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ABSTRACT: In this talk I will present joint work with E. Goren (McGill) concerning the theory of Hilbert modular forms in characteristic p. I will describe the kernel and the image of the q-expansion map and prove the existence of filtration for Hilbert modular forms. I will define operators U, V and Q_{c} and study the variation of the filtration under these operators. In particular, I will prove that every ordinary eigenform has filtration in a prescribed box of weights. Our methods are geometric -- comparing holomorphic Hilbert modular forms with rational functions on a moduli scheme with level-p structure, whose poles are supported on the non-ordinary locus.
ABSTRACT: Let K be a field complete for a discrete valuation and with algebraically closed residue field of positive characteristic p. We will consider a non degenerate pairing between the first (flat) cohomology group of an abelian variety A over K and the fundamental group of the dual abelian variety A'. This pairing extends to the p-primary components a pairing introduced by Shafarevich in 1962. We relate this pairing with Grothendieck's pairing between the component groups of A and A'.
ABSTRACT: Let q be a nonzero complex number. One usually calls q-difference operator the operator j_{q}(f)(x)=f(qx) or the associated q-derivation d_{q}(f)(x)=(f(qx)-f(x))/(q-1)x, both acting on a convenient ring of functions. The relation d_{q}x^{n}= (1+q+...+q^{n-1})x^{n-1} shows intuitively that d_{q} ® d/dx when q® 1: this phenomenon is known as confluence. In a recent paper J. Sauloy proves the confluence of the so-colled Birkhoff matrix to the complex monodromy.
We will introduce p-adic q-difference equations and their first properties: namely their weak Frobenius structure and transfert theorems. These results are motivated by the p-adic confluence to differential equations.
ABSTRACT: Suppose X is an arithmetic surface and let G be a finite group which acts tamely on X. We define Y to be the quotient scheme X/G and we let V be a representation of G over \overlineQ. Associated to this information, we can define an L-function L(Y,V,s) which is conjectured to satisfy a certain functional equation involving the conductor and the number e(Y,V). Calculating these e-constants is very difficult in general, but under additional hypotheses the situation can greatly simplify. My research assumes that V is an orthogonal representation, and in that case reduces the calculation of e(Y,V) to a calculation about how the representation acts at only a finite number of points.
ABSTRACT: Berthelot has constructed a p-adic cohomology theory, called rigid cohomology, for which he was able to prove finite dimensionality for any smooth variety. His theory also allows coefficients in an overconvergent F-isocrystal; on a curve, this more general finite dimensionality was proved by Crew assuming a certain conjecture on p-adic differential equations. This conjecture is now a theorem; we'll describe how, using fibrations in curves, it implies finite dimensionality for rigid cohomology of an arbitrary smooth variety with arbitrary coefficients.
ABSTRACT: Together with R. Greenberg and R. Pollack we have proved that if two ellipitc curves over Q are congruent modulo a prime p>2, which is a prime of supersingular reduction for both, then if the mu-invariants vanish and the Main conjecture holds for one of the curves then the same is true for the other curve. This result together with recent results of Kurihara allow us to verify the Main Conjecture for a large family of elliptic curves at primes of supersingular reduction.
ABSTRACT: We discuss the geometry of a certain class of (PEL) type Shimura varieties, in connection with Langlands conjectures. In particular, we study the Newton polygon stratification and the Oort's foliation of the reduction in positive characteristic of the Shimura varieties. Over some of the leaves of the foliation, we define a tower of finite etale covers, we call Igusa varieties. We construct a system of finite surjective morphisms from the product of the Igusa varieties with the pertinent Rapoport-Zink spaces to each Newton polygon stratum inside the reduction of the Shimura varieties. We also show that the above constructions extend Zariski locally to the corresponding formal schemes in characteristic zero. As a result, we are able to express the cohomology of the Shimura varieties, in terms of the cohomology with compact supports of the Igusa varieties and the Rapoport-Zink spaces.
ABSTRACT: Based on theory from the function field case, we expect every natural family of L-functions has an associated symmetry group controlling the distribution of zeros. For elliptic curves, we expect SO(even) if every curve is even, SO(odd) if all are odd, and O if the signs are equi-distributed. By studying the n-level density (defined by summing a test function over the zeros), we obtain a statistic which is different for each of the candidate groups. Previous investigations for families of elliptic curves have only succeeded in evaluating the 1-level density for functions supported in (-1,1), where the three candidates are indistinguishable. In this talk we calculate the 2-level density in a restricted range, often for families with forced rank over Q(t). We show the three orthogonal candidates have different 2-level densities for test functions supported in an arbitrarily small neighborhood of the origin. Assuming standard conjectures, we observe the expected orthogonal group.
ABSTRACT: Let d>2 and let A^{d} be the space of degree-d monic polynomials over Q parametrized by their coefficients. For any polynomial f over Z_{p} and Q of degree d, let L(f;T) be the L function of the exponential sum of f mod p. Let NP(f,p) denote the Newton polygon of L(f;T); and let HP(A^{d},p) be the Hodge polygon, which is the lower convex hull in the real plane R^{2} of the points (n, n(n+1)/2d) for n from 0 to d-1. We prove that for p large enough there exists a "generic Newton polygon" GNP(A^{d},p) and there is a Zariski dense open subset U defined over the rationals Q in A^{d} such that for all f in U(Q) and for p large enough we have NP(f,p)=GNP(A^{d},p). Furthermore, as p goes to infinity their limit exists and is equal to HP(A^{d},p). The last statement was a conjecture of Daqing Wan.