Finite-Prime Weil Quadratic Form

• Finite-prime Weil quadratic forms are cyclic quadratic modules defined by q(a)=a²/(2p^r) that underpin the arithmetic of vector-valued modular forms.
• They feature explicit Weil representations with closed-form generating weight formulas and bi-modal multiplicity distributions reflecting deep modular symmetries.
• Computational methods leveraging quadratic Gauss sums and algebraic reductions enable efficient determination of invariants and basis elements in these representations.

A finite-prime Weil quadratic form refers, in the context of the arithmetic theory of modular forms and automorphic representations, to a specific class of finite quadratic modules and their associated Weil representations, particularly those arising from cyclic groups of order $2p^r$ for a prime $p\geq 5$. The associated quadratic forms, their induced bilinear forms, and the symmetries of the resulting module algebra encode rich arithmetic and geometric information, especially concerning vector-valued modular forms of half-integral weight, their generating weights, and their module structures over rings of modular forms. Recent research has achieved explicit, closed-form descriptions of the corresponding Weil representations and their modular form modules, including limiting multiplicity distributions, geometric interpretations on stacks, and computational algorithms for invariants.

1. Finite Cyclic Quadratic Modules: Definition and Structure

For a prime $p\geq 5$ and integer $r\geq 1$, set $m=2p^r$. The cyclic quadratic module in question is $A=\mathbb{Z}/m\mathbb{Z}$. The quadratic form $q:A\to\mathbb{Q}/\mathbb{Z}$ is defined by $q(a) = a^2/(2p^r) \bmod 1$. The associated symmetric bilinear form is $b(a,b)=q(a+b)-q(a)-q(b) = ab/p^r \bmod 1$.

This module and form fit the general theory of finite quadratic modules $(A,q)$, where $A$ is a finite abelian group and $q$ satisfies $q(-x)=q(x)$ and yields a $\mathbb{Z}$-bilinear map $B(x,y)=q(x+y)-q(x)-q(y)$ (Ehlen et al., 2017). When the bilinear form $b$ is nondegenerate, $(A,q)$ is called nondegenerate. The level of $(A,q)$ is the minimal positive integer $N$ with $N q(x)\in\mathbb{Z}$ for all $x\in A$.

For these cyclic modules, the signature $\mathrm{sig}(A)=1$ mod $8$, which informs transformation properties of the associated representations under the modular group (Candelori et al., 2016).

2. The Weil Representation for Finite-Prime Modules

The Weil representation $\rho_A$ of the metaplectic group $\mathrm{Mp}_2(\mathbb{Z})$ acts naturally on the group algebra $\mathbb{C}[A]$ with the standard delta basis $\{\delta_x\}_{x\in A}$ and $e(x)=e^{2\pi i x}$. The action of the standard generators $T$ and $S$ is:

• $\rho_A(T)\cdot\delta_x = e(-q(x))\delta_x$
• $$\rho_A(S)\cdot\delta_x = \Omega_A(1) |A|^{-1/2}\sum_{y\in A} e(b(x,y))\delta_y$$

with $|A|=2p^r$ and $\Omega_A(1)=\sqrt{i}$ because $\mathrm{sig}(A)=1$ (mod 8). These formulas are specific instances of the general Weil representation on finite quadratic modules (Ehlen et al., 2017, Zhu, 17 Dec 2025). Notably, $\rho_A(S)^4 = i\cdot \mathrm{Id}$, reflecting the metaplectic double cover and parity structure (Candelori et al., 2016).

The explicit computation of these operators reduces in particular cyclic and diagonal cases to elementary expressions involving Gauss sums (see below for generalizations) (Zhu, 17 Dec 2025).

