Alternatively have a look at the program.

## On an elliptic curve over an imaginary quadratic field

In my Diplomarbeit I have proved the Hasse-Weil conjecture for an elliptic curve over an imaginary quadratic field which has no complex multiplication. In this talk I shall survey the background and some of the ingredients of my work.

## On p-adic functions satisfying Kummer type congruences

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.

## Smooth curves having a large automorphism p-group in characteristic p>0

Let k be an algebraically closed field of characteristic p>0 and C a connected nonsingular projective curve over k with genus g>1. Let G be a p-subgroup of the k-automorphism group of C such that |G| > 2pg/(p-1). Then, C -->C/G is an étale cover of the affine line Spec k[X] totally ramified at infinity. To study such actions, we focus on the second ramification group G_2 of G at infinity, knowing that G_2 actually coincides with the derived group of G. We first display realizations of such actions with G_2 abelian of arbitrary large exponent .

## Harder's conjecture and ratios of standard L-values

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).

## Recent progress on the local Langlands conjecture for $G_2$

I will describe recent work, joint with G.

## Kronecker limit formula for Fermat curves

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.

## Acid zeta function and Riemann hypothesis

The motivation of constructing the acid zeta function is to study the distribution of the Riemann zeta zeros. In this lecture, I will present theory of the acid zeta function and the adjoint acid zeta function, particularly, as one of the applications, we have some important reasons to doubt the truth of the Riemann Hypothesis.

## Bloch's exact sequence for surfaces over local fields

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 $m > 1$.

## Isogenies of prime degree over number fields

Let K be a number field and E an elliptic curve defined over K. The so-called exceptional set for (E,K) consisting of prime numbers p such that E has a p-isogeny defined over K is finite iff E does not have CM over K. In this talk, I will state a criterion which allows, in various situations, to explicitly determine the exceptional set when it is finite.

## Galois representations and the Tame Inverse Galois problem

In this talk we address the following strengthening of the Inverse Galois problem over $\mathbb{Q}$, introduced by B. Birch around 1994: Let $G$ be a finite group. Is there a tamely ramified Galois extension of $\mathbb{Q}$ with Galois group $G$? When $G$ is a linear group, this problem can be approached through the study of Galois representations attached to arithmetic-geometric objects. Let $\ell$ be a prime number.

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