Cyclic Quadratic Modules

• Cyclic quadratic modules are finite abelian groups equipped with quadratic forms that satisfy bilinearity and nondegeneracy, providing a framework for studying Weil representations.
• They decompose into cyclic modules, such as those of order 2p^r, with invariants like level, signature, and discriminant forms that influence half-integral weight modular forms.
• In the large parameter limit, the generating weights converge to a symmetric, bi-modal distribution, revealing deep arithmetic and representation-theoretic symmetries.

A cyclic quadratic module is a finite abelian group equipped with a quadratic form satisfying specific bilinearity and nondegeneracy conditions, central to the study of the Weil representation and vector-valued modular forms. In particular, modules of order $2p^r$ (with $p$ an odd prime, $r\ge1$) possess intricate arithmetic and representation-theoretic properties that control the structure of associated half-integral weight modular forms. The generating weights of these modules display remarkable uniformity in the large parameter limits, reflecting deep symmetries in the underlying arithmetic.

1. Definition and Invariants of Cyclic Quadratic Modules

Let $M$ be a finite abelian group, and $q: M \to \mathbb{Q}/\mathbb{Z}$ a function such that:

• For all $x \in M$ and integers $n$, $q(nx) = n^2 q(x)$ in $\mathbb{Q}/\mathbb{Z}$.
• The associated bilinear form $b(x,y) = q(x+y) - q(x) - q(y)$ is $\mathbb{Z}$-bilinear and nondegenerate, in the sense that $x \mapsto (y \mapsto b(x,y))$ yields an isomorphism $$M \cong \operatorname{Hom}(M, \mathbb{Q}/\mathbb{Z})$$.

Such a pair $(M,q)$ is a finite quadratic module. A module is cyclic of order $N$ if $M \cong \mathbb{Z}/N\mathbb{Z}$; every such module admits a generator $1$ with $q(k) = \frac{a k^2}{N} \bmod \mathbb{Z}$ for unique $a \in (\mathbb{Z}/N)^\times$, denoted $A_N^a$.

Key invariants include:

• Level: The minimal $N$ such that $N q(x) \in \mathbb{Z}$ for all $x$.
• Signature: $\operatorname{sig}(M) \in \mathbb{Z}/8$, extracted from the Gauss sum

$$\Omega_M(1) = \frac{1}{\sqrt{|M|}} \sum_{x \in M} e(q(x)) = (\sqrt{i})^{\,\operatorname{sig}(M)}$$

where $e(u) = e^{2\pi i u}$.

• Discriminant form: For an integral lattice $L$, the quadratic module $L^*/L$ with $q(x+L) = \frac{1}{2}(x,x) \bmod \mathbb{Z}$.

2. Structure of Even-Order Cyclic Modules of Order $2p^r$

Fixing an odd prime $p\ge5$ and $r\ge1$, the distinguished cyclic module is

$$D = A_{2p^r} = (\mathbb{Z}/2p^r\mathbb{Z},\, q(x) \equiv \tfrac{x^2}{4p^r} \bmod \mathbb{Z}).$$

By the Chinese Remainder Theorem, $$\mathbb{Z}/2p^r\mathbb{Z} \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/p^r\mathbb{Z}$$, and the form splits orthogonally:

$$q_{2p^r} \cong q_2 \oplus q_{p^r}, \quad \text{with} \quad q_2(x \bmod 2) = \tfrac{x^2}{4} \bmod 1,\; q_{p^r}(y \bmod p^r) = \tfrac{y^2}{p^r} \bmod 1.$$

In Jordan decomposition, this is $A_{2}^{\,1} \oplus A_{p^r}^{4}$.

The signature is $\operatorname{sig}(A_{2p^r}) = 1 \in \mathbb{Z}/8$ (odd), excluding the existence of integral-weight modular forms for its Weil representation.

3. The Weil Representation Associated to $D$

Let $\mathrm{Mp}_2(\mathbb{Z})$ denote the metaplectic double cover of $\mathrm{SL}_2(\mathbb{Z})$, generated by $(T,1)$ and $(S,\sqrt{\tau})$ with relations $S^2=Z$, $Z^2=(I,-1)$. The Weil representation $\rho_D$ is a unitary representation:

$$\rho_D: \mathrm{Mp}_2(\mathbb{Z}) \longrightarrow \mathrm{GL}(\mathbb{C}[D])$$

on the group algebra $\mathbb{C}[D]$ with basis $\{\delta_x\}_{x\in D}$, defined via:

• $\rho_D(T)\,\delta_x = e(-q(x))\,\delta_x$
• $$\rho_D(S)\,\delta_x = \frac{(\sqrt{i})^{\,\operatorname{sig}(D)}}{\sqrt{|D|}} \sum_{y \in D} e(b(x,y))\,\delta_y$$

Here $b(x,y) = q(x+y) - q(x) - q(y)$. The representation respects all relations of $\mathrm{Mp}_2(\mathbb{Z})$.

