Abstracts for Conference in Memory of Yuri Manin

I will talk mainly about two algebraic structures and the relations between them.  The first is that of a {\it Rankin-Cohen algebra}, which is a graded vector space with infinitely many bilinear 
operations satisfying the same algebraic identities as those satisfied by the Rankin-Cohen brackets in the theory of modular forms.  The second is that of {\it $\mathfrak{sl}_2$-algebra}, 

We prove that an algebraic flat connection has $ {\mathbb R}_{\rm{ an, \  exp}}$-definable flat sections if and only if it is regular singular with unitary monodromy eigenvalues at infinity, refining previous work of Bakker–Mullane. This provides e.g. an o-minimal characterization of classical properties of the Gauss-Manin connection.

Joint work with Moritz Kerz.

By Deligne's Hodge theory, the integral cohomology groups $H^n(\mathcal{X}^h,\mathbf{Z})$ of the $\mathbf{C}$-analytification $\mathcal{X}^h$ of a separated scheme $\mathcal{X}$ of finite type over $\mathbf{C}$ are provided with a mixed Hodge structure, functorial in $\mathcal{X}$. Given a non-Archimedean field $K$ isomorphic to the field of Laurent power series $\mathbf{C}((z))$, there is a functor $\mathcal{X}\mapsto \mathcal{X}^{\text{an}}_K$ that takes $\mathcal{X}$ to the non-Archimedean $K$-analytification of $\mathcal{X}_K = \mathcal{X}\otimes_{\mathbf{C}} K$.

The fundamental property of zeta functions and L-functions is that their meromorphic continuations provide a lot of information about the corresponding objects. Complex values of $s$ occur as a technical tool, with little direct arithmetic-geometric meaning. In the refined theory, $1/n^s$ are replaced by certain $q,t,a$-series, which are invariants of Lens Spaces $L(n,1)$ directly related to Elliptic Hall