tailieunhanh - Đề tài " On the nonnegativity of L(1/2, π) for SO2n+1 "

Let π be a cuspidal generic representation of SO(2n + 1, A). We prove that L( 1 , π) ≥ 0. 2 1. Introduction Let π be a cuspidal automorphic representation of GLn (A) where A is the ring of ad`les of a number field F . Suppose that π is self-dual. Then the e “standard” L-function ([GJ72]) L(s, π) is real for s ∈ R and positive for s | Annals of Mathematics On the nonnegativity of L 1 2 n for SO2rc 1 By Erez Lapid and Stephen Rallis Annals of Mathematics 157 2003 891-917 On the nonnegativity of L 1 n for SO2n 1 By Erez Lapid and Stephen Rallis Abstract Let n be a cuspidal generic representation of SO 2n 1 A . We prove that L 2 n 0. 1. Introduction Let n be a cuspidal automorphic representation of GLn A where A is the ring of adeles of a number field F. Suppose that n is self-dual. Then the standard L-function GJ72 L s n is real for s G R and positive for s 1. Assuming GRH we have L s n 0 for 2 s 1 except for the case where n 1 and n is the trivial character. It would follow that L 2 n 0. However the latter is not known even in the case of quadratic Dirichlet characters. In general if n is self-dual then n is either symplectic or orthogonal . exactly one of the partial L-functions LS s n A2 LS s n sym2 has a pole at s 1. In the first case n is even and the central character of n is trivial JS90a . In the language of the Tannakian formalism of Langlands Lan79 any cuspidal representation n of GLn A corresponds to an irreducible n-dimensional representation ip of a conjectural group Lp whose derived group is compact. Then n is self-dual if and only if p is self-dual and the classification into sym-plectic and orthogonal is compatible with and suggested by the one for finite dimensional representations of a compact group. Our goal in this paper is to show Theorem 1. Let n be a symplectic cuspidal representation of GLn A . Then L 2 n 0. We note that the same will be true for the partial L-function. The value L 1 n appears in many arithmetic analytic and geometric contexts - among them the Shimura correspondence Wal81 or more generally - the theta First named author partially supported by NSF grant DMS-0070611. Second named author partially supported by NSF grant DMS-9970342. 892 EREZ LAPID AND STEPHEN RALLIS correspondence Ral87 the Birch-Swinnerton-Dyer conjecture the Gross-Prasad conjecture GP94 .