Modular forms and ellipsoidal T-designs

Abstract

In recent work, Miezaki introduced the notion of a spherical T-design in $\mathbb {R}^2$, where T is a potentially infinite set. As an example, he offered the $\mathbb {Z}^2$-lattice points with fixed integer norm (a.k.a. shells). These shells are maximal spherical T-designs, where $T=\mathbb {Z}^+\setminus 4\mathbb {Z}^+$. We generalize the notion of a spherical T-design to special ellipses, and extend Miezaki’s work to the norm form shells for rings of integers of imaginary quadratic fields with class number 1.

1 Introduction and statement of results

Spherical t-designs were introduced in 1977 by Delsarte, Goethals and Seidel [5], and they have played an important role in algebra, combinatorics, number theory and quantum mechanics (for background see [2,3,4, 6, 9, 10]). A spherical t-design is a nonempty finite set of points on the unit sphere with the property that the average value of any real polynomial of degree $\le t$ over this set equals the average value over the sphere. Namely, if $S^{n-1}$ denotes the unit sphere in $\mathbb {R}^n$ centered at the origin, then a finite nonempty subset $X\subset S^{n-1}$ is a spherical t-design if

$$\begin{aligned} \dfrac{1}{|X|}\sum _{x\in X}P(x)= \dfrac{1}{{\text {Vol}}(S^{n-1})}\int _{S^{n-1}} P(x)\mathrm{{d}}\sigma (x) \end{aligned}$$

(1.1)

for all polynomials P(x) of degree $\le t$. The right-hand side of (1.1) is the usual surface integral over $S^{n-1}$. In general, a finite nonempty subset X of $S_{n-1}(r)$, the sphere of radius r centered at the origin, is a spherical t-design if $\frac{1}{r}X$ satisfies (1.1). Since a spherical t-design is also a spherical $t^{\prime }$-design for all $t^{\prime }\le t$, we say that X has strength t if it is the maximum of all such numbers.

Delsarte, Goethals and Seidel developed a very simple criterion for determining spherical t-designs. This criterion involves homogeneous harmonic polynomials of bounded degree. A polynomial in n variables is harmonic if it is annihilated by the Laplacian operator $\Delta :=\sum ^{n}_{i=1}\partial ^2/\partial x^2_i$, and they showed [5] that $X\subset S^{n-1}$ is a spherical t-design if

$$\begin{aligned} \sum _{x\in X}P(x) = 0 \end{aligned}$$

(1.2)

for all homogeneous harmonic polynomials P(x) of nonzero degree $\le t$. This criterion is a consequence of two results from harmonic analysis. The first result is the mean value property for harmonic functions [1, p. 5], which implies that the integral of a harmonic polynomial over a sphere centered at the origin vanishes, combined with the fact that homogeneous polynomials of fixed degree are spanned by certain harmonic polynomials [1, Theorem 5.7].

In view of this framework, it is natural to ask whether there are generalizations of spherical t-designs to other curves, surfaces and varieties. Here we consider certain ellipsoids¹ in dimension two. To be precise, for square-free $D\ge 1$ we define the norm r ellipses

$$\begin{aligned} C_D(r) := {\left\{ \begin{array}{ll} \{(x,y)\in \mathbb {R}^2 : x^2+Dy^2=r\} &{} \text { if } D\equiv 1,2\pmod {4},\\ \{(x,y)\in \mathbb {R}^2 : x^2+xy+\frac{1+D}{4}y^2=r\} &{} \text { if } D\equiv 3\pmod {4}. \end{array}\right. } \end{aligned}$$

(1.3)

Remark

These ellipses arise from certain imaginary quadratic orders.

For $D\equiv 1,2\pmod 4$, we say that a finite nonempty subset $X\subset C_D(r)$ is an ellipsoidal t-design if

$$\begin{aligned} \dfrac{1}{|X|}\sum _{(x,y)\in X}P(x,y)= \dfrac{1}{2\pi \sqrt{D}}\int _{C_D(r)}\dfrac{P(x,y)}{\sqrt{x^2/D^2+y^2}}\mathrm{{d}}\sigma (x,y) \end{aligned}$$

(1.4)

for all polynomials P(x, y) of degree $\le t$ over $\mathbb {R}$. For $D\equiv 3\pmod 4$, instead we require

$$\begin{aligned} \dfrac{1}{|X|}\sum _{(x,y)\in X}P(x,y) = \dfrac{\sqrt{D}}{\pi }\int _{C_D(r)}\dfrac{P(x,y)}{\sqrt{20x^2+(D^2+2D+5)y^2+(20+4D)xy}}\mathrm{{d}}\sigma (x,y). \end{aligned}$$

(1.5)

Here the right-hand sides are line integrals. As in the case of spherical t-designs, every ellipsoidal t-design is also an ellipsoidal $t^{\prime }$-design for all $t^{\prime }\le t$, and the maximum of all such t’s is called the strength of X.These definitions coincide with the notion of a spherical t-design when $D=1$.

