Exercise 61, § 1

Let $r\geqslant 1$ be an integer. Let $A = (a_{i,j})$ be a symmetric positive-definite matrix of type $(r, r)$ with integer coefficients, and let $N\geqslant 1$ be an integer such that the matrix $A^*=$ NA$^{-1}$ also has integer coefficients. We denote by Q (resp. $\widetilde{Q}$) the integral quadratic form associated with A (resp. with $A^{-1}$), that is to say that one has $Q(x) =^txAx$ for every $x\in \mathbf{Z}^r$. We denote by $(a, b)\mapsto \langle a|b\rangle_A$ the bilinear form associated with A, so that $\langle a|b\rangle_A=^taAb$.

We denote by $\mathbf{H}$ the set of complex numbers $z$ such that $\mathscr{I}(z)>0$. It is an open subset of $\mathbf{C}$. We denote by log$:\mathbf{H}\rightarrow \mathbf{C}$ the restriction to $\mathbf{H}$ of the principal determination of the logarithm (FVR, III, p. 10). Let $k\in \mathbf{N}$ be an odd integer. For $z\in \mathbf{H}$, we denote by $z^k=$ exp($k$ log($z$)).

a) The group SL$_2(\mathbf{Z})$ acts transitively on $\mathbf{H}$ by

$(a bc d)\cdot z=\frac{az + b}{cz + d},(a bc d)\in$ SL$_2(\mathbf{Z})$

(cf. TA, I, p. 150, exercise 5).

b) For every complex number $z\in \mathbf{H}$ and every $x\in \mathbf{Z}^r$, the series

$\sum_{m\in\mathbf{Z}^r}$ exp($i\pi Q(m)z$)

$m\equiv x$ mod$.N$

converges absolutely and uniformly on compact sets.

We denote by $\theta_A(z;x)$ or $\theta_Q(z;x)$ the sum of this series ("theta function associated with Q"). It is a holomorphic function on $\mathbf{H}$. c) For every $y\in \mathbf{Z}^r$, one has

$\sum_{_r}$ exp($i\pi Q(m+y)z$) $=\surd$det(A)1 $(\frac{i}{z})\sum^{r/2_r}$ exp$(-i\pi \frac{\widetilde{Q}(m)}{z}+m\cdot y)$,

$m\in \mathbf{Z}m\in \mathbf{Z}$

(apply the Poisson formula and exercise 1).

d) We denote by H the subgroup of $(\mathbf{Z}/N\mathbf{Z})^r$ formed by the elements $\alpha \in (\mathbf{Z}/N\mathbf{Z})^r$ such that $A\alpha = 0$ in $\mathbf{Z}/N\mathbf{Z}$. The group H is of cardinal det(A). Every character of H is of the form

$\psi_{\beta}(\alpha ) =$ exp$(2i\pi \frac{\langle\alpha|\beta\rangle_A}{N^2})$,

where $\beta \in H$.

e) For every $\alpha \in H$ and $z\in \mathbf{H}$, one has $\theta_Q(z+ 2;\alpha ) =$ exp$(\frac{2i\pi Q(\alpha)}{N^2})\theta_Q(z;\alpha )$. f) For every $\alpha \in H$ and $z\in \mathbf{H}$, one has

$\theta_Q(-\frac{1}{z};\alpha )=\surd$det(A)1 $(-iz)^{r/2}\sum\psi_{\beta}(\alpha )\theta_Q(z;\beta )$.

$\beta \in H$

g) Let $g=\in$ SL$_2(\mathbf{Z})$ be such that $c\not= 0,d >0$ and $b\equiv c\equiv 0$ mod$.2$. For

$\alpha \in H$ and $z\in \mathbf{H}$, one has

$\theta_Q(g\cdot z;\alpha ) =d^{r/2}$ det(A)1 $(cz+d)^{r/2}\sum\Phi (\alpha , \beta )\theta_Q(z;\beta )$,

$\beta \in H$

where

$\Phi (\alpha , \beta ) =\sum\psi_{\gamma}(\alpha )(\sum$ exp$(\frac{i\pi}{dN^2}(bQ(\delta ) + 2\langle \gamma |\delta \rangle_A-cQ(\gamma ))))$

$\gamma \in H\delta$ mod$. dN$

$\delta \equiv \alpha$ mod$.N$

(first consider $\theta_Q(\widetilde{g}\cdot z;\alpha )$ where $\widetilde{g}=g$ ; note the formula

1 0

$d\widetilde{g}\cdot z=b-(dz-c)^{-1}$, and apply the preceding question ; note that $\Phi (\alpha , \beta )$ depends only on $\beta$ modulo N).

h) Suppose moreover that $c\equiv 0$ mod$.2N$. Then

where

$\tau =\sum$ exp$(\frac{i\pi bQ(\delta)}{dN^2})$.

$\delta$ mod$. dN$ $\delta \equiv x$ mod$.N$

i) Suppose moreover that $d$ is odd. We have $\tau =$ exp$(\frac{i\pi abQ(\alpha)}{N^2})\tau_Q(c;d)$, where

$x\in (\mathbf{Z}/d\mathbf{Z})^r$ ("Gauss sum"). j) Let us write $d=p_1\cdots p_k$, where $p_1, . . .p_k$ are prime numbers. We denote

$k$ $(\frac{n}{d})=\prod\ell_{p_i}(n)$ $i=1$

where, for every odd prime number $p$, $\ell_p$ denotes the unique character of order 2 of $(\mathbf{Z}/p\mathbf{Z})^*$, extended to $\mathbf{Z}/p\mathbf{Z}$ by setting $\ell_p(0) = 0$ ("Jacobi symbol"). If $d$ and $2c$ det(A) are relatively prime, then

$\tau_Q(c, d) =($ det(A)$d)(\overline{\varepsilon}_d(\frac{2c}{d})\surd \overline{d})^r$,

where

$\{1$ if $d\equiv 1$ mod$.4$

$i$ if $d\equiv -1$ mod$.4$.

(Diagonalize the quadratic form and use the preceding exercise.)

k) If $g$ is congruent to the identity modulo 4N, then

This means that the mapping $z\mapsto \theta_q(z;\alpha )$ is a holomorphic modular form of weight $r/2$ and of level 4N. (FM, in preparation.)