Activities This Week

Multiple zeta values can be written as sums of series and as integrals. Their integral expression makes them into periods of the pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - {0,1,\infty}$. p-Adic multiple zeta values are defined as p-adic analogues of these integrals. We will show how to express them as sums of series, which allows in particular to compute them explicitly. We will mention the role of finite multiple zeta values defined by Kaneko and Zagier, and of a question asked by Deligne and Goncharov on a relation between the computation of p-adic multiple zeta values and their algebraic properties. To express the results we will introduce new objects in relation with motivic Galois theory of periods.

Given a lattice $\Lambda$ and a (perhaps long) vector $v \in \Lambda$, we consider two geometric quantities: - the projection $\Delta$ of $\Lambda$ along the line through $v$ - the “lift functional” which encodes how one can recover $\Lambda$ from the projection $\Delta$ Fixing $\Lambda$ and taking some infinite sequences of vectors $v_n$, we identify the asymptotic distribution of these two quantities. For example, for a.e. line $L$, if $v_n$ is the sequence of $\epsilon$-approximants to $L$ then the sequence $\Delta(v_n)$ equidistributes according to Haar measure, and if $v'_n$ is the sequence of best approximants to $L$ then there is another measure which $\Delta(v'_n)$ equidistributes according to. The basic tool is a cross section for a diagonal flow on the space of lattices, and after some analysis of this cross section, the results follow from the Birkhoff pointwise ergodic theorem.

Joint work with Uri Shapira.