- #1

binbagsss

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## Homework Statement

For the theta series given by : ##\theta(t) = \sum\limits_{n\in Z} e^{2\pi i n^2 t} ## in 1-d there is the transformation formula:

##\theta(-1/4t)=\sqrt{-2it\theta(t)}##

To prove this one uses the fact that this theta function is holomorphic and so by Riemann theorem it is sufficient to prove this on one line only- let this line chosen be an imaginary line ##t=ib##, so need to show:

##\sum\limits_{n\in Z} e^{\frac{- \pi n^2} {2b}} = \sqrt{2b}\sum\limits_{n\in Z} e^{- 2 \pi b n ^{2}} ## [1]

The proof then follows from the Poisson summation formula and that the Fourier transform of a Gaussian returns itself when the Fourier transform is defined as, taking the function gaussian :

##F(e^{- \pi x^{2}})= \int\limits^{\infty}_{-\infty} e^{- \pi x^{2}} e^{-2 i \pi xy } dx = e^{-\pi y ^2}## [2]

(And where the Poisson summation formula is ##\sum\limits_{n \in Z} F(f(n))=\sum\limits_{n \in Z} f(n) ##

and then take the Fourier transform of the LHS of [1] , and make the change of variables ##x'=x/(2b)^{1/2}## to make use of [2] shows

##F(e^{- \pi x^{2}/2b}) = \sqrt{2b} e^{- 2 \pi y^2 } ##

and then the result follows from the Poisson summation formula.

QUESTION

I want to generalise this to the m-d case where here the theta transformation formula is

##\theta(t,A)=\sum\limits_{x \in Z^m} e^{\pi i A[x] t} ##

where ##A[x] = x^t A x ##, ##x## a vector (sorry i don't know how to use vector notation)

## Homework Equations

The Poisson summation formula and Fourier transform generalised from 1-d , as above, to m-d are obtained by sending ##\sum_{n \in Z} \to \sum_{n \in Z^m} ## for the Poisson summation and sending ##e^{-2 \pi i xy} \to e^{ -2 \pi i x^t y } ## in the Fourier transform

## The Attempt at a Solution

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So the proof is to follow the above really. Again proving on an imaginary line ##t=ib## so we then want to show :

##\sum_{x \in Z^m} e^{- \pi A[x] / b } = \frac{\sqrt{b}^{m}}{\sqrt{det A}} \sum_{x \in Z^{m}} e^{- \pi b A^{-1} [x] }##

So again the idea is to show that ##F(e^{- \pi A[x] / b }) = \frac{\sqrt{b}^{m}}{\sqrt{det A}} e^{- \pi b A^{-1} [x]} ###

and then result follows from the poisson.

So we have that [2] holds in m-d and again need a transform of variable.

MY QUESTION

This is where I am stuck , in order to make use of the Gaussian result I need to transform ## \frac{x^t A x}{t} \to x^t x ## , I am unsure how to do this since the ##x^t x ## compared to simply just ##x## is throwing me, and then secondly I am unsure of the general rules for substitution when a matrix ##A## is involved, that will give arise to the required ##det A## that is needed.