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Dirichlet series of two variables, real analytic Jacobi–Eisenstein series of matrix index, and Katok–Sarnak type result

  • Yoshinori Mizuno EMAIL logo
Published/Copyright: July 12, 2018
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 6

Abstract

We study a real analytic Jacobi–Eisenstein series of matrix index and deduce several arithmetically interesting properties. In particular, we prove the followings: (a) Its Fourier coefficients are proportional to the average values of the Eisenstein series on higher-dimensional hyperbolic space. (b) The associated Dirichlet series of two variables coincides with those of Siegel, Shintani, Peter and Ueno. This makes it possible to investigate the Dirichlet series by means of techniques from modular form.

Keywords: Jacobi–Eisenstein series; Dirichlet series; quadratic forms; hyperbolic space
MSC 2010: 11F50; 11F37; 11F30

Dedicated to Professor Fumihiro Sato on the occasion of his 70th birthday

Communicated by Jan Bruinier

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 25800021

Award Identifier / Grant number: 17K05175

Funding statement: This work is supported by JSPS Grant-in-Aid for Young Scientists (B) 25800021 and JSPS Grant-in-Aid for Scientific Research (C) 17K05175.

A Appendix

A.1 The upper-half space Um, the hyperboloid Hm, and the unit ball Bm

We summarize from [31] the various models of hyperbolic manifolds and measures on each realizations. Put I1,m+1:=diag⁢(1,-Im+1), where Im is the identity matrix of size m. Denote by Em the m-dimensional Euclidean space, Sm:={x=(x1,…,xm+1)∈𝐑m+1;x12+…+xm+12=1} is the unit m sphere (cf. [31, p. 35]), and denote by Um the upper-half space (cf. [31, p. 116])

Um:={x=(x1,x2,…,xm)∈Em;xm>0}.

The volume element d⁢vUm and the line element d⁢sUm on Um are defined by (cf. [31, Theorems 4.6.6–4.6.7, pp. 133–134])

d⁢vUm:=d⁢x1⁢⋯⁢d⁢xmxmm,d⁢sUm2:=d⁢x12+…+d⁢xm2xm2.

Let π:Em→Sm∖{em+1:=(0,…,0,1)} be the stereographic projection. By [31, p. 107],

x:=(x1,…,xm)∈Em↦π(x):=(2⁢x11+|x|2,…,2⁢xm1+|x|2,|x|2-11+|x|2,)∈Sm.

This π is a bijection from Em onto Sm∖{em+1}. We put |x|2:=x12+…+xm2.

For β>0, put

Hm(β):={x=(x1,…,xm+1)∈𝐑m+1;x1>0,x12-x22-⋯-xm+12=β},
Km(β):={x=(x1,⋯,xm+1)∈𝐑m+1;x12-x22-⋯-xm+12=-β}.

It is known that Hm⁢(β)≅O+⁢(1,m)/diag⁢(1,S⁢O⁢(m)), Km⁢(β)≅O+⁢(1,m)/diag⁢(O+⁢(1,m-1),1) as homogeneous spaces. Here O+(1,m):={M∈SLm+1(𝐑);I1,m[M]=I1,m,(M)11≥1} and (M)11 is the (1,1)-component of M. Let Hm=Hm⁢(1) be the hyperboloid model of hyperbolic m space (cf. [31, p. 61]). A discrete group of isometries of Hm has a locally finite fundamental domain (cf. [31, p. 246]). Suppose that m≥2. We have the geodesic polar coordinate of Hm given by (cf. [31, p. 77])

xt=(x1x2x3x4⋮xmxm+1)=(cosh⁡η1sinh⁡η1⋅cos⁡η2sinh⁡η1⋅sin⁡η2⋅cos⁡η3sinh⁡η1⋅sin⁡η2⋅sin⁡η3⋅cos⁡η4⋮sinh⁡η1⋅sin⁡η2⁢⋯⁢sin⁡ηm-1⋅cos⁡ηmsinh⁡η1⋅sin⁡η2⁢⋯⁢sin⁡ηm-1⋅sin⁡ηm),

where 0≤η1<+∞, 0≤ηj≤π (j=2,3,…,m-1), 0≤ηm≤2⁢π. The volume element d⁢vHm and the line element d⁢sHm on Hm are defined by (cf. the proof of [31, Theorem 3.4.1, p. 78])

d⁢sHm2:=-d⁢x12+d⁢x22+…+d⁢xm+12,
d⁢vHm:=d⁢x2⁢⋯⁢d⁢xm+1x1=sinhm-1⁡η1⋅sinm-2⁡η2⁢⋯⁢sin⁡ηm-1⁢d⁢η1⁢⋯⁢d⁢ηm.

Denote by Bm the m-dimensional unit ball defined by (cf. [31, p. 119])

Bm:={x=(x1,…,xm)∈Em;|x|<1}.

The volume element d⁢vBm and the line element d⁢sBm on Bm are given by (cf. [31, Theorems 4.5.5–4.5.6, pp. 128–129])

d⁢vBm:=2m⁢d⁢x1⁢⋯⁢d⁢xm(1-|x|2)m,d⁢sBm2:=4⁢d⁢x12+…+d⁢xm2(1-|x|2)2.

We have the bijection ζ:Bm→Hm defined by (cf. [31, p. 122])

x=(x1,…,xm)∈Bm↦ζ⁢(x):=(1+|x|21-|x|2,2⁢xm1-|x|2,…,2⁢x11-|x|2)∈Hm,
y=(ym+1,…,y1)∈Hm↦ζ-1⁢(y):=(y11+ym+1,…,ym1+ym+1)∈Bm.

