Span of Restriction of Hilbert Theta Functions

Theta functions are classical examples of holomorphic modular forms. Given a positive definite, unimodular $\mathbb{Z}$-lattice $L$ of rank $8m$ with $m\in\mathbb{N}$, the associated theta function

(1.1)

$$\theta_{L}(\tau):=\sum_{\lambda\in L}q^{Q(\lambda)},~{}q:={\mathbf{e}}(\tau):=e^{2\pi i\tau},$$

is in $M_{4m}$, the space of elliptic modular forms of weight $4m$ on ${\mathrm{SL}}_{2}(\mathbb{Z})$. For example, the theta functions associated to the $E_{8}$ lattice and Leech lattice $\Lambda_{24}$ are explicitly given as

(1.2)

$$\theta_{E_{8}}(\tau)=E_{4}(\tau),~{}\theta_{\Lambda_{24}}(\tau)=E_{4}(\tau)^{3}-720\Delta(\tau),$$

where $E_{2k}(\tau)$ is the Eisenstein series of weight $2k$ and $\Delta(\tau)$ is the Ramanujan $\Delta$-function.

For $N\in\mathbb{N}$, we denote

(1.3)

$${\mathcal{M}}^{(N)}_{\mathbb{Q}}:=\bigoplus_{k\in\mathbb{N}}M_{Nk}$$

the finitely generated graded algebra of elliptic modular forms with weights divisible by $N$, and would like to consider the subalgebra ${\mathcal{M}}^{\theta}_{\mathbb{Q}}\subset{\mathcal{M}}^{(4)}_{\mathbb{Q}}$ generated by theta functions of unimodular lattices. Using the relation

(1.4)

$$\theta_{L_{1}\oplus L_{2}}(\tau)=\theta_{L_{1}}(\tau)\theta_{L_{2}}(\tau).$$

for any two unimodular lattices $L_{1},L_{2}$, we see that ${\mathcal{M}}^{\theta}_{\mathbb{Q}}$ is simply the span of such theta functions. Equation (1.2) and the fact ${\mathcal{M}}^{(4)}_{\mathbb{Q}}=\mathbb{C}[E_{4},\Delta]$ imply that

(1.5)

$${\mathcal{M}}^{\theta}_{\mathbb{Q}}={\mathcal{M}}^{(4)}_{\mathbb{Q}}.$$

The construction of theta functions also extends to the case of Hilbert modular forms. Let $F$ be a totally real field of degree $d$ with ring of integers ${\mathcal{O}_{F}}$, and denote $\alpha_{j}\in\mathbb{R}$ the real embeddings of $\alpha\in F$ for $1\leq j\leq d$. For $N\in\mathbb{N}$, denote ${\mathcal{M}}_{F}^{(N)}$ the algebra of holomorphic Hilbert modular forms of parallel weight $Nk$ for $k\in\mathbb{N}$. Given a totally positive definite, $\mathbb{Z}$-unimodular ${\mathcal{O}_{F}}$-lattice $L$ of rank $8m$ (see Definition 1), the associated theta function

(1.6)

$$\theta_{L}(\tau):=\sum_{\lambda\in L}\prod_{j=1}^{d}q_{j}^{Q(\lambda)_{j}},~{}\tau=(\tau_{1},\dots,\tau_{d})\in\mathbb{H}^{d},~{}q_{j}:={\mathbf{e}}(\tau_{j}),$$

is a Hilbert modular form of parallel weight $4m$ on ${\mathrm{SL}}_{2}({\mathcal{O}_{F}})$. It is well-known that such lattice exists precisely when

(1.7)

$$m\in\frac{1}{d_{2}}\mathbb{N},~{}d_{2}:=\gcd(2,d)$$

(see Prop. 2.1). However, their explicit constructions and classification have only been carried out when $d$ is small (see e.g. [Sch94, Wan14]). As a result, the relationship between ${\mathcal{M}}_{F}^{(4/d_{2})}$ and the subalgebra ${\mathcal{M}}^{\theta}_{F}$ generated by such $\theta_{L}$ is not clear.

