Jan’s lecture

$\DeclareMathOperator{\rank}{rank}$ $\DeclareMathOperator{\SL}{SL}$ $\DeclareMathOperator{\M}{M}$ $\DeclareMathOperator{\Stab}{Stab}$ $\DeclareMathOperator{\SO}{SO}$ $\DeclareMathOperator{\GSpin}{GSpin}$ $\DeclareMathOperator{\GL}{GL}$ $\DeclareMathOperator{\Symm}{Symm}$ $\DeclareMathOperator{\char}{char}$ $\DeclareMathOperator{\Ind}{Ind}$ $\DeclareMathOperator{\gen}{gen}$ $\DeclareMathOperator{\Sp}{Sp}$ $\DeclareMathOperator{\card}{\#}$ $\DeclareMathOperator{\level}{level}$ $\DeclareMathOperator{\sig}{sig}$ $\DeclareMathOperator{\diff}{\, d}$ $\DeclareMathOperator{\vol}{vol}$ $\DeclareMathOperator{\Tr}{tr}$

Jan’s Lecture 1

Let $(L,Q)$ be an even lattice of rank $n$. Assume $2 \mid n$. Let $Q: L \rightarrow \mathbb{Z}$ be a quadratic form. We define $(x,y) = Q(x+y)-Q(x)-Q(y)$. Let $L$ be a positive definite lattice.

Let $p$ be a prime. Let $L_{p} = L \otimes_{\mathbb{Z}} \mathbb{Z}_{p}$. Let $L_{\infty} = L \otimes_\mathbb{Z} \mathbb{R}$. We say that $(L_{1},Q_{1})$ is the genus of $(L_{2},Q_{2})$ ig \[\begin{equation} (L_{1} )_{p}\simeq (L_{2})_{p} \text{ for each } p \text{ and for } \infty. \end{equation}\]

Therefore, we can assume that $L_{1}$ and $L_{2}$ are embedded in the same quadratic vector $\mathbb{Q}$-space. The set $\gen(L)$ consists of finitely many isometry classes.

Theta series

The theta series assiciated with $L$ is \[\begin{equation} \Theta_{L}(\tau) = \sum_{x \in L}^{} e^{2 \pi i Q(x)\tau}, \tau \in \mathbb{H}. \end{equation}\] For $\mu \in L'/L$, we define \[\begin{equation} \Theta_{\mu + L}(\tau) = \sum_{x \in \mu + L}^{} e^{2 \pi i Q(x) \tau}. \end{equation}\]

One has \[\Theta_{\Lambda}(\tau) \in \mathcal{M}_{n/2}(\Gamma_{0}(N), \chi_{\Delta}).\] Here $\mathcal{M}_{n/2}$ is the space of modular forms, $\chi_{\Delta}(x) = (\Delta/x)$ is the Legendre symbol and \[N = \level(L) = \min \{\alpha \in \mathbb{Z}_{\geq 0} \mid \alpha Q(x) \in \mathbb{Z} \text{ for each } x \in L'\}.\] We also have \[\begin{equation} \Theta_{\mu + L} \in \mathcal{M}_{n/2} (\Gamma(N)). \end{equation}\]

When $L= E_{8}$, the even unimodular positive definite lattices of rank $8$, one has \[\begin{equation} \Theta_{E_{8}} (\tau) \in \mathcal{M}_{4}(\Gamma(1)) = \mathbb{C} \cdot E_{4}. \end{equation}\] Using this one-dimensionality, we have \[\begin{equation} \Theta_{E_{8}} = E_{4}. \end{equation}\]

Using this, one can say that the representation numbers defined as \[\begin{equation} r_L(m) = \# \{x \in L \mid Q(x) = m\} , \end{equation}\] satisfy \[\begin{equation} r_{E_{8}}(m) = 240 \sigma_{3}(m). \end{equation}\]

In general, one can get exact formulas for the average representation numbers. \[\begin{equation} E_{k}(\tau) = \sum_{\gamma \in \Gamma_{\infty} \setminus \Gamma(1)}^{} (c \tau + d)^{-k} \text{ for }k > 2 \in \mathcal{M}_{k/2}(\Gamma(1)). \end{equation}\] This is equal to \[\begin{equation} 1 + \frac{2k}{ B_{k}} \sum_{n=1}^{\infty} \sigma_{k-1}(n) q^{n}. \end{equation}\]