In analogy to Delsarte, Goethals and Seidel, we have a natural criterion for confirming ellipsoidal t-designs. To this end, we consider the 2-dimensional real vector space

$$\begin{aligned} H^{\mathbb {R}}_{D,j}[x,y] := {\left\{ \begin{array}{ll} \langle \text {Re}(x+\sqrt{-D}y)^j,\text {Im}(x+\sqrt{-D}y)^j\rangle &{} \text { if } D\equiv 1,2 \pmod {4},\\ \langle \text {Re}(x+\frac{1+\sqrt{-D}}{2}y)^j,\text {Im}(x+\frac{1+\sqrt{-D}}{2}y)^j\rangle &{} \text { if } D\equiv 3 \pmod {4}. \end{array}\right. } \end{aligned}$$

(1.6)

In terms of these vector spaces of polynomials, we have the following ellipsoidal t-design criterion.

Theorem 1.1

A finite nonempty set $X\subset C_D(r)$ is an ellipsoidal t-design if

$$\begin{aligned} \sum _{x\in X}P(x,y)=0 \end{aligned}$$

for all $P(x,y)\in H^{\mathbb {R}}_{D,j}[x,y]$ for all $0<j\le t$.

Remark

(1) Observe that if $X\subset S^{1}$ is a spherical t-design, then $Y=\{(x,y/\sqrt{D})|(x,y)\in X\}\subset C_D$ (resp. $Y=\{(x+y/\sqrt{D},2y/\sqrt{D}|(x,y)\in X\}\subset C_D$) is an ellipsoidal t-design for $D\equiv 1,2 \pmod {4}$ (resp. $D\equiv 3 \pmod {4}$). Therefore, the existence of a spherical t-design implies the existence of a corresponding ellipsoidal t-design. In fact, there is a one-to-one correspondence between spherical t-designs and ellipsoidal t-designs. However, the proof of Theorem 1.1 is not a direct consequence because care is required for justifying the role of the vector spaces $H^{\mathbb {R}}_{D,j}[x,y]$.

(2) Since there is one-to-one correspondence between spherical and ellipsoidal t-designs, we get a lower bound [5, p. 2] on the size of ellipsoidal t-design X,

$$\begin{aligned} |X|\ge t+1. \end{aligned}$$

Recently, Miezaki in [9] introduced a generalization of the notion of spherical t-designs. Instead of restricting to polynomials of degree $\le t$, he considered harmonic polynomials of degree $j\in T\subset \mathbb {N}$, where T is a potentially infinite set. The main theorem from [9] gives infinitely many spherical T-designs for $T :=\mathbb {Z}^{+}\setminus 4\mathbb {Z}^+$ in dimension two. Namely, he considered norm r shells, integer points on $x^2+y^2=r$ for fixed $r\in \mathbb {Z}^+$. He showed that these r-shells are spherical T-designs. Moreover, these sets have strength T, meaning that (1.2) fails if any multiple of 4 is added to T. His proof makes use of theta functions arising from complex multiplication by $\mathbb {Z}[i]$.

We generalize Miezaki’s work to ellipsoidal T-designs. We call $X\subset C_D$ an ellipsoidal T-design if the condition in Theorem 1.1 is satisfied for all polynomials in $H^{\mathbb {R}}_{D,j}[x,y]$ with $j\in T$. We say X has strength T if it is maximal among such sets. For each square-free positive integer D, let $\mathcal {O}_D$ be the ring of integers of $\mathbb {Q}(\sqrt{-D})$. In particular, this means that

$$\begin{aligned} \mathcal {O}_D={\left\{ \begin{array}{ll} \mathbb {Z}[\sqrt{-D}] \ \ \ \ \ &{} \text { if } D\equiv 1,2 \pmod {4}, \\ \mathbb {Z}[\frac{1+\sqrt{-D}}{2}] \ \ \ \ \ &{} \text { if } D\equiv 3 \pmod {4}. \end{array}\right. } \end{aligned}$$

(1.7)

We consider $D\in \{1,2,3,7,11,19,43,67,163\}$, the square-free positive integers for which $\mathcal {O}_D$ has class number 1. To make this precise, we define the norm r shells in $C_D(r)$ by

$$\begin{aligned} \Lambda _D^r:= \mathcal {O}_D \cap C_D(r). \end{aligned}$$

(1.8)

Generalizing Miezaki’s work for $D=1$, we obtain the following theorem.

Theorem 1.2

If $D\in \{1, 2, 3, 7, 11, 19, 43, 67, 163\},$ then every non-empty shell $\Lambda _D^r$ is an ellipsoidal $T_D$ design with strength $T_D$, where

$$\begin{aligned} T_D:={\left\{ \begin{array}{ll} \mathbb {Z}^+\setminus 4\mathbb {Z}^+ \ \ \ \ \ &{}{\text { if}}\ D=1,\\ \mathbb {Z}^+\setminus 6\mathbb {Z}^{+} \ \ \ \ &{}{\text { if}}\ D=3,\\ \mathbb {Z}^+\setminus 2\mathbb {Z}^+ \ \ \ \ &{}{\text { otherwise.}} \end{array}\right. } \end{aligned}$$

Remark

The method used here seems to be well-poised only for the dimension 2 cases. It would be interesting to obtain higher dimensional analogues.