On the other hand, we have the following diagonal restriction map

	$\displaystyle{\mathcal{M}}^{(N)}_{F}$	$\displaystyle\to{\mathcal{M}}^{(Nd)}_{\mathbb{Q}}$
	$\displaystyle f$	$\displaystyle\mapsto f^{\Delta}(\tau):=f(\tau^{\Delta}),$

where $\tau^{\Delta}=(\tau,\dots,\tau)\in\mathbb{H}^{d}$. In this note, we will investigate the question about the image of ${\mathcal{M}}^{\theta}_{F}$ under this map, which is denoted by $({\mathcal{M}}^{\theta}_{F})^{\Delta}$ and contained in ${\mathcal{M}}^{(4d/d_{2})}_{\mathbb{Q}}$. The main result is as follows.

Based on this, it is then natural to make the following conjecture.

To prove Theorem 1.1, we apply an instance of the Siegel-Weil formula to see that the Hecke Eisenstein series $E_{F,k}$ defined in (2.5) is contained in ${\mathcal{M}}^{\theta}_{F}$ for all $k\in(4/d_{2})\mathbb{N}$. Then we calculate the Petersson inner product between the diagonal restriction of $E_{F,k}$ and an elliptic cusp form. For $d=2$, this inner product is related to Fourier coefficients of half-integral weight modular forms by a result of Kohnen-Zagier [KZ84]. For $d\geq 3$, we give an expression for this inner product in terms of a sum over the double coset $\Gamma_{F,\infty}\backslash\Gamma_{F}/\Gamma_{\mathbb{Q}}$ (see Prop. 3.1). When $d=3$, we related this double coset to orders in a cubic field $F$ (see section 4). Using these results, we show that when $d=2,3$, ${\mathcal{M}}_{\mathbb{Q}}^{(4d/d_{2})}$ can be generated by $E_{F,k}^{\Delta}$ and $\theta_{L}^{\Delta}$ for a $\mathbb{Z}$-unimodular ${\mathcal{O}_{F}}$-lattice $L$.

The same approach can be used to check conjecture 1 numerically when $d\in\{4,5,6,8,10\}$. We list some results for $d=4,5,6$ and $F$ has small discriminants in the last section (see Theorem 6.1).

Acknowledgement: We thank Jan Bruinier and Tonghai Yang for helpful discussion about Prop. 2.3. The authors are supported by the LOEWE research unit USAG, and by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre TRR 326 “Geometry and Arithmetic of Uniformized Structures”, project number 444845124.

Let $F$ be a totally real field of degree $d$ with ring of integers ${\mathcal{O}_{F}}$ and different $\mathfrak{d}_{F}$. Denote ${\mathrm{Cl}}(F)$ the (wide) class group of $F$. Let $(V,Q)$ be an $F$-quadratic space of dimension $n$. We say that $V$ is totally positive if $V\otimes_{\iota(F)}\mathbb{R}$ is totally positive for every real embedding $\iota:F\hookrightarrow\mathbb{R}$. In that case, ${\mathrm{SO}}_{V}(\mathbb{R})$ is compact and the double quotient ${\mathrm{SO}}_{V}(F)\backslash{\mathrm{SO}}_{V}(\hat{F})/K$ is a finite set for any open compact subgroup $K\subset{\mathrm{SO}}_{V}(\hat{F})$. Here $\mathbb{A}_{F}$ and $\hat{F}$ are the adele and finite adele of $F$.

A finitely generated ${\mathcal{O}_{F}}$-module $L\subset V$ is called a (${\mathcal{O}_{F}}$-)lattice if $L\otimes_{\mathcal{O}_{F}}F=V$. We denote $\hat{L}:=L\otimes\hat{\mathbb{Z}}\subset\hat{V}=V\otimes\hat{\mathbb{Q}}$. If $Q(L)\subset\mathfrak{d}_{F}^{-1}$, we say that $L$ is $\mathbb{Z}$-even integral and call the lattice

(2.1)

$$L^{\prime}:=\{y\in V:(y,L)\subset\mathfrak{d}_{F}^{-1}\}$$

its $\mathbb{Z}$-dual. Viewed as a $\mathbb{Z}$-lattice with respect to $Q_{\mathbb{Q}}(x):=\mathrm{tr}_{F/\mathbb{Q}}Q(x)$, such $L$ is even integral with dual $L^{\prime}$.

