formula which relates (in some cases) the second derivative of L_p(k) at k=2 to the formal group logarithm of a Heegner point in E(Q). We discuss an attempt to understand this formula in the framework of the p-adic Birch and Swinnerton-Dyer conjectures via a regulator term coming from Nekovar duality for Selmer complexes. <

Wednesday June 16, 14.30-16.30, Aula Dottorato Remi Lodh Construction of some p-adic local systems
Abstract: Fontaine's theory associates p-adic Galois representations to filtered phi-modules. I'll explain how to construct p-adic local systems from certain p-adic analogues of variations of Hodge structures (more precisely: "filtered (phi,nabla)-modules"), if the variation is generically quasi-unipotent in some sense and if the Newton polygon of the variation is constant. This can be viewed as a special case of a possible generalization of Fontaine's theory, and is based on previous work of Katz, Brinon, and others.