Yasemin Kara

Ph.D. 2015 Cornell University


The laplacian on hyperbolic Riemann surfaces and Maass forms



This thesis concerns the spectral theory of the Laplacian on Riemann surfaces of finite type, with emphasis on the quotients of the upper half plane by congruence subgroups. In a first part we show, following Otal, that on a Riemann surface M of genus g with n punctures there are at most 2g - 2 + n eigenvalues $\lambda$ with $\lambda$ $\leq$ 1/4. In a second part, we focus on arithmetic surfaces. This subject is treated by Maass in a paper that is difficult to read. We work out some examples of his construction of Maass forms.