Example

We consider $D=3$, and $r=691$. Then we have

$$\begin{aligned} \Lambda ^{691}_3&=\{(11,19),(-11,-19),(19,11),(-19,-11),\\&\qquad (11,-30),(-11,30),(30,-19),(-30,19),\\&\qquad (30,-11),(-30,11),(19,-30),(-19,30)\}. \end{aligned}$$

We consider the polynomial $P(x,y)=2x^2+3462xy+1729y^2\in H^{\mathbb {R}}_{3,2}[x,y]$, and we find that $\sum _{(x,y)\in \Lambda ^{691}_3}P(x,y)=0$ which shows that $\Lambda ^{691}_3$ is an elliptical 2-design and $2\in T_3$. On the other hand, Theorem 1.2 implies that $\Lambda ^{691}_3$ is not an ellipsoidal 6-design.To see this we choose $Q(x,y)=2x^2+6x^5y-15x^4y^2-40x^3y^3-15x^2y^4+6xy^5+2y^6\in H^{\mathbb {R}}_{3,6}(x,y),$ and we find that $\sum _{(x,y)\in \Lambda ^{691}_3}Q(x,y)=-4818834696\not =0$.

In Sect. 2 we prove Theorem 1.1, criterion for confirming that a set is an ellipsoidal t-design, and in Section 3 we recall the theory of theta functions arising from complex multiplication, and we prove Theorem 1.2.

2 Criterion for ellipsoidal t-design

In this section we prove Theorem 1.1, criterion for confirming ellipsoidal t-designs. Throughout this section we assume that $D\ge 1$ is square-free and $j\ge 1$.

To prove that Theorem 1.1 is indeed a criterion for confirming ellipsoidal t-designs, we first need to show that the spaces $H^{\mathbb {R}}_{D,k}[x,y]$, for $0<k\le j$, generate all the polynomials of degree $\le j$ when restricted to $C_D(r)$. It suffices to show this for $P^{\mathbb {R}}_j[x,y]$, the set of homogeneous polynomials of degree j.

Lemma 2.1

If $D\ge 1$ is square-free and $j\ge 1$, then the following are true:

(1) If $D\equiv 3\mod 4$, then we have

$$\begin{aligned} P^{\mathbb {R}}_j[x,y] = \bigoplus ^{\lfloor j/2\rfloor }_{k=0} (x^2+Dy^2)^{k}H^{\mathbb {R}}_{D,j-2k}[x,y]. \end{aligned}$$

(2) If $D\equiv 3\mod 4$, then we have

$$\begin{aligned} P^{\mathbb {R}}_j[x,y] = \bigoplus ^{\lfloor j/2\rfloor }_{k=0} \Big (x^2+xy+\frac{1+D}{4}y^2\Big )^kH^{\mathbb {R}}_{D,j-2k}[x,y]. \end{aligned}$$

Proof

The lemma is well known for homogeneous harmonic polynomials (for example, see [1, Theorem 5.7]). Namely, if $H^{\mathbb {R}}_{k}[x,y]$ is the set of homogeneous harmonic polynomials of degree k then

$$\begin{aligned} P^{\mathbb {R}}_j(x,y) = \bigoplus ^{\lfloor j/2\rfloor }_{k=0} (x^2+y^2)^{k}H^{\mathbb {R}}_{j-2k}[x,y]. \end{aligned}$$

We extend it to general D. It is well known that $H^{\mathbb {R}}_j[x,y]=\langle \text {Re}(x+iy)^j, \text {Im}(x+iy)^j\rangle$, and so if we do the change of variable for $D\equiv 1,2\mod 4$ (resp. $D\equiv 3\mod 4$), $x^{\prime }=x$,$y^{\prime }=\sqrt{D}y$ (resp. $x^{\prime }=x+y/2$,$y^{\prime }=2y/\sqrt{D}$), then $H^{\mathbb {R}}_{j-2}(x^{\prime },y^{\prime })=\langle \text {Re}(x^{\prime }+ iy^{\prime })^j,\text {Im}(x^{\prime }+iy^{\prime })^j\rangle$ gives

$$\begin{aligned} P^{\mathbb {R}}_j[x^{\prime },y^{\prime }] = \bigoplus ^{\lfloor j/2\rfloor }_{k=0} (x'^2+y'^2)^{k}H^{\mathbb {R}}_{j-2k}[x^{\prime },y^{\prime }]. \end{aligned}$$

Therefore, if $D\equiv 1,2\mod 4$, then we have

$$\begin{aligned} P^{\mathbb {R}}_j(x,y) = \bigoplus ^{\lfloor j/2\rfloor }_{k=0} (x^2+Dy^2)^{k}H^{\mathbb {R}}_{D,j-2k}[x,y]. \end{aligned}$$

If $D\equiv 3\mod 4$, then we have

$$\begin{aligned} P^{\mathbb {R}}_j(x,y) = \bigoplus ^{\lfloor j/2\rfloor }_{k=0} \Big (x^2+xy+\frac{1+D}{4}y^2\Big )^kH^{\mathbb {R}}_{D,j-2k}[x,y]. \end{aligned}$$

$\square$

We now prove Theorem 1.1.