4. Half-Integral Weight Modular Forms and Generating Weights

For any finite-dimensional representation $\rho$ of $\mathrm{Mp}_2(\mathbb{Z})$, the module of holomorphic $\rho$-valued half-integral weight modular forms is

$$M(\rho) = \bigoplus_{k \in \tfrac{1}{2} \mathbb{Z}} M_k(\rho),$$

which is a free module of rank $\dim \rho$ over $\mathbb{C}[E_4, E_6]$, the ring of level-one modular forms. The $f_j \in M_{k_j}(\rho)$ with $$M(\rho) \cong \bigoplus_{j=1}^{\dim\rho} \mathbb{C}[E_4,E_6]\langle f_j\rangle$$ determine the generating weights $k_1 \le \cdots \le k_{\dim\rho}$, subject to the sum rule

$\sum_{j} k_j = 12\,\operatorname{Tr}(L),$

where $L$ is any exponent matrix with $\rho(T) = e^{2\pi i L}$.

For $\rho = \rho_{A_{2p^r}}$, $\operatorname{Tr}(L)$ is given by:

$$\operatorname{Tr}(L) = p^r - \left\{ \frac{p^r}{4} \right\} p^{\lfloor r/2 \rfloor} + \frac{p^{\lfloor (r+1)/2 \rfloor}-1}{2(p-1)} \begin{cases} h_p, & p \equiv 1 \pmod{4} \ 4h_p^* + 1, & p \equiv 3 \pmod{8} \ 2h_p + 1, & p \equiv 7 \pmod{8} \end{cases}$$

where $h_p$ is the class number of $\mathbb{Q}(\sqrt{-p})$.

It is established that $M_{1/2}(\rho_{A_{2p^r}}) = 0$ and $M_{1}(\rho_{A_{2p^r}}) = 0$ (by Skoruppa–Serre–Stark and parity), and a further check confirms $M_{3/2}$ is unobstructed. Thus all dimensions $\dim M_k$ can be extracted from the Euler characteristic. The multiplicity $m_k$ of each weight $k$, for $k \in \{3/2, 5/2, \dots, 21/2\}$, is described by explicit linear combinations involving $p^r$, $\operatorname{Tr}(L)$, and small correction factors:

$$\delta = \frac{1}{8}(2 + (\tfrac{-1}{p^r})), \quad \epsilon_\pm = \frac{1}{6}(1 \pm (\tfrac{p^r}{3})),$$

with mirror-symmetry $m_{k} = m_{23/2 - k}$, and $m_{1/2} = m_{23/2} = 0$.

5. Limiting Behavior of Generating Weights

As $p \to \infty$ (with $r$ fixed) or $r \to \infty$ (with $p$ fixed), $\dim \rho = 2p^r$ increases, and $h_p = O(p^{1/2+\varepsilon})$. One obtains:

$$\frac{\operatorname{Tr}(L)}{\dim \rho} \to \frac{1}{2}$$

Consequently, the normalized multiplicities $m_k / \dim \rho$ converge to a discrete, bi-modal distribution, symmetric and concentrated at the central weights:

$$\frac{m_k}{\dim\rho} \to \begin{cases} 0 & k = 1/2,\,23/2 \ \frac{1}{48} & k = 3/2,\,21/2 \ \frac{3}{48} & k = 5/2,\,19/2 \ \frac{5}{48} & k = 7/2,\,17/2 \ \frac{7}{48} & k = 9/2,\,15/2 \ \frac{8}{48} & k = 11/2,\,13/2 \ \end{cases}$$

This limiting pattern highlights a pronounced symmetric "hump" at weights $11/2$ and $13/2$. This universal emergence of a discrete, symmetric measure for the generating weights as parameters grow suggests a deeper structural or representation-theoretic phenomenon, analogous to automorphic forms over large-level metaplectic groups (Candelori et al., 2016).

6. Context and Open Directions

The explicit computation of generating weights for vector-valued modular forms in the context of Weil representations of even-order cyclic quadratic modules, particularly those of order $2p^r$, deepens the understanding of the interplay between arithmetic invariants (such as class numbers and Gauss sums) and the algebraic structure of modular forms. The observed measure convergence suggests new potential "limit objects" for these representation-theoretic constructions, indicating directions for further research into discrete distributions and automorphic phenomena in the context of large-level metaplectic and modular theory.