In this article, following an insight of Kontsevich, we extend the Weil conjecture, as well strong form Tate from realm algebraic geometry to broad noncommutative setting dg categories. Moreover, establish a functional equation for Hasse-Weil zeta functions, compute l-adic and p-adic absolute values eigenvalues cyclotomic Frobenius, provide complete description category numerical motives in ter...

In this paper we construct a new variant of the Weil representation, associated with a symplectic vector space (V, ω) defined over a finite field of characteristic two. Our variant is a representation ρ : AMp (V ) → GL (H), where the group AMp (V ) is forth cover of a group ASp (V ), which is a nontrivial gluing of the symplectic group Sp (V ) and the dual group V ∗. In particular, the group AS...

In this essay I hope to explain the classification of characters of algebraic tori defined over R, particularly Langlands’ parametrization in terms of homomorphisms from the Weil group into the associated L-group. I’ll follow more or less the treatment in §§2 and 3 of [Langlands:1974/1989], avoiding most subsequent ‘improvements’. The only slightly new thing is how I relate Langlands’ formula f...

We describe Somekawa’s K-group associated to a finite collection of semi-abelian varieties (or more general sheaves) in terms of the tensor product in Voevodsky’s category of motives. While Somekawa’s definition is based on Weil reciprocity, Voevodsky’s category is based on homotopy invariance. We apply this to explicit descriptions of certain algebraic cycles.

The relation between the cover (Tukey type) uniformities and the entourage (Weil type) ones, in the point-free context, is studied and a transparent translation is presented. In particular the natural uniformities on localic groups are discussed, and the uniformity of localic group homomorphisms is proved.

Deligne has shown that there is an equivalence from the category of ordinary abelian varieties over a finite field A: to a category of Z-modules with additional structure. We translate several geometric notions, including that of a polarization, into Deligne's category of Z-modules. We use Deligne's equivalence to characterize the finite group schemes over k that occur as kernels of polarizatio...

orbit closure, 32 character, 33cone, 9dual, 10lattice, 10strongly convex, 10convex function, 42 distinguished point, 28divisorCartier, 38T -invariant, 40principal, 39prime, 37Weil, 37T -invariant, 39principal, 38divisor class group, 38

We show that in characteristic 2, the Steinberg representation of the symplectic group Sp 2n (q), q a power of an odd prime p, has two irreducible constituents lying just above the socle that are isomorphic to the two Weil modules of degree (qn −1)/2.