Proof of Theorem 1.1

Lemma 2.1 shows that the set of polynomials when restricted to $C_D$ are generated by the spaces $H^{\mathbb {R}}_{D,j}[x,y]$ since $x^2+Dy^2=r$ (resp., $x^2+xy+\frac{1+D}{4}y^2=r$) on $C_D(r)$. Therefore, it suffices to show that if $P(x,y)\in H^{\mathbb {R}}_{D,j}[x,y]$, then the following are true:

(1) If $D\equiv 1,2\mod 4$, then we have

$$\begin{aligned} \int _{C_D(r)} \dfrac{P(x,y)}{\sqrt{x^2/D^2+y^2}}\mathrm{{d}}\sigma (x,y)=0. \end{aligned}$$

(2) If $D\equiv 3\mod 4$, then we have

$$\begin{aligned} \int _{C_D(r)}\dfrac{P(x,y)}{\sqrt{20x^2+(D^2+2D+5)y^2+(20+4D)xy}}\mathrm{{d}}\sigma (x,y)=0. \end{aligned}$$

As $H^{\mathbb {R}}_{D,j}[x,y]$ is a vector space, it is enough to show these claims for basis vectors. Since $X\subset C_D(r)$ is an ellipsoidal t-design if and only if $\frac{1}{r}\subset C_D(1)$ is an ellipsoidal t-design, it’s enough to consider $r=1$. For $D\equiv 1,2\pmod {4}$, $H^{\mathbb {R}}_{D,j}[x,y]=\langle \text {Re}(x+\sqrt{-D}y)^j,\text {Im}(x+\sqrt{-D}y)^j\rangle$. By the parametrization of $C_D(1):x^2+Dy^2=1$ as $\gamma :=\{(\cos {\theta },\sin {\theta }/\sqrt{D})|0\le \theta \le 2\pi \}$, we have

$$\begin{aligned}&\int _{C_D(1)} \dfrac{\text {Re}(x+\sqrt{-D}y)^j}{\sqrt{x^2/D^2+y^2}}\mathrm{{d}}\sigma (x,y) \\&\quad = \int ^{2\pi }_{0}\dfrac{\text {Re}(\cos {\theta }+\sqrt{-D}(\sin {\theta }/\sqrt{D}))^j}{\sqrt{\cos {\theta }^2/D^2+\sin {\theta }^2/D}}\sqrt{\sin {\theta }^2+\cos {\theta }^2/D}\mathrm{{d}}\theta \\&\quad =\sqrt{D}\int ^{2\pi }_{0}\text {Re}(\cos {\theta }+i\sin \theta )^j\mathrm{{d}}\theta \\&\quad =\sqrt{D}\int _{S^1}\text {Re}(x+i y)^j\mathrm{{d}}z =0. \end{aligned}$$

Since $\text {Re}(x+i y)^j$ is harmonic, the last integral over $S^1$ is 0.

A similar argument shows that

$$\begin{aligned} \int _{C_D(1)} \dfrac{\text {Im}(x+\sqrt{-D}y)^j}{\sqrt{x^2/D^2+y^2}}\mathrm{{d}}\sigma (x,y)=0. \end{aligned}$$

If $D\equiv 3\pmod {4}$, $H^{\mathbb {R}}_{D,j}[x,y]=\langle \text {Re}(x+\frac{1+\sqrt{-D}}{2}y)^j,\text {Im}(x+\frac{1+\sqrt{-D}}{2}y)^j\rangle$. By the parametrization of $C_D(1):x^2+xy+\frac{1+D}{4}y^2=1$ as $\gamma :=\{(\cos {\theta }-\sin {\theta }/\sqrt{D},2\sin {\theta }/\sqrt{D}):0\le \theta \le 2\pi \}$, we have

$$\begin{aligned}&\int _{C_D(1)} \dfrac{\text {Re}(x+(1+\sqrt{-D})y/2)^j}{\sqrt{20x^2+(D^2+2D+5)y^2+(20+4D)xy}}\mathrm{{d}}\sigma (x,y)\\&\quad = \int ^{2\pi }_{0}\frac{\text {Re}(\cos {\theta }-\sin {\theta }/\sqrt{D}+(1+\sqrt{-D}\sin {\theta }/\sqrt{D})^j}{\sqrt{4D\sin {\theta }^2+20\cos {\theta }^2+8\sqrt{D}\sin {\theta }\cos {\theta }}}\\&\qquad \times \sqrt{\sin {\theta }^2+5\cos {\theta }^2/D+2\sin {\theta }\cos {\theta }/\sqrt{D}}\mathrm{{d}}\theta \\&\quad = \dfrac{1}{2\sqrt{D}}\int ^{2\pi }_{0}\text {Re}(\cos {\theta }+i\sin \theta )^j\mathrm{{d}}\theta = \dfrac{1}{2\sqrt{D}}\int _{S^1}\text {Re}(x+i y)^j\mathrm{{d}}z=0. \end{aligned}$$

A similar argument shows that

$$\begin{aligned} \int _{C_D(1)}\dfrac{P(x)}{\sqrt{20x^2+(D^2+2D+5)y^2+(20+4D)xy}}\mathrm{{d}}\sigma (x,y)=0. \end{aligned}$$

