Abstracts for Number theory lunch seminar

Broadhurst and Kreimer proposed a Poincare series for the algebra of multiple zeta values based on large scale numerical computations. In this talk, I will explain how the coefficients in their series are related to period polynomials and to structural properties of the formal multizeta value (double shuffle) Lie algebra.

Let A/F be an abelian variety over a number field F. Let P be a point in A(F) and Lambda \subset A(F) be any subgroup of the Mordell-Weil group. I will discuss local conditions for P and Lambda (at primes v of the ring of integers of F) that imply that P is in Lambda. In addition to the local conditions an explicit upper bound on the multiplicities of the simple factors of A is necessary to show that P is in Lambda (I will present explicit counterxamples to this problem if the assumptions on the multiplicities of the simple factors of A are not met).

We show that for a prime $p$ the smallest $a$ with $a^{p-1}$ that is not congruent to 1 modulo ${p^2}$ does not exceed $(\log p)^{463/252 + o(1)}$ which improves the previous bound $O((\log p)^2)$ obtained by H. W. Lenstra in 1979. We also show that for almost all primes $p$ the bound can be improved to $(\log p)^{5/3 + o(1)}$.

We discuss a conjectural description of leading Taylor coefficients of Zeta functions of arithmetic schemes in terms of Weil-etale cohomology of motivic complexes. For varieties over finite fields this goes back to Milne, Lichtenbaum and Geisser, and for schemes of characteristic zero it amounts to more geometric and global reformulation of the Tamagawa number conjecture of Bloch, Kato, Fontaine and Perrin-Riou. We discuss some partial constructions of such a Weil-etale cohomology for $s=0$ (joint work with Morin) and for all $n$ for the Dedekind Zeta function.