Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers

Abstract

Abstract. Let X be a smooth projective geometrically connected curve over a finite field with function field K. Let G be a connected semisimple group scheme over X. Under certain hypotheses we prove the equality of two numbers associated with G. The first is an arithmetic invariant, its Tamagawa number. The second is a geometric invariant, the number of connected components of the moduli stack of Gtorsors on X. Our results are most useful for studying connected components as much is known about Tamagawa numbers.

Langlands Program, trace formulas, and their geometrization

“…In this case, it is sometimes said that G "satisfies the Hasse principle". This holds, for example, in the case that G is semi-simple and split over F q ; see [Ha] and [BeDh,Corollary 4.2]. Otherwise, the set of equivalence classes of principal G-bundles in theétale topology (equivalently, the fppf topology) is a union over ξ ∈ Ker 1 (F, G) of double quotients like (3.1) in which G(F ) is replaced by its form corresponding to ξ.…”

Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

“…(For generalizations of this result to other groups G, and to the case of smooth reductive group schemes over C, we refer the reader to the work of Behrend and Dhillon [11,12]. )…”

“…They asked for a deeper understanding of this observation and in particular for a geometric explanation, exploiting the analogy with equivariant cohomology, of the fact that the Tamagawa number of SL n is 1. Contributions since then towards such understanding have included work by Bifet, Ghione and Letizia [14,15], providing another inductive procedure for calculating the Betti numbers of the moduli spaces, which is in some sense intermediate between the arithmetic approach and the Yang-Mills approach, and more recently, work by Teleman, Behrend, Dhillon and others on the moduli stack of bundles over C (see [11,12,24,66]).…”

Geometrization of trace formulas

“…(see, e.g., [6]). For a class of kernels K described below, we will give a geometric interpretation of this sum as the trace of the Frobenius on the cohomology of an -adic sheaf on an algebraic stack over the diagonal in Bun G × Bun G .…”

“…We generalize this definition slightly, by allowing L to be a complex of local systems on X . 6 The resulting object L d will then be a complex of sheaves on X d .…”