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

By S. Shin, Nicolas Templier. 2012

102 citations recorded by the source corpus.

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).

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f55878dc905d9fbaf17c212c8a18d84f23691271

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In Algebra and Number Theory

  1. Paul D. Nelson
  2. Akshay Venkatesh
  3. P. Michel
  4. R. Munshi
  5. V. Blomer
  6. Nicolas Templier
  7. Gergely Harcos
  8. M. Young

In Discrete Mathematics and Combinatorics

  1. Akshay Venkatesh
  2. Paul D. Nelson

In Mathematical Physics

  1. Paul D. Nelson
  2. Akshay Venkatesh
  3. P. Michel
  4. R. Munshi
  5. V. Blomer
  6. Nicolas Templier
  7. Gergely Harcos
  8. Manfred Einsiedler

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