Elliptic surface

Abstract: In a recent paper, Rosen and Silverman showed that Tate's conjecture on the order of vanishing of L(E,s) implies Nagao's formula, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. The aim of this article is to extend their result to the case of elliptic threefolds, and deduce, from Tate's conjecture, a Nagao-type formula for the rank of an elliptic threefold E. This will require a two-pronged approach: on the one hand, we need some cohomological results in order to derive a Shioda-Tate-like formula for elliptic threefolds; on the other, we compute an "average" number of rational points on the singular fibers and relate this to the action of Galois on those fibers.

Abstract: This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra generated by the Hecke operators turns out to be a homomorphic image of the {\it quantum toroidal algebra}. The latter is a quantization, in the spirit of Drinfeld-Jimbo, of the universal enveloping algebra of the universal central extension of a "double-loop" Lie algebra. This yields, in particular, a new geometric construction of affine quantum groups of types A, D E in terms of Hecke operators for an elliptic surface.