Mac Williams identities and polarized Riemann-Roch conditions

The present note establishes the equivalence of Mac Williams identities for an additive code C and its dual to Polarized Riemann-Roch Conditions on their Zeta-functions. In such a way, the duality of additive codes appears to be a polarized form of the Serre duality on a smooth irreducible projective curve

Arakelov geometry over adelic curves

The purpose of this book is to build up the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for the researches of arithmetic geometry in several directions