Herbert Gangl

From polygons and symbols to polylogarithmic functions

A bstractWe present a review of the symbol map, a mathematical tool introduced by Goncharov and used by him and collaborators in the context of

Tame and wild kernels of quadratic imaginary number fields

\mathcal{N}

Classical and Elliptic Polylogarithms and Special Values of L-Series

= 4 SYM for simplifying expressions among multiple polylogarithms, and we recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon, and it is indicated how that recipe relates to a similar explicit formula for it previously given by Goncharov. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illustrate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of non-trivial elements in the kernel of the symbol map for arbitrary weight.

Generators and relations for K_2 O_F.

For all quadratic imaginary number fields F of discriminant d > −5000, we give the conjectural value of the order of Milnor’s group (the tame kernel) K2OF , where OF is the ring of integers of F. Assuming that the order is correct, we determine the structure of the group K2OF and of its subgroup WF (the wild kernel). It turns out that the odd part of the tame kernel is cyclic (with one exception, d = −3387).

Finite Polylogarithms, Their Multiple Analogues and the Shannon Entropy

The Dirichlet class number formula expresses the residue at s = 1 of the Dedekind zeta function ζ F(s) of an arbitrary algebraic number field F as the product of a simple factor (involving the class number of the field) with the determinant of a matrix whose entries are logarithms of units in the field. On the other hand, if F is a totally real number field of degree n, then a famous theorem by Klingen and Siegel says that the value ζ F (m) for every positive even integer in is a rational multiple of π mn In [52] and [53], a conjectural generalization of these two results was formulated according to which the special value ζ F (m) for arbitrary number fields F and positive integers m can be expressed in terms of special values of a transcendental function depending only on m, namely the m th classical polylogarithm function. These instances are expected to form part of a much more general picture in which a special value of an L-series of “motivic origin” is expressed in terms of some transcendental function. In this survey we collect some pieces fitting into and illustrating this picture.

Double zeta values and modular forms.

Tates algorithm for computing K_2 O_F for rings of integers in a number field has been adapted for the computer and gives explicit generators for the group and sharp bounds on their order---the latter, together with some structural results on the p-primary part of K_2 O_F due to Tate and Keune, gives a proof of its structure for many number fields of small discriminants, confirming earlier conjectural results. For the first time, tame kernels of non-Galois fields are obtained.

Quelques calculs de la cohomologie de GLN(Z) et de la K-théorie de Z

We show that the entropy function—and hence the finite 1-logarithm—behaves a lot like certain derivations. We recall its coho-mological interpretation as a 2-cocycle and also deduce 2n-cocycles for any n. Finally, we give some identities for finite multiple polylogarithms together with number theoretic applications.