The syntomic regulator forK₄of curves

Abstract

Let C be a curve defined over a complete discrete valuation subfield of ‫ރ‬ p . Assuming that C has good reduction over the residue field, we compute the syntomic regulator on a certain part of K (3) 4 (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 (3) 4 (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.

On the syntomic regulator for K 1 of a surface

“…To end this section we recall the definition and properties of the double and triple indices, discussed in [Bes00c,BdJ04] respectively. The theory has a local side and a global side.…”

“…We finally note that the triple index has the following equivariance property (see [BdJ04,Lemma 7.1]): If α : e → e is an endomorphisms of degree n, then the formula α * F, α * G; α * H e = n F, G; H e holds. In particular, when we specialize to the case α = φ and apply to the pairing •, • we obtain the following formula.…”

“…Formulas for the syntomic regulator are to be thought of as p-adic analogues of formulas for the Beilinson regulator [Bei85]. Several such formulas exists [Bes00c,BdJ03,BdJ04] for zero and one-dimensional varieties.…”

“…Local indices [Bes00c,BdJ04] are a generalization of the notion of residue. Consider the field of Laurent series K((z)) and the polynomial ring K((z))[log(z)] in the formal variable log(z), which admits a differential by setting d(log z) = dz/z.…”

Coleman–Gross height pairings and the p-adic sigma function

“…We begin by recalling the definition of the constant term of a Coleman function [8], [7,Definition 3.1]. Finally, there is a compatible action of Gal.…”