Talk 9 : Theta series

Two approaches.

• Siegel, ... : Generating functions for counting problems

• Geometrically, from Jacobi map on abelian varieties.

1. \(\Theta\)-functions and quadratic forms.

Quadratic form \(Q\) = Vector space with positive definite, symmetric, bilinear inner product \(\cdot\).

Let \(\Gamma\) be a lattice such that if \(x\cdot y \in \mathbb{Z}\) for every \(x\) if and only if \(y\in \Gamma\) (self-dual). It corresponds to a quadratic form over \(\mathbb{Z}\) (even diagonal).

Question. In how many ways does \(Q\) represent a given integer ?

Let \(r(Q,a) = |\{\overline{x}\in \Gamma : \overline{x}\cdot \overline{x} = a\}|<\infty\) since \(Q\) is positive definite.

Generating function \(\Theta_\Gamma(t) = \sum_{x\in \Gamma}e^{-\pi t x\cdot x}\), where \(t\in \mathbb{R}\). We have \(\Theta_\Gamma(t) = \sum_{a\in \mathbb{Z}}r(Q,a)q^a\) where \(q=e^{-\pi t}\).

Exercise. \(\Theta_\Gamma(t) = t^{-n/2}\mathrm{vol}(V/\Gamma)\Theta_\Gamma(t^{-1})\). If \(\Gamma\) was not self-dual, the right \(\Theta\) would be over \(\Gamma'=\{y:y.x\in \mathbb{Z} \ \forall X\in \Gamma\}\), the dual of \(\Gamma\). The proof is using Poisson summation formula.

Note. \(r(Q,a)\leq C q^{n/2}\) so the series converges for \(|q|<1\). Hence more generally we can take \(q=e^{2\pi i z}\) with \(z\in \mathbb{H}\) and it will converge.

Theorem.

• 1. \(\dim(V)\) is divisible by \(8\).
• 2. \(\Theta_\Gamma\) is a modular form of weight \(n/2\) (i.e. \(\Theta_\Gamma(-1/z) = (-iz)^{n/2}\Theta_\Gamma(z)\)).

Corollary. There exists a cusp form of weight \(n/2=2k\) such that \(\Theta_\Gamma = E_{2k}+f_\Gamma\) (because \(\Theta_\Gamma(\infty)=1\), so \(\Theta-E_{2k}\in S_{2k}\)).

So we get \(r_Q(a) = \frac{4k}{B_k}\sigma_{2k-1}(a)+O(a^k)\).

Note. If \(n=8\), there are no cusp form of weight \(8/2=4\) so \(\Theta_\Gamma=E_4\).

Genus of \(\Gamma\). The set of quadratic forms equivalent to \(Q\) (equivalently lattices equivalent to \(\Gamma\)) over \(\mathbb{Q}\).

The Minkowski-Siegel mass formula ``computes'' \(\sum_{\Gamma'\in \mathrm{genus}/\mathbb{Z}-\mathrm{equiv}} \frac{1}{|\mathrm{Aut}(\Gamma')|}=: M_\Gamma\).

We have the Siegel-Weil identity : \(\sum_{\Gamma'\in \mathrm{genus}(\Gamma)}\frac{1}{|\mathrm{Aut}(\Gamma')|}\Theta_{\Gamma'}= M_\Gamma\cdot E_{2k}\) (on average, over a genus, \(f_\Gamma\) disappears). It is an example of \(\Theta-lift\).

For \(n=8\) : the genus of self dual lattices has only one isometry class : root lattice of \(E_8\), and \(|\mathrm{Aut}(\Gamma)|=|W_{E_8}|=2^{14}3^55^27\).

Siegel. Natural generalization : representing a quadratic form in \(m\) variables by a quadratic form in \(n\) variables

\[ X^t\overset{\text{pos. def.}}{\overbrace{\underset{n\times n}{\underbrace{Q}}}}\underset{n\times m}{\underbrace{X}} = \overset{\text{pos. semi-def}}{\overbrace{\underset{m\times m}{\underbrace{A}}}}. \] Want \(X\) to have coeffs in \(\mathbb{Z}\). $r(Q,a) is a special case when \(A=(a)\), the quadratic form is \(ax^2\).

\[\Theta^n (z,Q):= \sum_{A \text{pos. semi-def. }n\times n\text{ mat }/\mathbb{Z}} r(Q,A)\exp(\pi i\mathrm{trace}(Az)),\] \(z\in \mathcal{H}_g\), the Siegel upper-half plane.

Key point. The automorphy factor by \(\begin{bmatrix}A&B\\C&D\end{bmatrix}\) : \(C\tau+D\). One approach (classical), scalar-valued autom factor : \(\det(C\tau+D)^{k}\)

Koecher effect : When \(g>1\), ``holom at \(\infty\)'' condition is automatic ! (you cannot have poles, every singularity is removable for functions of several complex variables).

Definition. Siegel modular form of weight \(k\) is a holomorphic function \(f: \mathcal{H}_g\rightarrow \mathbb{C}\) such that \(f(\gamma(\tau)) = \det(C\tau+D)^{k}f(\tau)\), where \(\gamma = \begin{bmatrix}A&B\\C&D\end{bmatrix}\).