$\square$

3 Ellipsoidal T-designs

Here we prove Theorem 1.2, the construction of ellipsoidal T-designs arising from the ring of integers of imaginary quadratic fields with class number 1. We use the theory of theta functions with complex multiplication. Throughout, we shall assume that $D\in \{1,2,3,7,11,19,43,67,163\}.$

3.1 Theta functions

Given an n-dimensional lattice $\Lambda$ and a polynomial P(x) of degree j in n variables, the theta function of P(x) over the lattice $\Lambda$ is defined by the Fourier series (note $q:=e^{2\pi iz}$)

$$\begin{aligned} \Theta (\Lambda ,P;z):=\sum _{x\in \Lambda } P(x)q^{N(x)} = \Theta (\Lambda ,P;z)=\sum ^\infty _{n=0} a(\Lambda ,P,n)q^n, \end{aligned}$$

(3.1)

where N(x) is the standard norm in $\mathbb {R}^n$. The theta functions for $\Lambda _D= \mathcal {O}_{D}$ play an important role in the study of ellipsoidal T-designs. Namely, if $\Theta (\Lambda _D,P;z)=\sum ^\infty _{r=0} a(\Lambda _D,P,r)q^r,$ then

$$\begin{aligned} a(\Lambda _D,P,r)= \sum _{(x,y)\in \Lambda ^r_D}P(x,y). \end{aligned}$$

(3.2)

The theta function $\Theta (\Lambda _D,P;z)\in \mathcal {M}_k(\Gamma _0(4D),\chi )$, the space of holomorphic modular forms with weight $k=j+1$ and nebentypus $\chi (A)=(\frac{-D}{d}),$ where $A=\Big (\begin{array}{cc} a &{}\quad b \\ c &{}\quad d \end{array}\Big )$ [7, Theorem 10.8]. Moreover, $\Theta (\Lambda _D,P;z)$ is a cusp form when $j>0$.

To ease the study of these theta function, it is convenient to introduce the following the polynomials for each $j\ge 1$:

$$\begin{aligned} R_{D,j}(x,y):={\left\{ \begin{array}{ll} \text {Re}(x+\sqrt{-D}y)^j \ \ \ \ \ &{} \text { if } D\equiv 1,2\pmod {4}, \\ \text {Re}(x+\frac{1+\sqrt{-D}}{2}y)^j \ \ \ \ \ &{} \text { if } D\equiv 3\pmod {4}, \\ \end{array}\right. } \end{aligned}$$

(3.3)

and

$$\begin{aligned} I_{D,j}(x,y):={\left\{ \begin{array}{ll} \text {Im}(x+\sqrt{-D}y)^j \ \ \ \ \ &{} \text { if } D\equiv 1,2\pmod {4}, \\ \text {Im}(x+\frac{1+\sqrt{-D}}{2}y)^j \ \ \ \ \ &{} \text { if } D\equiv 3\pmod {4}. \\ \end{array}\right. } \end{aligned}$$

(3.4)

By definition, we have that $H^{\mathbb {R}}_{D,j}[x,y]=\langle R_{D,j}(x,y),I_{D,j}(x,y)\rangle$. In particular, $\Theta (\Lambda _D,R_{D,j};z)$ and $\Theta (\Lambda _D,I_{D,j};z)$ are cusp forms. Theorem 1.1 together with the discussion above gives the following lemma which transforms the problem of determining ellipsoidal T-designs into the vanishing of certain coefficients of special theta functions.

Lemma 3.1

The norm r shell $\Lambda ^r_D=\Lambda _D\cap C_D(r)$ is an ellipsoidal T-design if and only if $a(\Lambda _D,R_{D,j},r)=0$ and $a(\Lambda _D,I_{D,j},r)=0$ for all $j\in T$.

We require some standard facts from the theory of newforms. Since $\mathcal {O}_D$ has class number 1, each Hecke character mod $\mathcal {O}_D$ is defined by its values on principal ideals. Let $(\alpha )\subset \mathcal {O}_D$ be a principal ideal. Let $u_D$ be the number of units in $\mathcal {O}_D$, namely

$$\begin{aligned} u_D:={\left\{ \begin{array}{ll} 4 \ \ \ \ \ &{} \text { if } D=1, \\ 6 \ \ \ \ \ &{} \text { if } D=3, \\ 2 \ \ \ \ \ &{} \text { otherwise. } \end{array}\right. } \end{aligned}$$

(3.5)

For each positive $j_D\equiv 0\pmod {u_D}$, define Hecke characters mod $\mathcal {O}_D$ by:

$$\begin{aligned} \zeta _{j_D}((\alpha ))=\Big (\dfrac{\alpha }{|\alpha |}\Big )^{j_D} \end{aligned}$$

Then by [8, Theorem 4.8.2], we have the following well known lemma about the modular form

$$\begin{aligned} f_{j_D}(\zeta _{j_D};z):={\left\{ \begin{array}{ll} \Theta (\Lambda _D,(x+\sqrt{-D}y)^j;z) &{} \text { if } D\equiv 1,2\pmod {4}, \\ \Theta \Big (\Lambda _D,\Big (x+\frac{1+\sqrt{-D}}{2}y\Big )^j;z\Big ) &{} \text { if } D\equiv 3\pmod {4} \end{array}\right. } \end{aligned}$$

