On the estimation of eigenvalues of Hecke operators

Abstract

Introduction.Let A denote the adele ring of the rational numbers Q and suppose that we have a cuspidal automorphic representation re of GL 2 (A).(For the terminology and the details, we refer the reader to Gelbart [3] and Jacquet and Langlands [5]).Langlands [9] has described how one can attach an L-function to ic.To describe this construction briefly, one can associate to 7r, a family of local representations n p for each prime/? of Q.This family is uniquely determined by n such that (i) Tip is irreducible for every /?, (ii) for all but finitely many primes p, % p is unramified (that is, the restriction of % p to GL 2 (Z^) contains the identity representation exactly once), (iii) 7T can be factored as the restricted infinite tensor product % = ®p7C p .Let S denote the set of primes p for which % p is ramified.For p <£ S 9 it is known that n p corresponds canonically to a semisimple conjugacy class a p in GL 2 (C), where o p contains a matrix of the form lo ßp\-If r denotes any finite dimensional complex representation of GL 2 (C), one can attach an L-series L(s, n, r) as follows.L(s, a, r) = \\L(s, % p , r), p where L(s, 7C P , r) = det(l -r(a p )p-^ whenever n p is unramified.If % p is ramified, and r is standard, we refer to J-L [5] for the definition of L(s, n p , r).It is known [9] that each L(s, n, r)

Locations

• Rocky Mountain Journal of Mathematics - View - PDF