Surveys in Differential Geometry

Volume 9 (2004)

Functoriality and small eigenvalues of Laplacian on Riemann surfaces

Pages: 385 – 400



F. Shahidi (Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A.)


The purpose of this article is to survey the recent progress made on estimating positive eigenvalues of Laplacian on hyperbolic Riemann surfaces in the case of congruence subgroups in connection with the Selberg conjecture, as well as certain related ones. The results are obtained as consequences of establishing certain important cases of Langlands’ functoriality conjecture.

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