As a convention, the trivial lattice in the trivial $F$-vector space is totally positive and $\mathbb{Z}$-unimodular. Consider the monoid

(2.2)

$${\mathcal{U}}^{+}_{F}:=\{(L,Q):L\text{ is an even }\mathbb{Z}\text{-unimodular }{\mathcal{O}_{F}}\text{-lattice and totally positive definite}\}$$

with respect to $\oplus$, and denote ${\mathcal{U}}^{+,n}_{F}\subset{\mathcal{U}}^{+}_{F}$ the subset of lattices of rank $n$. We first have the following result.

For each $L\in{\mathcal{U}}^{+,n}_{F}$, let $\theta_{L}(\tau)$ be the associated theta function defined in (1.6). It is a Hilbert modular form of parallel weight $n/2$ for ${\mathrm{SL}}_{2}({\mathcal{O}_{F}})$. Now, the Siegel-Weil formula [Sie66, Wei65] gives us the following result.

We can rewrite the Hecke-Eisenstein series $E_{F,k}$ as

$$E_{F,k}(\tau):=1+\sum_{\begin{subarray}{c}{\mathcal{A}}=[\mathfrak{a}]\in{\mathrm{Cl}}(F)\\ (c,d)\in\mathfrak{a}^{2}/\mathcal{O}_{F}^{\times}\\ c\neq 0\\ {\mathcal{O}_{F}}c+{\mathcal{O}_{F}}d=\mathfrak{a}\end{subarray}}\left(\frac{{\mathrm{Nm}}(\mathfrak{a})}{{\mathrm{Nm}}(c)}\right)^{k}\prod_{j=1}^{d}(\tau_{j}+d_{j}/c_{j})^{-k}$$

For any $\beta\in F$, there is unique ${\mathcal{A}}=[\mathfrak{a}]$ and $(c,d)\in\mathfrak{a}^{2}/\mathcal{O}_{F}^{\times}$ with $c\neq 0$ such that $\mathfrak{a}={\mathcal{O}_{F}}c+{\mathcal{O}_{F}}d$ and $\beta=d/c$. Therefore, we denote

(2.7)

$$A_{\beta}:=\frac{{\mathrm{Nm}}(c)}{{\mathrm{Nm}}(\mathfrak{a})}\in\mathbb{Z}-\{0\}.$$

It is easy to check this definition does not depend on the choice of the representative $\mathfrak{a}$, and

(2.8)

$$A_{\beta+a,k}=A_{\beta,k}$$

for all $a\in\mathbb{Z}$. Then we have

(2.9)

$$E_{F,k}(\tau)=1+\sum_{\beta\in F}A_{\beta}^{-k}\prod_{j=1}^{d}(\tau_{j}+\beta_{j})^{-k}.$$

In this section, let $F/\mathbb{Q}$ be totally real with degree $d\geq 3$. We will give an expression for the Petersson inner product between the diagonal restriction of the Hecke Eisenstein series $E_{F,k}$ and an elliptic cusp form $f$ of weight $dk$.

