τ(n)e(nz). The two equations were proven for τ(n) by Mordell, using what are now known as the Hecke operators. The inequality was proven by Deligne as a consequence of his proof of the Weil conjectures. Those results determine everything about af (n) except for the distribution of the af (p) ∈ [−2, 2]. Define θf (p) ∈ [0, π] by af (p) = 2 cos θf (p). It is conjectured that for each f the θf (p)...

In this article we develop a broad generalization of the classical Bost-Connes system, where roots of unit are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unit, Weil restriction, algebraic numbers, Weil numbers, CM fields, germs, completion of Weil numbers, etc. Making use of the Tannakian formalism, we categorify th...

In this talk the definition of polarization of abelian variety and Weil paring will be given. After introducing some proposition of Weil pairing, we will show the Zarhin’s trick about principal polarization.

We describe how to prove the Mordell-Weil theorem for Jacobians of hyperelliptic curves over Q and how to compute the rank and generators for the Mordell-Weil group.

In this paper, we define the Schrödinger-Weil representation for the Jacobi group and construct covariant maps for the Schrödinger-Weil representation. Using these covariant maps, we construct Jacobi forms with respect to an arithmetic subgroup of the Jacobi group.

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...

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 investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find m...