Sato–Tate theorem for families and low-lying zeros of automorphic L-functions

Abstract

We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let G be a reductive group over a number field F which admits discrete series representations at infinity. Let LG = Ĝ ⋊ Gal(F/F) be the associated L-group and r: LG → GL(d, ℂ) a continuous homomorphism which is irreducible and does not factor through Gal(F/F). The families under consideration consist of discrete automorphic representations of G(𝔸F) of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato–Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321–331, 1987) and Serre (J Am Math Soc 10(1):75–102, 1997). As an application we study the distribution of the low-lying zeros of the associated family of L-functions L(s, π, r), assuming from the principle of functoriality that these L-functions are automorphic. We find that the distribution of the 1-level densities coincides with the distribution of the 1-level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz–Sarnak heuristics.We provide a criterion based on the Frobenius–Schur indicator to determine this symmetry type. If r is not isomorphic to its dual r∨then the symmetry type is unitary.Otherwise there is a bilinear form on ℂd which realizes the isomorphism between r and r∨. If the bilinear form is symmetric (resp. alternating) then r is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).