Lemma 3.2

Assuming the notations above, we have

$$\begin{aligned} f_{j_D}(\zeta _{j_D};z)=\sum _{(\alpha )\subset \mathcal {O}_D} \zeta _{j_D}((\alpha ))N(\alpha )^{j/2} q^{N(\alpha )} \in \mathcal {S}_{k_D}(\Gamma _0(N),\chi ), \end{aligned}$$

the space of cusp forms of weight $k_D=j_D+1$ with nebentypus $\chi \pmod {N}$. Here $N:=|\Delta _{\mathcal {O}_D}|$, the absolute value of the discriminant of $\mathcal {O}_D$. Moreover, $f_{j_D}(\zeta _{j_D};z)$ is a newform.

3.2 Other propositions and lemmas

Recall that $\Lambda ^r_D=C_D(r)\cap \mathcal {O}_D$. Using well known facts about the positive definite binary quadratic forms corresponding to class number 1 norm forms, we have the following lemma.

Lemma 3.3

Suppose r is a positive integer. Then $\Lambda ^r_D$ is nonempty if and only if $ord_p(r)$ is even for every prime $p\not \mid r$ for which $\Lambda ^p_D$ is nonempty.

Rewriting (3.2), we have

$$\begin{aligned} a(\Lambda _D,P,r)= \sum _{(x,y)\in \Lambda ^r_D}P(x,y). \end{aligned}$$

(3.6)

Lemma 3.1 implies that $\Lambda ^r_D$ is an ellipsoidal T-design if and only if $a(\Lambda _D,R_{D,j},r)$ and $a(\Lambda _D,I_{D,j},r)$ vanish for all $j\in T$. Since $\Lambda ^r_D$ is antipodal (i.e. $-\Lambda ^r_D=\Lambda ^r_D$ for all r), $a(\Lambda _D,R_{D,j},r)$ and $a(\Lambda _D,I_{D,j},r)$ are 0 for all $j\in \mathbb {Z}^+\setminus 2\mathbb {Z}^+$. Therefore, we have that following proposition.

Proposition 3.1

Suppose $r\in \mathbb {Z}^+$ such that $\Lambda ^r_D$ is nonempty. Then $\Lambda ^r_D$ is an ellipsoidal $\mathbb {Z}^+\setminus 2\mathbb {Z}^+$-design.

Our objective is to find maximal set $T_D$ for which $\Lambda ^r_D$ is ellipsoidal T-design. By proposition above we have that $\mathbb {Z}^+\setminus 2\mathbb {Z}^+\subset T_D$. So we only look for all even j which can be in $T_D$.

Proposition 3.2

Suppose $j\equiv 0\pmod {2}$, and $r\in \mathbb {Z}^+$. Then the following are true:

(1) We have that $a(\Lambda _D,I_{D,j},r)=0.$

(2) We have that $a(\Lambda _D,R_{D,j},r)= {\left\{ \begin{array}{ll} \sum _{(x_0,y_0)\in \Lambda ^r_D}(x+\sqrt{-D}y)^j \ \ \ \ &{} \text { if } D\equiv 1,2\pmod {4}, \\ \sum _{(x_0,y_0)\in \Lambda ^r_D}\Big (x+\frac{1+\sqrt{-D}}{2}y\Big )^j \ \ \ \ &{} \text { if } D\equiv 3\pmod {4} \end{array}\right. }$

Proof

Part (2) is an obvious consequence of part (1). So it is enough to prove part(1). The idea is to show that points in $\Lambda ^r_D$ occur in pairs on which value of $I_{D,j}$ cancel. If $D\equiv 1,2\pmod {4}$, then $I_{D,j}=\mathrm{Im}(x+\sqrt{-D}y)^j$. In this case $(a,b), (a,-b)\in \Lambda ^r_D$ such that $I_{D,j}(a,b)+I_{D,j}(a,-b)=0$. This is true because each term of $I_{D,j}(x,y)$ has odd power in both the variables x, y. If $D\equiv 3\pmod {4}$, then $I_{D,j}=\mathrm{Im}((x+\frac{1}{2}y)+\frac{\sqrt{-D}}{2}y)^j$. In this case $(a,b),(a+b,-b)\in \Lambda ^j_D$ such that $I_{D,j}(a,b)+I_{D,j}(a+b,-b)=0$. This is because each term of $I_{D,j}(x,y)$ has odd power in $x+y/2,y.$ $\square$

We notice that if $(x_0,y_0)\in \mathcal {O}_D,$ then we have

$$\begin{aligned} \sum _{\alpha _D \in \mathcal {O}_D:|\alpha _D|=1}R_{D,j}(\alpha _D(x_0,y_0))=R_{D,j}(x_0,y_0)\sum _{\alpha _D \in \mathcal {O}_D:|\alpha _D|=1}\alpha ^j_D. \end{aligned}$$

(3.7)

Proposition 3.3

If $r\ge 1$, $1\le j\not \equiv 0\pmod {u_D}$, and $\Lambda ^r_D$ nonempty, then $a(\Lambda ^r_D,R_{D,j},r)=0$