For $\alpha\in M_{m,n}(F)$ and $1\leq j\leq d$, we write $\alpha_{j}\in M_{m,n}(\mathbb{R})$ with $1\leq j\leq d$ for the real embeddings of $\alpha$. We identify $\mathbb{P}^{1}(F)\cong B(F)\backslash{\mathrm{SL}}_{2}(F)$ via

(3.1)

$$\beta\mapsto\begin{cases}\left(\begin{smallmatrix}*&*\\ 1&\beta\end{smallmatrix}\right)&\beta\in F,\\ \left(\begin{smallmatrix}*&*\\ 0&1\end{smallmatrix}\right)&\beta=\infty.\end{cases}$$

Let $S_{0}\cup\{\infty\}\subset\mathbb{P}^{1}(F)$ be a set of representatives of the double coset $B(F)\backslash{\mathrm{SL}}_{2}(F)/{\mathrm{SL}}_{2}(\mathbb{Z})$. Then $S_{0}\subset F-\mathbb{Q}$ and we can use (2.9) to express the diagonal restriction of $E_{F,k}$ as

(3.2)

$$E_{F,k}^{\Delta}(\tau)=E_{dk}+\sum_{\beta\in S_{0}}E_{F,k,\beta}(\tau),~{}E_{F,k,\beta}(\tau):=\sum_{\gamma\in{\mathrm{SL}}_{2}(\mathbb{Z})}A_{-\gamma^{-1}\cdot(-\beta_{j})}^{-k}\prod_{j=1}^{d}(\tau-\gamma^{-1}\cdot(-\beta_{j}))^{-k}$$

with $\tau\in\mathbb{H}$. Note that $E_{dk}$ is just the elliptic Eisenstein series of weight $dk$.

Let $f(\tau)=\sum_{n\geq 1}c_{n}q^{n}\in S_{dk}$ be a cusp form. We are interested in estimating its inner product with $E^{\Delta}_{F,k}$. By the usual unfolding process, we obtain

	$\displaystyle\langle E^{\Delta}_{F,k},f\rangle$	$\displaystyle=\sum_{\beta\in S_{0}}\int_{\Gamma_{\infty}\backslash\mathbb{H}}E^{\infty}_{F,k,\beta}(\tau)\overline{f(\tau)}v^{dk}\frac{dudv}{v^{2}}$
		$\displaystyle=\sum_{\beta\in S_{0}}\int_{0}^{\infty}\sum_{n\geq 1}\overline{c_{n}}a_{F,k,\beta}(n,v)e^{-2\pi nv}v^{dk-1}\frac{dv}{v},$

where $\Gamma_{\infty}:=B(\mathbb{Q})\cap{\mathrm{SL}}_{2}(\mathbb{Z})$ and

(3.3)

$$\begin{split}E^{\infty}_{F,k,\beta}(\tau)&:=\sum_{\gamma\in\Gamma_{\infty}}A_{-\gamma^{-1}\cdot(-\beta)}^{-k}\prod_{j=1}^{d}(\tau-\gamma^{-1}\cdot(-\beta_{j}))^{-k}\\ &=2A_{\beta}^{-k}\prod_{j=1}^{d}(\tau+\beta_{j}+b)^{-k}=\sum_{n\in\mathbb{Z}}a_{F,k,\beta}(n,v){\mathbf{e}}(nu).\end{split}$$

for $\beta=d/c\in S_{0}$. Here we have $r_{-\gamma\cdot(-\beta)}=r_{\beta}$ for all $\gamma\in\Gamma_{\infty}$ by (2.8). It is easy to see that

(3.4)

$$\begin{split}a_{F,k,\beta}(n,v)&=2A_{\beta}^{-k}\int_{\mathbb{R}}\prod_{j=1}^{d}(u+iv+\beta_{j})^{-k}{\mathbf{e}}(-nu)du\\ &=4\pi i(-A_{\beta})^{-k}\sum_{z\in Z(\beta)}\mathrm{Res}_{x=z}\left({\mathbf{e}}(nx)\prod_{j=1}^{d}(x-(\beta_{j}+iv))^{-k}\right),\end{split}$$

where $Z(\beta):=\{\beta_{j}+iv:1\leq j\leq d\}\subset\mathbb{H}$ since

(3.5)

$$\sum_{z\in Z(\beta)}\mathrm{Res}_{x=z}\left({\mathbf{e}}(nx)\prod_{j=1}^{d}(x-z_{j})^{-k}\right)=\frac{1}{2\pi i}\int_{\mathbb{R}}{\mathbf{e}}(nx)\prod_{j=1}^{d}(x-z_{j})^{-k}dx.$$