We also define the map π~ by changing the sign of the last component of the map π by

x:=(x1,…,xm)∈Bm↦π~⁢(x):=(2⁢x11+|x|2,…,2⁢xm1+|x|2,1-|x|21+|x|2)∈Sm.

For y∈Hm with the geodesic polar coordinate, x=ζ-1⁢(y)∈Bm has the form

xt=(x1x2⋮xm-1xm)=(tanh⁡(η1/2)⋅sin⁡η2⁢⋯⁢sin⁡ηm-1⋅sin⁡ηmtanh⁡(η1/2)⋅sin⁡η2⁢⋯⁢sin⁡ηm-1⋅cos⁡ηm⋮tanh⁡(η1/2)⋅sin⁡η2⋅cos⁡η3tanh⁡(η1/2)⋅cos⁡η2)∈Bm.

If we apply the map π~ to this x=ζ-1⁢(y)∈Bm, the result is

(π~∘ζ-1(y))t=(π~(x))t=(tanh⁡η1⋅sin⁡η2⁢⋯⁢sin⁡ηm-1⋅sin⁡ηmtanh⁡η1⋅sin⁡η2⁢⋯⁢sin⁡ηm-1⋅cos⁡ηm⋮tanh⁡η1⋅sin⁡η2⋅cos⁡η3tanh⁡η1⋅cos⁡η21/cosh⁡η1)∈Sm.

A.2 Evaluation of a determinant

To deduce (5.3), it is sufficient to prove the following lemma.

Lemma A.1.

For any natural number n and variables xi (1≤i≤m) and x, put

An⁢(x1,x2,…,xn;x):=det⁡(xi⁢xj+δi,j⁢x)1≤i,j≤n.

Then one has

An⁢(x1,x2,…,xn;x)=xn+(∑i=1nxi2)⁢xn-1.

Proof.

We use induction on n. The case n=1 is trivial. Suppose that the statement holds true for the case n-1. It is easy to see that

∂x⁡An⁢(x1,x2,…,xn;x)=∑l=1nAn-1⁢(x1,…,xl-1,xlˇ,xl+1⁢…,xn;x).

Here xlˇ indicates that xl is omitted. By the induction hypothesis, we have

∂x⁡An⁢(x1,x2,…,xn;x)=n⁢xn-1+(n-1)⁢(∑i=1nxi2)⁢xn-2.

In view of An⁢(x1,x2,…,xn;0)=0, one concludes

An⁢(x1,x2,…,xn;x)=xn+(∑i=1nxi2)⁢xn-1.

This proves the case n as desired. ∎

Applying this Lemma to n=m, xi=yi (1≤i≤m) and x=y2, with y2+∑i=1myi2=α2, we get

det⁡(yi⁢yj+δi,j⁢y2)1≤i,j≤m=y2⁢m+(∑i=1myi2)⁢y2⁢m-2=(y2+∑i=1myi2)⁢y2⁢m-2=α2⁢y2⁢m-2.

This implies (5.3).

Acknowledgements

The author would like to thank Prof. A. Ichino and Prof. T. Ikeda for suggestions from adelic setting point of view and to Prof. O. Richter for answering a question about his paper. The author would like to thank the referee for detailed reading of this manuscript and for suggesting improvements about the presentation of this paper. The author would like to thank Prof. F. Sato and Prof. K. Sugiyama for useful discussion about this topic.

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Received: 2017-06-10
Revised: 2018-04-09
Published Online: 2018-07-12
Published in Print: 2018-11-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The Mackey problem for free locally convex spaces
  3. Restricted averaging operators to cones over finite fields
  4. Asymptotic expansion of holonomy
  5. Compactness criteria for real algebraic sets and Newton polyhedra
  6. Unit-graphs and special unit-digraphs on matrix rings
  7. ϕ-amenability and character amenability of Fréchet algebras
  8. Dirichlet series of two variables, real analytic Jacobi–Eisenstein series of matrix index, and Katok–Sarnak type result
  9. Idempotence of finitely generated commutative semifields
  10. N-Lusin property in metric measure spaces: A new sufficient condition
  11. Group schemes and local densities of ramified hermitian lattices in residue characteristic 2. Part II
  12. Homogeneous Finsler spaces and the flag-wise positively curved condition
  13. Index of Grassmann manifolds and orthogonal shadows
  14. Jantzen filtration and strong linkage principle for modular Lie superalgebras
  15. On the maximum conjecture
https://doi.org/10.1515/forum-2017-0119
Keywords for this article
Jacobi–Eisenstein series; Dirichlet series; quadratic forms; hyperbolic space

Articles in the same Issue

  1. Frontmatter
  2. The Mackey problem for free locally convex spaces
  3. Restricted averaging operators to cones over finite fields
  4. Asymptotic expansion of holonomy
  5. Compactness criteria for real algebraic sets and Newton polyhedra
  6. Unit-graphs and special unit-digraphs on matrix rings
  7. ϕ-amenability and character amenability of Fréchet algebras
  8. Dirichlet series of two variables, real analytic Jacobi–Eisenstein series of matrix index, and Katok–Sarnak type result
  9. Idempotence of finitely generated commutative semifields
  10. N-Lusin property in metric measure spaces: A new sufficient condition
  11. Group schemes and local densities of ramified hermitian lattices in residue characteristic 2. Part II
  12. Homogeneous Finsler spaces and the flag-wise positively curved condition
  13. Index of Grassmann manifolds and orthogonal shadows
  14. Jantzen filtration and strong linkage principle for modular Lie superalgebras
  15. On the maximum conjecture
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