Proof

The idea is that if $(x_0,y_0)\in \Lambda ^r_D$ then $\alpha _D(x_0,y_0) \in \Lambda ^r_D$ where $\alpha _D$ is a unit in $\mathcal {O}_D$. Therefore enough to show that the sum in RHS of (3.7) is 0. For $D=1$, number of units in $\mathcal {O}_D,$ $u_D=4$ which are $\{1,-1,i,-i\}$. We have $1^j+(-1)^j+i^j+(-i)^j=0.$ For $D=3$, number of units in $\mathcal {O}_D,$ $u_D=6$ which are $\{ \pm 1 , \frac{\pm 1\pm \sqrt{-3}}{2}\}$. A brute force calculation shows the result. For other D, the number of units in $\mathcal {O}_D,$ $u_D=2$ which are $\{1,-1\}$. For all j odd, $(1)^j+(-1)^j=0$ $\square$

From here on we will only consider the theta function $\Theta \Big (\Lambda _D,\frac{1}{u_D}R_{D,j};z\Big )$ so let’s give its coefficients a shorthand.

$$\begin{aligned} \Theta \Big (\Lambda _D,\frac{1}{u_D}R_{D,j};z\Big )=\sum ^\infty _{r=0} a(D,j,r)q^r. \end{aligned}$$

(3.8)

Proposition 3.2 together with Lemma 3.2 give us that if $j\equiv 0\pmod {u_D}$, then the theta function $\Theta \Big (\Lambda _D,\frac{1}{u_D}R_{D,j};z\Big )\in \mathcal {S}_{j+1}(\Gamma _0(N),\chi )$ is a Hecke eigenform. So we have the following lemma.

Lemma 3.4

Suppose $j\in u_D\mathbb {Z}^+$. Then the following is true:

(1) If $\gcd {(r_1,r_2)}=1$ then

$$\begin{aligned} a(D,j,r_1r_2)=a(D,j,r_1)a(D,j,r_2). \end{aligned}$$

(2) For p prime and $\alpha >0$, we have

$$\begin{aligned} a(D,j,p^{\alpha })= a(D,j,p)a(D,j,p^{\alpha -1})-\chi (p)p^ja(D,j,p^{\alpha -2}). \end{aligned}$$

(3) For p prime and $\alpha >0$, we have

$$\begin{aligned} a(D,j,p^{\alpha })= a(D,j,p)^{\alpha }\pmod {p}. \end{aligned}$$

Suppose p be a prime such that $\Lambda ^p_D$ be nonempty. Let $(x_p,y_p)\in \Lambda ^p_D$ and $j\equiv 0\pmod {u_D}$. When $p=D$ then it ramifies in $\mathcal {O}_D$ and there are exactly $u_D$ points in $\Lambda ^p_D$. From (3.7) we have $a(D,j,p)=R_{D,j}(x_p,y_p).$ If $p\not =D$ then it’s unramified and we get exactly $2u_D$ solutions. In this case $a(D,j,p)=2R_{D,j}(x_p,y_p).$

Lemma 3.5

Suppose $j\in u_D\mathbb {Z}^+$ and p be an odd prime such that $\Lambda ^p_D$ is nonempty. Let $(x_p,y_p)\in \Lambda ^p_D$ then $R_{D,j}(x_p,y_p)\not \equiv 0\pmod {p}$. In particular, a(D, j, p) is non-zero.

Proof

We will consider two cases, $D\equiv 1,2\pmod {4}$ and $D\equiv 3\pmod {4}$. Proof is essentially same in both the cases.

If $D\equiv 1,2 \pmod {4}$ then $p=x_p^2+Dy_p^2$, in particular $x_p\not \equiv 0\pmod {p}$. we consider the binomial expansion

$$\begin{aligned} R_{D,j}(x_p,y_p)&= \text {Re}(x_p+\sqrt{-D}y_p)^j \\&= \frac{1}{2}\sum ^{j/2}_{n=0}\left( {\begin{array}{c}j\\ 2n\end{array}}\right) x^{j-2n}_p(-1)^n(Dy^2_p)^n\\&= \frac{1}{2}\sum ^{j/2}_{n=0}\left( {\begin{array}{c}j\\ 2n\end{array}}\right) x^{j-2n}_p(-1)^n(p-x^2_p)^n \\&\equiv \frac{1}{2}x^j_p\sum ^{j/2}_{n=0}\left( {\begin{array}{c}j\\ 2n\end{array}}\right) \equiv 2^{j-2}x^j_p \not \equiv 0 \pmod {p} \end{aligned}$$