(Siegel)

Assume $L$ is positive definite unimodular of rank $n$. Then \[\begin{equation} \sum_{M \in \gen(L)/\sim}^{} \frac{1}{\# O(M)} \Theta_{M}(\tau) = \Big( \sum_{M \in \gen(L)/\sim}^{} \frac{1}{\card O(M)}\Big) E_{n/2}(\tau). \end{equation}\]

Define $h(\tau)$ to be the difference of lhs and rhs. Let $k = n/2$. We want to show that $h(\tau) = 0$. Observe that $h \in \mathcal{M}_{n/2}(\Gamma(1))$ and in fact $h \in \mathcal{S}_{n/2}(\Gamma(1))$ (the space of cusp forms).

Let $p$ be a prime and let $T_{p}$ be the $p$th Hecke operator on $\Gamma_{n/2}(\Gamma(1))$. Here we know that \[\begin{equation} T_{p}E_{k} = \sigma_{k-1}(p) \cdot E_{k}. \end{equation}\]

We then show the following lemma.

\[\begin{equation} T_{p}(\text{ LHS }) = \sigma_{k-1}(p) (\text{ LHS }). \end{equation}\]

And then we need the following lemma.

If $0 \neq f \in \mathcal{S}_{k}(\Gamma(1))$ and if $T_{p}(f)= \lambda_{p}\cdot f$, then \[\begin{equation} |\lambda_{p}| \leq 2 p^{ (k-1)/2}. \end{equation}\]

Then, since we have \[\begin{equation} |\lambda_{p}| \leq p^{k/2} + p^{k/2-1} , \end{equation}\] and $k = n/2 \geq 4$. So \[\begin{equation} \sigma_{k-1} = p^{k-1} + 1 > p^{\frac{n}{4} }+ p^{\frac{n}{4}-1} \implies h=0. \end{equation}\]

Jan’s Lecture 2

Adelic version of the SW formula.

The following is our setup. Let $(V,Q)$ be a quadratic space over $\mathbb{Q}$. More precisely, let $Q:V \rightarrow \mathbb{Q}$ be a quadratic form, non-degenerate. We write \[\begin{equation} \sig(Q) = (b^{+},b^{-}), b^{+} + b^{-} = l. \end{equation}\] We define $O(V)$ to be the orthogonal group of $(V,Q)$. We have a quadratic character $\chi_{V} : \mathbb{A}^{\times}/\mathbb{Q}^{\times} \rightarrow \{ \pm 1\}$ , defined using the Hilbert symbol $\chi_{V}(x) = (x,(-1)^{l/2} \det (V))_{\mathbb{Q}}$

Let $W = \mathbb{Q}^{2r}$ (row-vectors). Each $x \in \mathbb{Q}^{2r}$ is written as $x=(x,x')$. We define \[\begin{equation} \langle(x,x') , (y,y')\rangle = x (y ')^{\dagger} - x' y^{\dagger}. \end{equation}\] We write $(W, \langle\ , \ \rangle)$ to be the symplectic space of dimension $2r$.

We denote $G= \Sp(W) = \Sp_{r}$. We have $\Sp_{r}(\mathbb{Q})= \{ g \in \GL_{2r}(\mathbb{Q}) \mid g^{\dagger} J g = J \}$
where \[\begin{equation} J = \begin{bmatrix} 0 & -1_{r} \\ 1_{r} & 0 \end{bmatrix}. \end{equation}\] We denote $P = \{ \Big( \begin{smallmatrix} * & * \\ 0_{r} & * \end{smallmatrix} \Big)\} \subseteq \Sp_{r}$ to be the matrices in a certain block form. This is called the Siegel Parabolic subgroup.

