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Kronecker limit formalism and moonshine type groups

Published in Courses
Tuesday, 02-02-2016

Topic and importance:

Kronecker’s limit formula refers to the evaluation of the first two terms in the Laurent expansion of the non-holomorphic Eisenstein series E(s,z) associated to the discrete group PSL(2,Z) at s=1.  In this case, the first term is a pole with residue equal to 1/V, where V is the co-volume of PSL(2,Z), and the second term involves the weight twelve cusp form, suitably normalized, associated to PSL(2,Z). 
More generally, one can consider generalizations of Kronecker’s limit formula to be any result which studies the Laurent expansion of any zeta or L function about its first pole.  There are various examples where such considerations exist, including the study of spectral zeta functions (which yields determinants of the Laplacian and analytic torsion) and various types of Eisenstein series (which yields constructions of certain modular forms).  Going further, the variation of the Kronecker limit function may be of additional interest, one example of which is the manifestation of Dedekind sums. 

Syllabus:

In these talks the speakers will outline their point-of-view of Kronecker’s limit functions and Dedekind sums associated to various Eisenstein series for co-finite quotients of the hyperbolic upper half plane.  Specific attention will be paid to the so-called “groups of moonshine type”, which we define as normalizer subgroups associated to Gamma_{0}(N).  In lieu of a problem solving session, the speakers will offer a number of open problems.  

Read 499 times Last modified on Thursday, 23 June 2016 11:48
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