Functoriality and small eigenvalues of Laplacian on Riemann surfaces

Abstract

Abstract. 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.

On the Ramanujan Conjecture for Quasisplit Groups

“…Its archimedean counterpart, the Selberg conjecture [79], states that the positive eigenvalues of the hyperbolic Laplacian on the space of cuspidal functions (functions vanishing at all the cusps) on a hyperbolic Riemann surface parametrized by a congruence subgroup must all be at least 1/4 (cf. [76,79,94]). While for the holomorphic modular cusp forms, this is a theorem ([25], also see [8,12]), the case of Maass forms is far from resolved and both conjectures are yet unsettled and out of reach.…”

On Selberg's small eigenvalue conjecture and residual eigenvalues

“…Selberg's conjecture would follow from the general Langland's functoriality conjectures [29] concerning symmetric tensor powers of automorphic L-functions. See [48] for a survey of the methods leading to such results.…”

Distribution of Angles Between Geodesic Rays Associated With Hyperbolic Lattice Points

“…I am grateful to Professor Peter Sarnak for his interest and for bringing to my attention references [6] and [15], and to Professor Freydoon Shahidi for kindly providing me a reprint of [15].…”

“…For N = 1 formula (1.6) has been proved using Kloosterman sum estimates in [8]. Better error terms with exponent as low as 2 3 can be obtained using Selberg's theory on the spectral decomposition of L 2 (Γ(N )\H) (see [13] for exponent ) and lower bounds for the first eigenvalue of the Laplacian on Γ(N )\H (see [14], [9], and [15] for a review of recent developments). Similar results hold when Γ(N ) is replaced by any of the congruence groups…”