If $D\equiv 1,2 \pmod {4}$ then $p=(x_p+y_p/2)^2+Dy_p^2/4$, in particular $x_p+y_p/2\not \equiv 0\pmod {p}$. we consider the binomial expansion

$$\begin{aligned} R_{D,j}(x_p,y_p)&= \text {Re}\Big (x_p+y_p/2+\sqrt{-D}y_p/2\Big )^j\\&= \frac{1}{2} \sum ^{j/2}_{n=0}\left( {\begin{array}{c}j\\ 2n\end{array}}\right) \Big (x_p+\frac{y_p}{2}\Big )^{j-2n}(-1)^n\Big (\frac{Dy^2_p}{4}\Big )^n \\&= \frac{1}{2} \sum ^{j/2}_{n=0}\left( {\begin{array}{c}j\\ 2n\end{array}}\right) \Big (x_p+\frac{y_p}{2}\Big )^{j-2n}(-1)^n\Big (p-\Big (x_p+\frac{y_p}{2}\Big )^2\Big )^n \\&\equiv \frac{1}{2} \Big (x_p+\frac{y_p}{2}\Big )^j\sum ^{j/2}_{n=0}\left( {\begin{array}{c}j\\ 2n\end{array}}\right) \equiv 2^{j-2}\Big (x_p+\frac{y_p}{2}\Big )^j \not \equiv 0 \pmod {p} \end{aligned}$$

$\square$

Proposition 3.4

For prime 2, $\Lambda ^2_D$ is nonempty only for $D=1,2,7$. In this case a(D, j, 2) does not vanish for all $j\in 2\mathbb {Z}^+$. Moreover, we have that $a(7,j,2)\equiv 1\pmod {2}$

Proof

For $D=1,2$, $2|\Delta _{\mathcal {O}_D}(=-4D)$ so the ideal (2) is ramified in $\mathcal {O}_D$, in particular there are elements of norm 2. For $D\in \{3,7,11,19,43,67,163\}$, $2\not \mid \Delta _{\mathcal {O}_D}(=-D)$. So the ideal (2) is unramified in $\mathcal {O}_D$. Here we need to check whether 2 splits or not. We have the condition that 2 splits if and only if $-D\equiv 1\pmod {8}$. Only $D=7$ satisfies the condition.

A brute force calculation shows that $a(1,j,2)=(1+i)^j\not =0$, $a(2,j,2)=i^j2^{j+1}\not =0,$ and $a(7,j,2)= 4\mathrm{Re}\Big (\frac{1+\sqrt{-7}}{2}\Big )^j\not =0.$

We prove that $a(7,j,2)\equiv 1\pmod {2}$ using induction on even j. First, note that $a(7,2,2)=-3\equiv 1\pmod {2}$. Now we assume that $a(7,j,2)\equiv 1\pmod {2}$, which implies that $\mathrm{Re}\Big (\frac{1+\sqrt{-7}}{2}\Big )^j=(2k+1)/2$ for some k. The norm of $\Big (\frac{1+\sqrt{-7}}{2}\Big )^j$ is even, so we get that $\mathrm{Im}\Big (\frac{1+\sqrt{-7}}{2}\Big )^j=\sqrt{7}(2k'+1)/2$ for some $k'$. An easy calculation shows that $a(7,j+2,2)=-3\mathrm{Re}\Big (\frac{1+\sqrt{-7}}{2}\Big )^j-\sqrt{7}{\mathrm{Im}} \Big (\frac{1+\sqrt{-7}}{2}\Big )^j\equiv 1\pmod {2}.$ $\square$

3.3 Proof of Theorem 1.2

Propositions 3.1, 3.2 and 3.3 together imply that $a(\Lambda _D,R_{D,j},r)$ and $a(\Lambda _D,I_{D,j},r)$ vanish for all $j\not \equiv 0\pmod {u_D}$, which implies that every nonempty shell $\Lambda ^r_D$ is an ellipsoidal $T_D$-design (remember that $T_D=\mathbb {Z}^+\setminus u_D\mathbb {Z}^+$).

Now we prove the maximality of $T_D$. We show that $a(D,j,r)\not =0$ (note that $a(D,j,r)= \frac{1}{u_D}a(\Lambda _D,R_{D,j},r)$) for all $j\not \in T_D$ and $\Lambda ^r_D$ nonempty. By Lemma 3.4, enough to take r to be a prime power. Suppose p be a prime and $\alpha \ge 1$ be such that $\Lambda ^{p^{a}}_D\not =\phi$. There are two cases possible, either $\Lambda ^p_D$ is empty or it is not. First suppose $\Lambda ^p_D$ is nonempty. If p is 2 then $a(D,j,2)\not =0$ by Proposition 3.4. By part(2) of Lemma 3.4, we have that $a(D,j,2^{\alpha })=a(D,j,2)^{\alpha }\not =0$ for $D=1,2$ since $\chi (2)=0$. When $D=7$ then part(3) of Lemma 3.4, we have $a(7,j,2^{\alpha })\not =0$. If p is an odd prime, then Lemma 3.5 implies that $a(D,j,p)\not =0$. Now using part(3) of Lemma 3.4 again, we have $a(D,j,p^{\alpha })\not =0$. Suppose $\Lambda ^p_D$ is empty then $a(D,j,p)=0$ and Lemma 3.3 implies $\alpha$ is even. Now by part(2) of Lemma 3.5, we get $a(D,j,p^{\alpha })=p^{j\alpha /2}\not =0$ (note that this case includes 2 too). So we get that $a(D,j,p^{\alpha })\not =0$ whenever $\Lambda ^{p^{\alpha }}_D$ is nonempty.