Useful generalization : vector-valued siegel modular form :

if \(g>1\), let \(\rho : \mathrm{GL}_g(\mathbb{C})\rightarrow \mathrm{GL}(V)\) be a representation (classical : \(\rho = \det^k\), enough to deal with irreps ). A weight \(\rho\) modular form is a holom map \(f:\mathcal{H}_g\rightarrow V\) such that \(f(\gamma(\tau)) = \rho(C\tau+D)f(\tau)\) for \(\gamma \in \mathrm{Sp}_{2g}(\mathbb{Z})\) (or congruence subgroup).

Fourier expansions \(f(\tau) = \sum_{n}\underset{\in V}{\underbrace{a(n)}} \underset{\text{``q''}}{\underbrace{e^{2\pi i \mathrm{trace}(n\tau)}}}\), where the sum is taken over \(n\) elements of \(\mathrm{GL}_n(\mathbb{Q})\) such that \(2n\in \mathrm{M}_g(\mathbb{Z})\) with even diagonal.

Geometric picture.

Why abelian varieties ? because of abelian integrals.

Abel : Tries to compute \(\int f(x,y(x))\ \mathrm{d}x\) when \(y\) satisfies an algebraic equation \(F(x,y(x)) = 0\) for \(F\in \mathbb{C}[x,y]\).

Example. \(\int \frac{\mathrm{d}x}{\sqrt{x^2+ax+b}}.\) Arc length of an ellipse. If \(y\) satisfies a quadratic equation then we get \(\int \frac{\mathrm{d}t}{\sqrt{(1-t^2)(1-k^2t^2)}} = \frac{\Theta_3(0)\Theta_1(v)}{\Theta_2(0)\Theta_0(v)}\) for some \(v\), where \(\Theta_{0,1,2,3}\) are the Jacobi Theta functions (Exercise for the future).

For \(y^2=x^3+ax+b\). A point on an elliptic curve is of the form \((\mathcal{p},\mathcal{p}')\) since \(X\cong \mathbb{C}/\Lambda\), the curve is its own Jacobian, so the integral \(\int \frac{y}{x}\ \mathrm{d}x\) is some log of \(\mathcal{p}\).

In modern terms \(F(x,y(x))=0\) is a curve in \(\mathbb{P}^2\) choose a basis of the homology \(H_1(X,\mathbb{Z})\) \(\gamma_1,\dots,\gamma_g\) and a basis of the De Rham cohomology \(H^1(X,\mathbb{Z})\) \(\omega_1,\dots,\omega_g\). The \(\mathbb{Z}\)-span of \(\int_{\gamma_i}\omega_j\) is denoted by \(\Lambda\), its period lattice.

Let \(X\) be a complex projective curve. We can map points \(P\in X\) to \((\int_{P_0}^Pw_1,\dots,\int_{P_0}^P \omega_g)\) mod \(\Lambda\), this gives a map \(X\rightarrow \mathbb{C}^g/\Lambda\), the abelian variety \(\mathbb{C}^g/\Lambda\) is called \(\mathrm{Jac}(X)\) the Jacobian of \(X\). The map is Abel-Jacobi map.

Theorem. Abel : This map is injective. Jacobi : This map is surjective.

Riemann's \(\Theta\)-functions. For \(\tau\in \mathcal{H}_g\). Recall \(\Gamma\backslash \mathcal{H}_g = \mathcal{M}_g\) the moduli space of \(\mathcal{C}^g/\Lambda\) that have complex structure.

\[Theta(z,\tau) = \sum_{n\in \mathbb{Z}^g}e^{2\pi i (u^t\tau u + 2 u^t z)},\] it converges uniformlu on compact subsets of \(\mathbb{C}^g\times \mathcal{H}_g\).

We can express \(\Theta^n(Q,Z)\) (the theta series) in terms of such \(\Theta\) function, and get the functional equation from it.

\(\Theta\)-functions.

\(\Theta\begin{bmatrix}a\\b\end{bmatrix}(z,\tau) = e^{2i\pi (a^t\tau a + 2a^t(z+b))}\Theta(z+\tau a+b,\tau)\).

analogy for \(g=1\) : \(j : \Gamma\backslash\mathbb{H} \rightarrow \mathbb{P}^1\), modular functions are \(\mathbb{C}(j)\).

One can consider sections of a line bundle on \(\Gamma\backslash\mathcal{H}_g\), embed it into a projective space, those \(\Theta\) functions should be generating the ring of modular functions.

Sections of line (vector) bundles :

• 1. on \(\mathbb{C}^g/\Lambda\)
• 2. On \(\Gamma\backslash \mathcal{H}_g\), take universal bundle on \(\Gamma\backslash \mathcal{H}_g\), its sections (which are Siegel modular forms) corresponds to sections of the bundle \(\Gamma\backslash \mathcal{H}_g\times \mathcal{C}^g/\Lambda\).

Step 1: Understand line bundles on \(\Lambda\backslash \mathbb{C}^g\)

Summary : Suppose \(H\) is a hermitian form taking \(\mathbb{Z}\)-values on \(\Lambda\) (get that if \(\Lambda = period matrix\)), equivalently this is a polarization. Elements of \(H^2(X,\mathbb{Z})\) correspond to line bundles on \(\Lambda\backslash \mathbb{C}^g\).

Lefschetz Theorem. \(\Theta\) functions give enough sections for an embedding.

Appell-Humbert Theorem. Every hypersurface on \(\mathbb{C}^g/\Lambda\) is the zero locus of a \(\theta\) function.