Let L(E/Q, s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the central value L(E, 1, χ) of twists of L(E/Q, s) by Dirichlet characters χ. We discuss the vanishing and non-vanishing frequencies of these values as χ ranges over characters of fixed order greater than 2. We also examine thee square-free part of the algebraic part of L(E/F, 1) for abelian fields F/Q when these values are non-zero.

Abstract: In modern algebraic geometry we encounter the notion of derived intersection of subschemes. This is a sophisticated way to encode what happens when two subschemes Y_1 and Y_2 of a given scheme X intersect non-transversely. The classical intersection multiplicity can be extracted from the derived intersection.

We prove cases of Rietsch mirror conjecture that the quantum connection for projective homogeneous varieties is isomorphic to the pushforward D-module attached to Berenstein-Kazhdan geometric crystals. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. In particular we link the purity of Berenstein-Kazhdan crystals to the Ramanujan property of certain Hecke eigensheaves. The isomorphism of D-modules comes from global rigidity results where a Hecke eigenform is determined by its local ramification. We reveal

This talk is in natural in the context of the Zagier conjecture. We express values of the Kronecker double series at CM points in terms of values some version (Bloch-Wigner) of dilogarithm in algebraic numbers. As zeta-function of the Hilbert class field of quadratic field can be expressed as combination of the Kronecker double series at CM points my result gives explicit form of the Zagier conjecture. My technique is rather elementary and the proof is based on the introduction some new function (elliptic (1,1)-logarithm) and comparisons with it.

In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields. I will discuss the definition of a "short interval" on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting.

In this talk we discuss some aspects concerning the arithmetic of systems of quadratic forms. This includes a result on the frequency of counterexamples to the Hasse principle for del Pezzo surfaces of degree four (joint work with J. Jahnel), and a result on the representability of integers by systems of three quadratic forms (joint work with L. B. Pierce and M. M. Wood).

Abstract: Motivated by understanding the action of Hecke operators on special sub-varieties of Shimura varieties, we consider the simplest possible case: the action of Hecke operators on the j-line, namely on the moduli space of elliptic curves, and in particular the action on singular moduli. Our interest is in this action considered in the p-adic topology. The emerging picture is surprisingly rich and the answers involve Serre-Tate coordinates, the Gross-Hopkins period map and finally involves random walks on GL_n. This is joint work with Payman Kassaei (King's College).

Abstract: In 1981, Sansuc obtained a formula for Tamagawa numbers of reductive groups over number fields, modulo some then unknown results on the arithmetic of simply connected groups which have since been proven, particularly Weil's conjecture on Tamagawa numbers over number fields. One easily deduces that this same formula holds for all linear algebraic groups over number fields. Sansuc's method still works to treat reductive groups in the function field setting, thanks to the recent resolution of Weil's conjecture in the function field setting by Lurie and Gaitsgory.

A natural question is whether there exists a continuous p-adic analogue for the classical local Langlands correspondence for GL_n(F) . Namely, for a finite extension F of Q_p, we want to associate continuous p -adic representations of GL_n(F) to n-dimensional p-adic representations of the Weil group of F. The particular case, where F=Q_p and n=2 , is now known. One of the main tools for establishing this correspondence was the existence of GL_2(Q_p)-invariant norms in certain representations of GL_2(Q_p).

Abstract: This talk will be about joint work with Eyal Goren about the structure of Picard modular surfaces at a prime p which is inert in the underlying quadratic imaginary field. The main tool for studying the bad reduction of Shimura varieties is the theory of local models (due to de Jong and Rapoport-Zink). Our results concern global geometric questions which go beyond the theory of global models. For example, we are able to count supersingular curves on the Picard surface. We also study certain foliations in its tangent bundle that have not been studied before, and

In this expository talk we review the Selberg trace formula for compact Riemannian surfaces, and its application to the Weyl law via the heat kernel.

Abstract: my talk will be devoted to a basic theory of extensions of complete real-valued fields L/K. Naturally, one says that L is topologically-algebraically generated over K by a subset S if L lies in the completion of the algebraic closure of K(S). One can then define topological analogues of algebraic independence, transcendence degree, etc. These notions behave much more wierd than their algebraic analogues. For example, there exist non-invertible continuous K-endomorphisms of the completed algebraic closure of K(x). In my talk, I will tell which part

Abstract: Inseparable morphisms proved to be an important tool for the study of algebraic varieties in characteristic p. In particular, Rudakov-Shafarevitch, Miyaoka and Ekedahl have constructed a dictionary between "height 1" foliations in the tangent bundle and "height 1" purely inseparable quotients of a non-singular variety in characteristic p. In a joint work with Eyal Goren we use this dictionary to study the special fiber S of a unitary Shimura variety of signature (n,m), m < n, at a prime p which is inert in the underlying imaginary quadratic field. We

I discuss some class of function of several elliptic variables, this functions generalize multiple polylogarithms of D. Zagier. I show some applications of developed formalism. This is a joint work with F. Brown.