• DOI:10.1007/S00222-015-0583-Y
• Corpus ID: 119317992

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

@article{Shin2012SatoTateTF,
  title={Sato–Tate theorem for families and low-lying zeros of automorphic \$\$L\$\$L-functions},
  author={Sug Woo Shin and Nicolas Templier},
  journal={Inventiones mathematicae},
  year={2012},
  volume={203},
  pages={1-177}
}

• S. Shin, Nicolas Templier
• Published 9 August 2012
• Mathematics
• Inventiones mathematicae

We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let $$G$$G be a reductive group over a number field $$F$$F which admits discrete series representations at infinity. Let $$^{L}G=\widehat{G} \rtimes \mathrm{Gal}(\bar{F}/F)$$LG=G^⋊Gal(F¯/F) be the associated $$L$$L-group and $$r:{}^L G\rightarrow \mathrm{GL}(d,\mathbb {C})$$r:LG→GL(d,C) a continuous homomorphism which is irreducible and does not factor through $$\mathrm{Gal}(\bar{F}/F)$$Gal(F¯/F). The families under consideration consist of discrete automorphic representations of $$G(\mathbb {A}_F)$$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 $$L$$L-functions $$L(s,\pi ,r)$$L(s,π,r), assuming from the principle of functoriality that these $$L$$L-functions are automorphic. We find that the distribution of the $$1$$1-level densities coincides with the distribution of the $$1$$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 $$r$$r is not isomorphic to its dual $$r^\vee $$r∨ then the symmetry type is unitary. Otherwise there is a bilinear form on $$\mathbb {C}^d$$Cd which realizes the isomorphism between $$r$$r and $$r^\vee $$r∨. If the bilinear form is symmetric (resp. alternating) then $$r$$r is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).