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The Kronecker limit formulas and geometric invariants for real quadratic fields, part II

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Speaker: 
William Duke
Affiliation: 
UCLA
Date: 
Wed, 2017-05-24 11:00 - 11:50
Location: 
MPIM Lecture Hall

The  Kronecker Limit formulas lead to some beautiful relations between quadratic fields, L functions, elliptic curves and, of course, modular forms! Associated to the ideal classes of a  quadratic field are some well known  geometric invariants attached to a modular curve: CM points in the imaginary quadratic case and closed geodesics in the real quadratic case. These talks will concentrate on the case of real quadratic fields and genus characters associated to two negative discriminants.We will explain that there are surfaces associated to ideal classes with some lovely arithmetic and distribution properties that complement the cm points and closed geodesics.

The first talk will review classical applications of the Kronecker limit formulas and give some new variations in the case of real quadratic  genus characters associated to two negative discriminants. We will  give a  generalization of the Katok Sarnak formula which relates integrals of Maass forms over the various geometric invariants to products of Fourier coefficients of weight 1/2 Maass forms.

The second talk will outline the construction of the surfaces and give some of their properties. The surfaces are bounded by closed geodesics and their areas are determined by certain minus continued fractions.Finally, we discuss the distribution properties of these surfaces and show how that is related  to the generalization of the Katok Sarnak formula  and known sub-convexity estimates for L-functions.

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