Note that \[\begin{equation} P = M N \end{equation}\] where \[\begin{equation} M =\{m(\alpha) = \Big( \begin{smallmatrix} \alpha & 0 \\ 0 & \alpha^{-1} \end{smallmatrix}\Big) \in \mid \alpha \in \GL_{r} \} \end{equation}\] and \[\begin{equation} N =\{n(\beta) = \Big( \begin{smallmatrix} 1_{r} & \beta \\ 0 & 1_{r} \end{smallmatrix}\Big) \in \mid \beta \in \Symm_{r} \} \end{equation}\]

We have $K_{\infty} = U(r)$ which is \[\begin{equation} U(r) = \{\Big( \begin{smallmatrix} a & b \\ -b & a \end{smallmatrix}\Big) \mid z = a + i b, z z^{\dagger} = 1 \} \end{equation}\]

Let $\omega = \omega_{\psi}$ be the Weil representation of $G \times H$ associated with the additive character $\psi: \mathbb{A}/\mathbb{Q} \rightarrow \mathbb{C}^{(1)}$.

$\omega$ can be realized on the Schwartz-Bruhat functions \[\begin{equation} \mathcal{S}(V(\mathbb{A})^{r}) \simeq \mathcal{S}(V(\mathbb{A}_{f})^{r}) \otimes \mathcal{S}(V(\mathbb{R})^{r}). \end{equation}\]

With $\varphi \in \mathcal{S}(V(\mathbb{A}))$, $$ is characterized by \[\begin{equation} (\omega(h) \varphi )(x) = \varphi(h^{-1} x ), \text{ for } h \in H(\mathbb{A}), \end{equation}\] \[\begin{equation} (\omega(m(\alpha))\varphi)(x) = \chi_{v}( \det (\alpha)) |\det \alpha|_{\mathbb{A}}^{l/2} \varphi(x \alpha), \end{equation}\] and \[\begin{equation} ( \omega(n(b)) \varphi)(x) = \psi( \Tr(b Q(x))) \cdot \varphi(x), \end{equation}\] where $Q(x) = 1/2 (x_{i,x_{j}})_{i,j}$

Here $x = (x_{1},\dots,x_{r}) \in V^{r}(\mathbb{A})$ and $h^{-1} x = (h^{-1 x_{1}, \dots, h^{-1} x_{r}})$. The most interesting action is \[\begin{equation} (\omega(J) \varphi)(x) = \gamma(V)^{r} \int_{ V(\mathbb{A})^{r}}^{} \varphi(y) \psi(-\Tr(x,y)) \diff y, \end{equation}\] where $\gamma(V)$ is some eigth root of unity.

Theta functions

The average value with respect to $H$ is \[\begin{equation} I(g,\varphi) = \int_{H(\mathbb{Q}) \setminus H(\mathbb{A})}^{} \theta(g,h,\varphi) \diff h, \end{equation}\] where $\diff h$ is the unique Haar measure on $H(\mathbb{A})$ such that $\vol(H(\mathbb{Q}) \setminus H(\mathbb{A})) = 1$ . In general, we don’t know if this integral converges. However, we can use the criterion given by Weil.

(Weil convergence criterion)

The integral $I(g, \varphi)$ converges for all $\varphi \in \mathcal{S}(V(\mathbb{A})^{r})$ if 1. $V$ is anisotropic over $\mathbb{Q}$ and 2. The $l$-Witt rank of $V$ is $> r+ 1$.

The character $\psi: \mathbb{A}/\mathbb{Q} \rightarrow \mathbb{C}^{(1)}$ is a product of local characters. That is, \[\begin{equation} \psi = \prod_{p \leq \infty}^{} \psi_{p}. \end{equation}\] Here for $p$ non-Archimidean, one has maps like $\psi_{p}: \mathbb{Q}_{p} \rightarrow \mathbb{C}^{(1)}$ that come from $\mathbb{Q}_{p}/\mathbb{Z}_{p} \rightarrow \mathbb{Q}/\mathbb{Z}$. For $p= \infty$, we have \[\begin{equation} \psi_{\infty}(x) = e^{2 \pi i x}. \end{equation}\]

The exercise is to figure out all these details.

