Abstracts for Number theory lunch seminar

We introduce p-adic Kummer spaces of continuous functions on $Z_p$ that satisfy certain Kummer type congruences. We will classify these spaces and show their properties, for instance, ring properties and some decompositions. This theory can be transferred to values of Dirichlet L-functions at negative integer arguments in residue classes. That leads to a conjecture about their structure supported by several computations using a link to p-adic functions that are related to Fermat quotients.

I will explain how the Bloch-Kato conjecture leads to the following conclusion: any large prime dividing a critical value of the L-function of a classical Hecke eigenform of level 1, should also divide a certain ratio of critical values for the standard L-function of a related genus 2 (and in general vector-valued) Hecke eigenform F. This can be proved in the scalar-valued case, and there is experimental evidence in the vector-valued case (where the relation between f and F is a congruence of Hecke eigenvalues conjectured by Harder).

We will consider the n-th Fermat curve together with a cover of projective space. There is a (non congruence) subgroup of the full modular group associated to this cover for which the modular forms are known. We will describe a connection of non-holomorphic Eisenstein series and certain modular forms. From that we can derive the scattering constants that have some applications in Arakelov theory.

Let $k$ be a local field such that $[k:Q_p] < \infty$ and $X$ be a proper smooth variety over $k$ with good reduction. Define $SK_1(X):=Coker(\partial: \bigoplus_{All C on X} K_2^M(k(C)) \to \bigoplus_{All points x on X} k(x)^*)$, where $\partial$ is the tame-symbol map. There is a reciprocity homomorphism   $\rho_S:SK_1(S) \to \pi_1^{ab}(S)$   to the abelianized fundamental group of S. During my last stay in MPIM, I proved class field theory for S=elliptic fibration, by which I mean $\rho_S/m$ is bijective for any $m > 1$.