Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series
Heilbronn Number Theory Seminar
22nd November 2017, 4:00 pm – 5:00 pm
Howard House, 4th Floor Seminar Room
The higher-order Eisenstein series are constructed as twists of the non-holomorphic Eisensten series with powers of modular symbols associated to a weight two holomorphic cusp form. As expected, their pole at s=1 of is a source of rich features, one of them being higher-order modular Dedkind symbols. Namely, generalizing classical construction of Dedekind sums based on the Kronecker's limit formula, we start with the next to lowest term in the Laurent series expansion of the higher-order Eisenstein series and deduce the higher-order eta function, which, rather surprisingly, does not depend upon the power of the twist. The higher-order eta function has a multiplier system which can be expressed in terms of a map from the Fuchsian group to the set of real numbers and which we call higher-order modular Dedekind symbol. We show that the higher-order modular Dedekind symbol possesses many interesting arithmetic features which are related to the study of periods. As a further example of arithmeticity, we show that the higher-order modular Dedekind symbols associated to the genus one congruence groups are rational numbers. The talk is based on a recent joint work with Jay Jorgenson and Cormac O'Sullivan.