Jan’s Lecture 3

Siegel Eisenstein series

We have $G(\mathbb{A}) = P(\mathbb{A}) \cdot K$ where $K = \prod_{p \leq \infty }^{} K_{p}$ where \[\begin{equation} \text{ for }p< \infty, K_{p} = \Sp_{r}(\mathbb{Z}_{p}) \text{ and } p = \infty, K_{p} = U(r) \subseteq \Sp_{r}(\mathbb{R}). \end{equation}\]

For $g \in G(\mathbb{A})$, write $q = n m(\alpha) k$ for $n \in N$, $m(\alpha) \in M$ and $k \in K$. Put $|a(g)|= |\det a|_{\mathbb{A}}$. Let $g \mapsto |\alpha(g)|$ is norm on $G(\mathbb{A})$ . For $\varphi \in \mathcal{S}(V(\mathbb{A}))$, we put for $s \in \mathbb{C}$ \[\begin{equation} \phi(g,s) = (\omega(g) \varphi)(0) |a(g)|^{s-s_{0}}, \end{equation}\] where $s_{0} = l/2 - s_{r}, s_{r} = (r+1)/2$.

For $\phi(n(b) n(\alpha) g , s) = \chi_{V}(\det(a)) |\det \alpha|^{s} \phi(g,s)$ . This implies \[\begin{equation} \phi(g,s) \in I_{r}(s_{0},\chi_{v}) = \Ind^{G(\mathbb{A})}_{P(\mathbb{A})} (\chi_{V} |\cdot |^{s}) \end{equation}\] This is to say, $\phi$ is a section of the induced representation (I don’t understand what that means).

The map $\lambda : \mathcal{S}(V(\mathbb{A})^{r}) \rightarrow I_{r}(s_{0}, \chi_{V})$ defined as \[\begin{equation} \varphi \mapsto \phi(s_{0}) = (\omega(g) \varphi)(0) \end{equation}\] is a $G(\mathbb{A})$-intertwining operator.

For any $\phi(s) \in I_{r}(s,\chi_{V}), g \in G(\mathbb{A})$ the Eisenstein series is defined as \[\begin{equation} E(g,s,\Phi) = \sum_{\gamma \in P(\mathbb{Q}) \setminus G(\mathbb{Q})}^{} \phi(\gamma g,s) \end{equation}\] converges for $\Re(s) > s_{r}$ if $\phi$ is standard, i.e. $\phi(k,s)$ is independendent of $s$.

(Siegel, Weil, Kudla-Rallis)

Assume that Weil-convergence criteria holds. Let $\gamma \in \mathcal{S}(V(\mathbb{A})^{r})$ and put $\phi(s) = \lambda(\phi)(s) = \phi(g,s) \in I_{r}(s,\chi_{V})$.

Then, $E(g,s,\phi)$ is holomorphic at $s=s_{0}$.
$E(g,s_{0},\Phi) = \kappa I(g,\phi)$ where \[\begin{equation} \kappa = \begin{cases} 1 & \text{ if } l > r+1 \\ 2 & \text{ when } l \leq r+1 \end{cases}. \end{equation}\]

To get the classical formula from yesterday, put $(V,Q)$ positive definite, $(b^{+},b^{-}) = (n,0)$, $l=n$ even $L V $ is even unimodular. We put $r=1$, $\varphi = \varphi_{f} \cdot \varphi_{\infty}$ where \[\begin{equation} \varphi_{f} = \char(\hat{L}), \hat{L} = \prod_{p < \infty} L_{p} \subseteq V(\mathbb{A}_{f}), \end{equation}\] and $\varphi_{\infty}(x) = e^{- \pi (x,x)} \text{ is a Gaussian }.$ Put $K_{\infty} \simeq \SO(2) \subseteq \SL_{2}(\mathbb{R}) \simeq \Sp_{2}(\mathbb{R})$. We have for $k \in \SO(2)$ a rotation by $\theta$ \[\begin{equation} \omega(k) \varphi_{\infty} = e^{i \theta n /2 } \varphi_{\infty}. \end{equation}\] Then, we get \[\begin{equation} \nu^{-n/4} \theta( (g_{\tau},1,1,\dots),h,\varphi) = \theta_{hL} (\tau), \end{equation}\] where $\tau \in \mathbb{H}$, $g_{\tau} i = \tau$ and $h \in H(\mathbb{A})$.

