Convolution Structures and Arithmetic Cohomology

In the beginning of 1998 Gerard van der Geer and Ren e Schoof posted a beautiful preprint (cf. [2]). Among other things in this preprint they de ned exactly h(L) for Arakelov line bundles L on an \arithmetic curve", i.e. a number eld. The main advantage of their de nition was that they got an exact analog of the Riemann-Roch formula h(L) h(K L) = degL+1 g: Before that h(L) was de ned as an integer and the Riemann-Roch formula above was only true approximately (cf. [6]). However van der Geer and Schoof gave no interpretation for h(L) except via duality. They indicated this as one of the missing blocks of their theory. In this paper we go even further to develop the interpretations for H(L) and H(L) as well as their dimensions. The main features of our theory are the following. 1) H is de ned by a procedure very similar to Ĉech cohomology. 2) We get separately Serre's duality and Riemann-Roch formula without duality. 3) We get the duality of H(L) and H(K L) as Pontryagin duality of convolution structures. 4) The Riemann-Roch formula of van der Geer and Schoof follows automatically from our construction by an appropriate dimension function. The paper is organized as follows. In section 2 we de ne our basic objects (ghost-spaces) and their dimensions. In section 3 we introduce some short exact sequences of ghost-spaces. In section 4 we develop the duality theory of ghost-spaces. In section 5 we apply the theory to arithmetic and obtain our main results. In section 6 we discuss possible directions in which the theory can grow. Acknowledgments. The author thanks Adrian Ocneanu and Yuri Zarhin for their interest and stimulating discussions. The author is