Suppose $\beta_{j}$’s are all distinct. Then

	$\displaystyle\sum_{z\in Z(\beta)}\mathrm{Res}_{x=z}$	$\displaystyle\left({\mathbf{e}}(nx)\prod_{j=1}^{d}(x-(\beta_{j}+iv))^{-k}\right)=\frac{1}{\Gamma(k)}\sum_{j=1}^{d}\left(\frac{d}{dx}\right)^{k-1}\left(\frac{{\mathbf{e}}(nx)}{\prod_{j^{\prime}=1,~{}j^{\prime}\neq j}^{d}(x-(\beta_{j^{\prime}}+iv))^{k}}\right)\mid_{x=\beta_{j^{\prime}}+iv}$
		$\displaystyle=\frac{{\mathbf{e}}(niv)}{\Gamma(k)}\sum_{j=1}^{d}\sum_{\ell=0}^{k-1}(2\pi in)^{k-1-\ell}{{\mathbf{e}}(n\beta_{j})e^{-2\pi nv}}\binom{k-1}{\ell}\left(\frac{P_{d-1,k,\ell}}{Q_{d-1,k+\ell}}\right)(\beta_{j}-\beta_{1},\dots,\beta_{j}-\beta_{d}),$

where $P_{m,k,\ell},Q_{m,r}\in\mathbb{Q}[x_{1},\dots,x_{m}]$ are symmetric polynomials of degrees $(m-1)\ell$ and $mr$ defined by

(3.6)

$$\begin{split}P_{m,k,\ell}(x_{1},\dots,x_{m})&:=(x_{1}\dots x_{m})^{k+\ell}(\partial_{x_{1}}+\dots+\partial_{x_{m}})^{\ell}(x_{1}\dots x_{m})^{-k},\\ Q_{m,r}(x_{1},\dots,x_{m})&:=(x_{1}\dots x_{m})^{r}.\end{split}$$

Note that

(3.7)

$$\frac{P_{m,k,\ell}}{Q_{m,k+\ell}}(x_{1},\dots,x_{m})=(-1)^{\ell}{\ell!}\sum_{r=(r_{j})\in\mathbb{N}^{m},~{}\sum_{j}r_{j}=\ell}\left(\!\!\binom{k}{r}\!\!\right)\prod_{j=1}^{m}x_{j}^{-k-r_{j}},$$

where $\left(\!\!\binom{k}{r}\!\!\right):=\frac{k^{(r_{1})}\dots k^{(r_{m})}}{r_{1}!\dots r_{m}!}$ for $r=(r_{1},\dots,r_{m})\in\mathbb{N}^{m}$ with $k^{(n)}:=k(k+1)\dots(k+n-1)$. Substituting this into the unfolding gives us the following result.

When $d=3$, we can identify the double coset $B(F)\backslash{\mathrm{SL}}_{2}(F)/{\mathrm{SL}}_{2}(\mathbb{Z})-\{\infty\}$ with orders in ${\mathcal{O}_{F}}$ in the following way. Let

$$\mathcal{Q}_{F}:=\{f(X,Y)=AX^{3}+BX^{2}Y+CXY^{2}+DY^{3}\in\mathbb{Z}[X,Y]:f(\beta,1)=0\text{ for some }\beta\in F\backslash\mathbb{Q}\}$$

be the set of integral binary cubic forms with a root in $F-\mathbb{Q}$. A form is primitive if its coefficients have no common factor. There is a natural action of ${\mathrm{SL}}_{2}(\mathbb{Z})$ on $\mathcal{Q}_{F}$ that preserves the discriminant

