Vorträge in der Woche 21.07.2008 bis 27.07.2008

Montag, 21.07.2008: Reciprocity Laws on Riemann surfaces

Ivan Horozov

I will present a new reciprocity law on a Riemann surface involving a generating series of iterated integrals. At degree 1, it leads to sum of residues is equal to 0. At degree 2, it leads to reciprocity between differential forms of 3rd kind, found in algebraic geometry books or to Weil reciprocity. For degree 3 or higher we obtain infinitely many relations among iterated integrals. This approach has consequences for the Manin's non-commutative modular symbol. More precisely, we define a non-commutative modular symbol, involving both Eisenstein series and cusp forms so that this refinement of Manin's symbol has good algebraic, topological and arithmetic properties.