The Behavior of General Theta Functions Under the Modular Group and the Characters of Binary Modular Congruence Groups. I

Abstract

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Kloosterman sums over finite Frobenius rings

“…Here τ is a multiplicative character of R. Such twisted Kloosterman sums were first considered, over a finite field Z/pZ, by Davenport [1]. Kloosterman himself studied them, in some later work [10], over the modular ring Z/qZ with q a power of a prime. Twisted Kloosterman sums over modular rings Z/qZ, where q is a power of an odd prime, in which the twist τ is the quadratic character, are known as Salié sums.…”

A quantum ergodic theorem for mapping class groups action on character variety

“…These projective representations can be lifted to linear representations of SL 2 (Z) first defined by Kloosterman in [Klo46], and usually referred as Weil representations and satisfy the exact Egorov identity. This fact can be compared to the case of higher dimensional torus where the equivalent projective representations of the symplectic groups have to be centrally extend by Z/2Z to be lifted to linear ones when N is even.…”

Representations of automorphism groups of finiteo-modules of rank two

“…First results concerning the representation theory of the group SL 2 (Z/p Z) were obtained by Kloosterman in his Annals papers [14,15]. Kloosterman constructed some of the representations of SL 2 (Z/p Z) using the Weil representation (see also [30] for a description of Kloosterman's work).…”