Geometric theta series

Here is the setup. Let $(V,Q)$ be a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. Let $2 \mid n$. Let $L V $ be an even lattice. Let $\SO(V), \SO(L)$ be the Lie group and discrete subgroup. Let $H = \GSpin(V) =\{g \in C^{0}(V)^{\times} \mid g Vg^{-1} = V \}$. Then \[\begin{equation} 1 \rightarrow \mathbb{G}_{m} \rightarrow H \rightarrow \SO(V) \rightarrow 1. \end{equation}\]
We then look at \[\begin{equation} D = \{z \in V_{\mathbb{C}} \mid (z,z) = 0, (z,\overline{z}) < 0\} / \mathbb{C}^{\times}. \end{equation}\] We write $D = D^{+} \cup D^{-} \subseteq P(V_{\mathbb{C}})$. Let $K \subseteq H(\mathbb{A}_{f})$ be a compact open. We denote $X_{K}(\mathbb{C}) = H(\mathbb{Q}) \setminus D \times H(\mathbb{A}_{f})/K$. The complex space of a Shimura variety defined over $\mathbb{Q}$. This is quasi-projective of dimension $n$.

If $F/\mathbb{Q}$ is a real quadratic field. Let $V= \{x \in \mathcal{M}_{2}(F) \mid x^{\dagger} = x'\}$. Write $Q(x) = \det$ and $(V,Q)$ be a rational quadratic structure of signature $(2,2)$. Then $D^{+} = \mathbb{H} \times \mathbb{H}$. Take $H(\mathbb{Q}) = \{ g\in \GL_{2}(F) \mid \det g \in \mathbb{Q}^{\times}\}$. We put $K = \Stab(\M_{2}(\mathcal{O}_{F}) \cap V) \subseteq H(\mathbb{A}_{f})$.

Jan’s Lecture 3

Consider $(V,Q)$ of signature $(n,2)$. We have an exact sequence. \[\begin{equation} 1 \rightarrow \mathbb{G}_{m} \rightarrow \GSpin( V ) \rightarrow \SO(V) \rightarrow 0. \end{equation}\]

We consider \[\begin{equation} X_{K}(\mathbb{C}) = H(\mathbb{Q}) \setminus D \times H(\mathbb{A}_{f}) / K, \end{equation}\] where \[\begin{equation} H(\mathbb{A}) = \cup_{j=1}^{\infty} H(\mathbb{Q}) H(\mathbb{R})^{+} h_{j} K, \end{equation}\] for some $h_{j} \in H(\mathbb{A}_{f})$.

Then $X_{K}(\mathbb{C}) = \cup_{j}^{\infty} X_{j}$ , and $X_{j} = \Gamma_{j} \setminus D^{+}$ and \[\begin{equation} \Gamma_{j} = H(\mathbb{Q}) \cap (H(\mathbb{R})^{+} h_{j} K h_{j}^{-1}). \end{equation}\]

Special cycles

Let $x=(x_{1},\dots,x_{t}) \in V(\mathbb{Q})$
where $Q(x) = \tfrac{1}{2}( (x_{i}),x_{j})_{ij} > 0$. We have $V_{x} = x^{\perp} \subseteq V$. The signature is assumed to be $(n-r,2)$.

Let $H_{x}$ stabilizer of $x$ in $H$. This acts on $D_{x} = \{z \in D \mid z \perp x\} \subseteq D$. For $h \in H(\mathbb{A}_{f})$, one puts $K_{h,x} = H_{x}(\mathbb{A}_{f}) \cap h K h^{-1}$. We set $H_{x}(\mathbb{Q}) \setminus D_{x} \times H_{x}(\mathbb{A}_{f} / K_{h,x}) \rightarrow X_{K},$ where $[z,h_{1}] \rightarrow [z,h_{1}h]$ has codimension $r$ (or maybe $t$? illegible). We get $Z(x,h) \subset X_{k}$.

Weighted special cycles

Take $\Gamma \in \Symm_{r}(\mathbb{Q})$, $\Gamma > 0$ and $\varphi \in \mathcal{S}(V(\mathbb{A}_{f})^{r})$. If there exists $x \in V^{r}$ such that $Q(x) = \Gamma$, put \[\begin{equation} Z(\Gamma, \Phi) = \sum_{h \in H_{x}(\mathbb{A}_{f}) \setminus H(\mathbb{A}_{f}) / K }^{} \varphi (h^{1} x ) Z(x,h) \in Z^{r}(X_{K})_{\mathbb{C}}, \end{equation}\] otherwise put $Z(\Gamma,\varphi) = 0$.

