### Convolution structures and arithmetic cohomology, by Alexandr Borisov

In the beginning of 1998 Gerard van der Geer and Rene Schoof posted a
beautiful preprint in which they defined exactly $h^0(L)$ for Arakelov line
bundles $L$ on an "arithmetic curve", i.e. a number field. In this paper we go
further to define $H^0(L)$ and $H^1(L)$ as well as their dimensions. The $H^1$
is defined by a procedure very similar to the usual Cech cohomology. We get
Serre's duality as Pontryagin duality of convolution structures. We get
separately Riemann-Roch formula and Serre's duality. Instead of using the
Poisson summation formula, we basically reprove it. The whole theory is pretty
much parallel to the geometric case.

Alexandr Borisov <borisov@math.psu.edu>