(4.1)

$$\begin{split}\Delta(f)&:=A^{6}((\beta_{1}-\beta_{2})(\beta_{1}-\beta_{3})(\beta_{2}-\beta_{3}))^{2}\\ &=18ABCD+B^{2}C^{2}-4AC^{3}-4B^{3}D-27A^{2}D^{2},\end{split}$$

and the subset of primitive forms. The quantity

(4.2)

$$P(f):=B^{2}-3AC>0$$

is the leading coefficient of the Hessian of $f$, which is a positive definite quadratic form and a coinvariant of $f$. For every $f\in\mathcal{Q}_{F}$, Prop. 2 in [Cre99] gives us $f^{\prime}\sim_{{\mathrm{SL}}_{2}(\mathbb{Z})}f$ satisfying

(4.3)

$$P(f^{\prime})\leq\sqrt{\Delta(f^{\prime})}=\sqrt{\Delta(f)}.$$

Given $\beta\in F-\mathbb{Q}$, we can associate to it a primitive element $f_{\beta}\in\mathcal{Q}_{F}$ defined by

(4.4)

$$f_{\beta}(X,Y):=A_{\beta}\prod_{j=1}^{3}(X-\beta_{j}Y)=A_{\beta}X^{3}+B_{\beta}X^{2}Y+C_{\beta}XY^{2}+D_{\beta}Y^{3}\in\mathcal{Q}_{F}.$$

Note that $f_{\beta}(\beta,1)=0$ and the right action of ${\mathrm{SL}}_{2}(\mathbb{Z})$ on $B(F)\backslash{\mathrm{SL}}_{2}(F)$ corresponds to its natural action on $\mathcal{Q}_{F}$.

To any binary cubic form $f$ with non-zero discriminant and $f(\beta,1)=0$ we can associate the free $\mathbb{Z}$-module of rank 3

(4.5)

$$\mathcal{O}_{f}:=\mathbb{Z}+\mathbb{Z}A\beta+\mathbb{Z}(A\beta^{2}+B\beta+C)\subset\mathbb{Q}(\beta),$$

which is also a commutative ring. A classical result of Delone and Faddeev tells us that this gives a bijection between ${\mathrm{GL}}_{2}(\mathbb{Z})$-classes of binary cubic forms with non-zero discriminants and isomorphism classes of commutative rings that are free $\mathbb{Z}$-modules of rank 3 [DF64]. If we restrict $\beta$ to be in a fixed field $F$, then $\mathcal{O}_{f}$ is an order in ${\mathcal{O}_{F}}$, and $\mathcal{O}_{f_{1}},\mathcal{O}_{f_{2}}\subset{\mathcal{O}_{F}}$ are the same if and only if $f_{1},f_{2}\in\mathcal{Q}_{F}$ are ${\mathrm{GL}}_{2}(\mathbb{Z})$-equivalent (see e.g. [Nak98, Lemma 3.1]). Furthermore, we have

(4.6)

$$\Delta(f)=\Delta(\mathcal{O}_{f})=D_{F}[{\mathcal{O}_{F}}:\mathcal{O}_{f}]^{2}$$

with $\Delta(\cdot)$ the discriminant. For $s=[\beta]\in\mathbb{P}^{1}(F)/{\mathrm{SL}}_{2}(\mathbb{Z})-\{\infty\}$, we then denote

(4.7)

$$\mathcal{O}_{s}:=\mathcal{O}_{f_{\beta}},\Delta(s):=\Delta(\mathcal{O}_{s}).$$

The discussions above lead to the following result.

Finally, the following Dirichlet series

(4.8)

$$\begin{split}\eta_{F}(s)&:=\sum_{\mathcal{O}\subset{\mathcal{O}_{F}}\text{ order}}[{\mathcal{O}_{F}}:\mathcal{O}]^{-s}=\sum_{\mathcal{O}\subset{\mathcal{O}_{F}}\text{ order}}\frac{D_{F}^{s/2}}{\Delta(\mathcal{O})^{s/2}}.\end{split}$$

can be factorized in the following way by a result of Datskovsky and Wright [DW86] (see [Nak98, Lemma 3.2])

(4.9)

$$\eta_{F}(s)=\frac{\zeta_{F}(s)}{\zeta_{F}(2s)}\zeta(2s)\zeta(3s-1).$$