We can extend this definition to $\Gamma \geq 0$ by \[\begin{equation} [Z(\Gamma,\varphi) ] \cdot [Z^{v}] \in H^{2r}(X_{K},\mathbb{C}), \end{equation}\] where $r = \rank(\Gamma)$.

Generating series

One has \[\theta_{r}^{\coh}(\tau , \varphi ) = \sum_{ \Gamma \geq 0}^{} [Z(T,\varphi)] q^{\Gamma}.\] Here $\Gamma \in \Symm_{r}(\mathbb{Q}), \Gamma \geq 0$ We define $q^{\Gamma} = e^{2 \pi i \Tr(\Gamma \tau)}$.

(Kudla-Millson, 1990-1997)

We have $\theta_{r}^{\coh}(\tau, \varphi)$ is a holomorphic Siegel modular form of weight $1+n/2$, of genus $r$ (illegible) and values in $H^{2r}(X_{K},\mathbb{C})$.

(Idea)

Use Kudla-Millson Schartz forms. \[\begin{equation} \varphi^{r}_{KM} (x,z) \in [S(V(\mathbb{R})^{r}) \otimes A^{2r}(D)], \end{equation}\] for \[\begin{equation} x \in V(\mathbb{R})^{r}, z \in D. \end{equation}\]

Here

$\varphi_{KM}^{r}( \times , \cdot )$ is Poincare dual to $\mathcal{D}_{+} \subset D$
Weight is $k=1+n/2$ under $U(r) \subset \Sp_{r}(\mathbb{R})$
\[\theta_{r}(\tau , h, \varphi) = \det (v)^{-k/2} . \sum_{x \in V(\mathbb{Q})^{r}}^{} \varphi(h^{-1} x ) \cdot (\omega(g_{\tau}) \varphi^{r}_{KM})(x).\] Here $\mathbb{H}_{r}, z \in D, h \in H(\mathbb{A}_{f})$ and $\varphi \in \mathcal{S}(V(\mathbb{A}_{f})^{r})^{K}$. We also have $g_{\tau} \cdot i = \tau = k+ i v \in \Sp_{r}(\mathbb{R})$. The above is a non-holomorphic Siegel modular form for $\Sp_{r}$ of weight $k$ with values in $A^{2r}(X_{K})$. \[\begin{equation} [\Theta_{r}(\tau, z,h,\varphi)] = \theta_{r}^{\coh}(\tau,\varphi). \end{equation}\]

Formula

Degree map \[\begin{align} \deg : H^{2r}(X_{K},\mathbb{C}) & \rightarrow \mathbb{C} \\ [\alpha] & \mapsto \int_{X_{K} (\mathbb{C})} \alpha \wedge \Omega^(n-r). \end{align}\] Here $\Omega$ is the Kahler form. Also we have \[\begin{equation} X_{K}(\mathbb{C}) = H(\mathbb{Q}) \setminus D_{+} \times H(\mathbb{A_{f}} / K ) \end{equation}\]

From Kudla-Millson theorem, we have \[\begin{equation} \deg \theta_{r}^{\coh} (\tau , \varphi ) = \sum_{\Gamma \geq 0}^{} \deg [ Z (\Gamma, \Phi)] q^{\Gamma} \end{equation}\] is holomorphic SMF of weight $k$ for $\Sp_{r}$.

(Kudla)

Assume that Weil convergence criterion holds. Then $\deg \theta_{r}^{\coh}(\tau, \varphi) = \vol(X_{K}(\mathbb{C}), \Omega^{r}) \cdot E(\tau, s , \lambda(\varphi \otimes \Phi_{k}) )$ .

Jan’s Lecture 1

Theta series

Jan’s Lecture 2

Adelic version of the SW formula.

Theta functions

Jan’s Lecture 3

Siegel Eisenstein series

Geometric theta series

Jan’s Lecture 3

Special cycles

Weighted special cycles

Generating series

Formula

References