In these notes we construct a quantization functor, associating a Hilbert space H(V ) to a finite dimensional symplectic vector space V over a finite field Fq. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp (V ). The main technical result is a proof of a stronger form of the Stonevon Neumann theorem for the Heisenberg group over Fq. Our result an...

On June 23, 1993, Andrew Wiles unveiled his strategy for proving the Shimura-Taniyama-Weil conjecture for semistable elliptic curves defined over the field Q of rational numbers. Thanks to the work of Gerhard Frey, JeanPierre Serre and Kenneth Ribet, this was known to imply Fermat’s Last Theorem. Six years later, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor have finally ann...

We construct a Petersson-Weil type Kähler form on the moduli spaces of Higgs bundles over a compact Kähler manifold. A fiber integral formula for this form is proved, from which it follows that the Petersson-Weil form is the curvature of a certain determinant line bundle, equipped with a Quillen metric, on the moduli space of Higgs bundles over a projective manifold. The curvature of the Peters...

Weil-Felix reaction with blood of patients suffering ̂ from scrub typhus during its course shows significant agglutination titres against yproteus XK strains. In Bombay. Patel (1940) reported on two cases but the titres were ^r below the diagnostic level. Later, two more cases clinically suspected and serologically confirmed were reported by Patel (1943) and Patel (1943). Systematic Weil-Felix r...

We construct explicit examples of E8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E8 and have maximal Galois action. Our main objects of study are del Pezzo surfaces of degree 1 ove...

In these notes we construct a quantization functor, associating an Hilbert space H(V ) to a finite dimensional symplectic vector space V over a finite field Fq. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp (V ). The main technical result is a proof of a stronger form of the Stone-von Neumann theorem for the Heisenberg group over Fq. Our result ...

Purpose of review A relatively neglected topic to date has been the occurrence of concentrated epidemics within generalized epidemic settings and the potential role of targeted interventions in such settings. We review recent studies in high-risk groups as well as findings relating to geographical heterogeneity and the potential for targeting ‘high-transmission zones’ in the 10 countries with h...

In this paper, we construct explicit eigenfunctions of the local Weil representation on unitary group of one variable in the p-adic case where p is odd. The idea is to use the lattice model, and the results will be used to compute special values of certain Hecke L-functions in separated papers. We also recover Moen’s results on when a local theta lifting from U(1) to itself does not vanish. 0. ...

In [12] and [13], Silverman discusses the problem of bounding the Mordell-Weil ranks of elliptic curves over towers of function fields. We first prove generalizations of the theorems of those two papers by a different method, allowing non-abelian Galois groups and removing the dependence on Tate’s conjectures. We then prove some theorems about the growth of Mordell-Weil ranks in towers of funct...