Dirichlet series, asymptotics, and statistics based on functoriality from GL(2)
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Dirichlet series, asymptotics, and statistics based on functoriality from GL(2) Huixue Lao1 and Yangbo Ye2* * Correspondence:
[email protected] Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA Full list of author information is available at the end of the article 2
Abstract Let πi , i = 1, 2, 3, be unitary automorphic cuspidal representations of GL2 (QA ) with Fourier coefficients λπi (n). Consider an automorphic representation which is equivalent to ∧2 (Sym3 π1 ), π1 π2 , π1 Sym2 π2 , ∧2 (π1 π2 ), or π1 × π2 × π3 . Since ) is known to be complicated, a simpler Dirichlet series the Dirichlet series of L(s, × −s λ(n)n is defined and analytically continued in each case, which is closely related to L(s, × ) and catches the essence of the underlying functoriality. Asymptotics of n≤x λ(n) are proved. As applications, certain means, variance, and covariances of |λπi (n)|k for k = 2, 4, 6 and |λπi (nj )|2 for j = 2, 3, 4 are computed. These statistics provide a deep insight of the distribution of the GL(2) Fourier coefficients λπi (n). Keywords: Automorphic representation, Cuspidal representation, Functoriality, Dirichlet series, GL(2), Fourier coefficient, Asymptotic expansion, Statistics Mathematics Subject Classification: 11F70, 11F66, 11F30
1 Introduction Proven functoriality of automorphic representations can be described by relations of local parameters of their unramified local representations. These local relations in turn imply a matching of unramified local L-factors and hence a matching of partial Euler products of the corresponding automorphic L-functions.1 As an example consider three unitary automorphic cuspidal representations π1 , π2 and π3 of GL2 (QA ). Denote the local parameters by α1 (p), α2 (p) for π1p , β1 (p), β2 (p) for π2p , and γ1 (p), γ2 (p) for π3p for all p ∈ / S, where S is a finite set consisting of ∞ and primes p at which at least one of π1 , π2 and π3 is ramified. Then the local factors of π1 , π2 and π3 at p ∈ / S are 1 The first author is partially supported by Natural Science Foundation of Shandong Province (Grant No. ZR2018MA003).
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H. Lao , Y. Ye Res. Number Theory (2020)6:37
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Lp (s, π1 ) =
2 i=1
Lp (s, π2 ) =
2 j=1
Lp (s, π3 ) =
2 k=1
1 , 1 − αi (p)p−s 1 , 1 − βj (p)p−s 1 , 1 − γk (p)p−s
(1.1)
while the local L-factor of the triple product π1 × π2 × π3 is Lp (s, π1 × π2 × π3 ) =
2 2 2 −1 1 − αi (p)βj (p)γk (p)p−s .
(1.2)
i=1 j=1 k=1
The matching of the Euler products outside S is thus between LS (s, π ) = for = 1, 2, 3 and LS (s, π1 × π2 × π3 ) = Lp (s, π1 × π2 × π3 ).
p∈S /
Lp (s, π )
(1.3)
p∈S /
The matching between Dirichlet series of the partial L-functions is, however, much more complicated in general. As in the case of (1.1) and (1.2) for p ∈ / S, the Dirichlet coefficient λtriple (pn ) in Lp (s, π1 × π2 × π3 ) =
2 2 2 ∞ (αi (p)βj (p)γk (p))n i=1 j=1 k=1 n=0
pns
=:
∞ λtriple (pn ) n=
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