A note on polylogarithms on curves and abelian schemes

Cohomology classes defined by polylogarithms have been one of the main tools to study special values of L-functions. Most notably, they play a decisive role in the study of the Tamagawa number conjecture for abelian number fields ([Be2], [Del], [HuW] [Hu-Ki]), CM elliptic curves ([Den], [Ki2]) and modular forms ([Be1], [Ka]). Polylogarithms have been defined for relative curves by Beilinson and Levin (unpublished) and for abelian schemes by Wildeshaus [Wi] in the context of mixed Shimura varieties. In general, the nature of these extension classes is not well understood. The aim of this note is to show that there is a close connection between the polylogarithm extension on curves and on abelian schemes. It turns out that the polylog on an abelian scheme is roughly the push-forward of the polylog on a sub-curve. If we apply this to the embedding of a curve into its Jacobian, we can give a more precise statement: the polylog on the Jacobian is the cup product of the polylog on the curve with the fundamental class of the curve (see theorem 3.2.1). With this result it is possible to understand the nature of the polylog extension on abelian schemes in a better way. The polylog extension on curves has the advantage of being a one extension of lisse sheaves. Thus, itself can be represented by a lisse sheaf. The polylog extension on the abelian scheme on the contrary is a 2d− 1 extension, where d is the relative dimension of the abelian scheme. The contents of this note is as follows: To simplify the exposition we only treat the étale realization. First we define the polylog extension on curves and abelian schemes in a unified way for integral coefficients. To our knowledge this and the construction on curves is not published but goes back to an earlier version of [Be-Le1]. The case of abelian schemes is treated in [Wi] (for Qlsheaves), which we mildly generalize to Z/lZand Zl-sheaves. All the main ideas are of course already in [Be-Le1]. The second part gives three important properties of the polylog extension, namely compatibility with base change, norm compatibility and the splitting principle. In the last part we show that the push-forward of the polylog on a subcurve of an abelian scheme gives the polylogarithmic extension on the abelian