Arithmetic of Gamma, Zeta and Multizeta Values for Function Fields

We explain work on the arithmetic of Gamma and Zeta values for function fields. We will explore analogs of the gamma and zeta functions, their properties, functional equations, interpolations, their special values, their connections with periods of Drinfeld modules and t-motives, algebraic relations they satisfy and various methods showing that there are no more relations between them. We also briefly describe work on multizeta and many open problems in the area. In the advanced course given at Centre de Recerca Matemática (CRM), Barcelona consisting of twelve hour lectures during 22 February-5 March 2010, we described the results and discussed some open problems regarding the gamma and zeta functions in the function field context. The first four sections, dealing with gamma, roughly correspond to the first four lectures of one and half hour each, and the last three sections, dealing with zeta, cover the last three two hour lectures. Typically, in each part, we first discuss elementary techniques, then easier motivating examples with Drinfeld modules in detail, and then outline general results with higher dimensional t-motives. The section 4 is independent of section 3, whereas the last part (last three sections) are mostly independent of the first part, except that the last two sections depend on section 3. At the end, we include a guide to the relevant literature. We will assume that the reader has basic familiarity with the language of function fields, cyclotomic fields and Drinfeld modules and t-motives, though we will give quick reviews at the appropriate points. We will usually just sketch the main points of the proofs, leaving the details to references. We will use the following setting and notation, sometimes it will be specialized. Fq: a finite field of characteristic p having q elements X: a smooth, complete, geometrically irreducible curve over Fq K: the function field of X ∞: a closed point of X, i.e., a place of K d∞: the degree of the point ∞ A: the ring of elements of K with no pole outside ∞ K∞: the completion of K at ∞ C∞: the completion of an algebraic (‘separable’, equivalently ) closure of K∞ K, K∞: the algebraic closures of K, K∞ in C∞ F∞: the residue field at ∞ Av: the completion of A at a place v 6=∞ g: the genus of X h: the class number of K * Supported in part by NSA grants H98230-08-1-0049, H98230-10-1-0200.