On the space of generalized theta-series for certain quadratic forms in any number of variables

Ketevan Shavgulidze

doi:10.1515/ms-2017-0205

Article

On the space of generalized theta-series for certain quadratic forms in any number of variables

Ketevan Shavgulidze

Published/Copyright: January 22, 2019

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From the journal Mathematica Slovaca Volume 69 Issue 1

Abstract

An upper bound of the dimension of vector spaces of generalized theta-series corresponding to some nondiagonal quadratic forms in any number of variables is established. In a number of cases, an upper bound of the dimension of the space of theta-series with respect to the quadratic forms of five variables is improved and the basis of this space is constructed.

MSC 2010: Primary 11E20; Secondary 11F27; 11F30

Keywords: quadratic form; spherical polynomial; generalized theta-series

(Communicated by Federico Pellarin)

Acknowledgement

I am very grateful to the anonymous reviewers for valuable comments concerning this work.

References

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Appendix

Now consider a full example with small ν (ν = 4) and small r (r = 3) to clarify the whole picture.

For quadratic form Q1(x1,x2,x3)=b11x12+b22x22+b33x32+b12x1x2 we have |A| = det A = 2b₃₃(4b₁₁b₂₂− b122), A₁₁ = 4b₂₂b₃₃, A₁₂ = −2b₁₂b₃₃, A₂₂ = 4b₁₁b₃₃, A₁₃ = A₂₃ = 0, A₃₃ = 4b₁₁b₂₂− b122.

Let

P(X)=P(x1,x2,x3)=∑k=0ν∑i=0k∑j=0iakix1ν−kx2k−ix3i

be a spherical function of order ν with respect to the ternary quadratic form Q₁(x₁, x₂, x₃) and

L=(a00a10a11a20a21a22a30…aνν)T

be a column vector, where a_ki (ν ≥ k ≥ i ≥ 0) are the coefficients of polynomial P(x₁, x₂, x₃).

The condition (1) for the quadratic form Q₁(x₁, x₂, x₃) takes the form

1|A|∑k=1ν−1∑i=0k−1(A11(ν−k+1)(ν−k)ak−1i+2A12(ν−k)(k−i)aki +2A13(ν−k)(i+1)aki+1+A22(k−i)(k−i+1)ak+1i+2A23(k−i)(i+1)ak+1i+1+A33(i+2)(i+1)ak+1i+2)x1ν−k−1x2k−i−1x3i=0.

In the matrix equation

S⋅L=0,

for ν = 4 the matrix S has the following form

S=12A116A1202A2202A3300000000006A1108A12006A2202A33000000006A1104A12002A2206A33000000002A11006A1200012A2202A330000002A11004A120006A2206A330000002A11002A120002A22012A33.

Consider all possible polynomials P_ki, with even indices i and k = ν − 1, ν; their number is 5 for ν = 4:

P30=b12(b122−2b11b22)4b223x14+b122−b11b22b222x13x2+3b122b22x12x22+x1x23,P32=b12(b122−4b11b22)24b222b33x14+b122−4b11b2212b22b33x13x2+b122b22x12x32+x1x2x32,P40=−b11(b122−b11b22)b223x14−4b11b12b222x13x2−6b11b22x12x22+x24,P42=(b122−4b11b22)(b122−2b11b22)24b223b33x14+b12(b122−4b11b22)6b222b33x13x2+b122−4b11b224b22b33x12x22−b11b22x12x32+x22x32,P44=(b122−4b11b22)216b222b332x14+3(b122−4b11b22)2b22b33x12x32+x34.

Now we construct the corresponding generalized theta-series:

ϑ(τ,P30,Q1)=∑n=1∞(∑Q1(x)=nP30(x))zn=b12(b122−2b11b22)2b223zb11+⋯+0zb22+…+0zb33+⋯+b12(b122−2b11b22)b223zb11+b33+⋯+0zb22+b33+…,ϑ(τ,P32,Q1)=∑n=1∞(∑Q1(x)=nP32(x))zn=b12(b122−4b11b22)12b222b33zb11+⋯+0zb22+…+0zb33+⋯+(b12(b122−4b11b22)6b222b33+2b12b22)zb11+b33+⋯+0zb22+b33+…,ϑ(τ,P40,Q1)=∑n=1∞(∑Q1(x)=nP40(x))zn=−2b11(b122−b11b22)b223zb11+⋯+2zb22+…+0zb33+⋯−4b11(b122−b11b22)b223zb11+b33+⋯+4zb22+b33+…,ϑ(τ,P42,Q1)=∑n=1∞(∑Q1(x)=nP42(x))zn=(b122−4b11b22)(b122−2b11b22)12b223b33zb11+⋯+0zb22+…+0zb33+…+((b122−4b11b22)(b122−2b11b22)6b223b33−4b11b22)zb11+b33+…+4zb22+b33+…,ϑ(τ,P44,Q1)=∑n=1∞(∑Q1(x)=nP44(x))zn=(b122−4b11b22)28b222b332zb11+⋯+0zb22+…+2zb33+⋯+4((b122−4b11b22)216b222b332+3(b122−4b11b22)2b22b33+1)zb11+b33+…+4zb22+b33+….

These generalized theta-series are linearly independent since the determinant of the fifth order constructed from the coefficients of these theta-series is not equal to zero. By virtue of (8) we have dim T(4,Q1)≤(r−1)(r+2)2=5. Hence these theta-series form the basis of the space T(4, Q₁).

We have the following

Theorem A

LetQ₁(X) be the nondiagonal ternary quadratic form, given byQ1(X)=b11x12+b22x22+b33x32+b12x1x2,then dim T(4, Q₁) = 5 and the generalized theta-series with spherical polynomialsP_ki (k = 3 or 4; i is even):

ϑ(τ,P30,Q1);ϑ(τ,P32,Q1);ϑ(τ,P40,Q1);ϑ(τ,P42,Q1);ϑ(τ,P44,Q1)

form the basis of the spaceT(4, Q₁).

Received: 2017-02-11

Accepted: 2018-02-21

Published Online: 2019-01-22

Published in Print: 2019-02-25

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Articles in the same Issue

https://doi.org/10.1515/ms-2017-0205

Keywords for this article

quadratic form; spherical polynomial; generalized theta-series