On K_2 of certain families of curves

We construct families of smooth, proper, algebraic curves in characteristic 0, of arbitrary genus g, together with g elements in the kernel of the tame symbol. We show that those elements are in general independent by a limit calculation of the regulator. Working over a number field, we show that in some of those families the elements are integral. We determine when those curves are hyperelliptic, finding, in particular, that over any number field we have non-hyperelliptic curves of all composite genera g with g independent integral elements in the kernel of the tame symbol.Comment: Revised version: improved exposition. some sections spli

Etale cohomology, cofinite generation, and p-adic L-functions

For a prime number p and a number field k, we first study certain etale cohomology groups with coefficients associated to a p-adic Artin representation of its Galois group, where we twist the coefficients using a modified Tate twist with a p-adic index. We show that those groups are cofinitely generated and explicitly compute an additive Euler characteristic. When k is totally real and the representation is even, we relate the order of vanishing of the p-adic L-function at a point of its domain and the corank of such a cohomology group with a suitable p-adic twist. If the groups are finite, then the value of the p-adic L-function is non-zero and its p-adic absolute value is related to a multiplicative Euler characteristic. For a negative integer n (and for 0 in certain cases), this gives a proof of a conjecture by Coates and Lichtenbaum, and a short proof of the equivariant Tamagawa number conjecture for classical L-functions that do not vanish at n. For p=2 our results involving p-adic L-functions depend on a conjecture in Iwasawa theory.Comment: Updated version. The final version will appear in the Annales de l'Institut Fourie

The syntomic regulator for K4K_4 of curves

Let $C$ be a curve defined over a discrete valuation field of characteristic zero where the residue field has positive characteristic. Assuming that $C$ has good reduction over the residue field, we compute the syntomic regulator on a certain part of K_4^[(3)}(C). The result can be expressed in terms of $p$-adic polylogarithms and Coleman integration. We also compute the syntomic regulator on a certain part of K_4^[(3)}(F) for the function field $F$ of $C$. The result can be expressed in terms of $p$-adic polylogarithms and Coleman integration, or by using a trilinear map ("triple index") on certain functions.Comment: 61 page