3. Vector-Valued Modular Forms and Module Structure

For $k\in\frac{1}{2}\mathbb{Z}$, the space of vector-valued modular forms of weight $k$ transforming under $\rho_A$ can be constructed as the sheaf $V_k$ on the metaplectic orbifold, which in this context is identified with the weighted projective line $\mathbb{P}(8,12)$. The graded module of such forms,

$$M(\rho_A) = \bigoplus_{k\in\frac12\mathbb{Z}} M_k(\rho_A),$$

is a free module of rank $2p^r$ over $M(1)=\mathbb{C}[E_4,E_6]$, the ring of (scalar) modular forms for $\mathrm{SL}_2(\mathbb{Z})$. If $k_1\leq k_2\leq\cdots\leq k_{2p^r}$ are the generating weights,

$\sum_j k_j = 12\cdot \mathrm{Tr}(L),$

where $L$ is an exponent matrix determined by the action of $T$. The standard exponents satisfy $0 \leq k_j \leq 23/2$ (Candelori et al., 2016).

4. Explicit Formulas for Generating Weights and Multiplicities

The generating weights have closed-form expressions as functions of $p$ and $r$ via an explicit trace-of-exponents formula:

$$\mathrm{Tr}(L) = p^r - \{p^r/4\}\cdot p^{\lfloor r/2\rfloor} + \frac{p^{\lfloor (r+1)/2\rfloor}-1}{2(p-1)}\cdot C_{p,r}$$

where $C_{p,r}$ encodes data from Dirichlet class numbers and $p$ modulo $8$.

The dimension-generating series is

$$\sum_{k\in\frac12\mathbb{Z}} \dim M_k(\rho_A) t^k = \frac{t^{k_1} + \cdots + t^{k_{2p^r}}}{(1-t^4)(1-t^6)}.$$

Multiplicity formulas for each allowed half-integral weight $k$ (excluding the parity-forbidden $k=1$ and negative integers) are presented as:

$$\begin{align*} &m_{1/2}=m_{23/2}=0, \ &m_{3/2} = \frac{13}{24}(p^r+1) - \frac{1}{2}\mathrm{Tr}(L) - \delta - \epsilon_+,\ &\cdots\ &m_{21/2}= -\frac{11}{24}(p^r-1) + \frac{1}{2}\mathrm{Tr}(L) - \delta + \epsilon_-, \end{align*}$$

where $\delta = (2 + (-1|p^r))/8$ and $\epsilon_\pm = (1 \pm (p^r|3))/6$ (with appropriate Legendre symbols). Full explicit expressions are given in [(Candelori et al., 2016), Table 4.1].

5. Limiting Profile and Distribution of Generating Weights

By analyzing the lower-order terms and taking $p\to\infty$ (for fixed $r$), or $r\to\infty$ (for fixed $p$), the multiplicity ratios $m_k/(2p^r)$ stabilize to a bi-modal profile:

$$\begin{array}{ll} 0 & \text{for } k=1/2,\,23/2 \ 1/48 & \text{for } k=3/2,\,21/2 \ 3/48 & \text{for } k=5/2,\,19/2 \ 5/48 & \text{for } k=7/2,\,17/2 \ 7/48 & \text{for } k=9/2,\,15/2 \ 8/48 & \text{for } k=11/2,\,13/2 \ \end{array}$$

As a consequence, the generating weights of the free module $M(\rho_A)$ "pile up" around $11/2$ and $13/2$ in the large-dimension limit, offering a refined structural view of the modular form landscape for these representations (Candelori et al., 2016).

6. Quadratic Gauss Sums and the Weil Representation: Prime and Composite Moduli

The matrix elements of the Weil representation, especially for cyclic quadratic modules over $\mathbb{Z}/p\mathbb{Z}$, are expressible by explicit quadratic Gauss sums. For a symmetric integral $n\times n$ matrix $G$ and prime $p$, reduction brings $Q(x)=\sum a_i x_i^2/2$ modulo $p$. General Gauss sums $G_p(\lambda)$ factor as products of classical 1D Gauss sums, with

$$g_p(a;\ell) = \sum_{x\in\mathbb{F}_p} e((a x^2 + \ell x)/p) = \epsilon_p (a/p)\sqrt{p} e(-\ell^2 (4a)^{-1}/p),$$

where $(a/p)$ is the Legendre symbol, and $\epsilon_p=1$ or $i$ for $p\equiv 1,3 \bmod 4$ (Zhu, 17 Dec 2025).

More generally, for $D=\mathbb{Z}^n/p\mathbb{Z}^n$ and quadratic form $Q$, the Weil representation acts by:

$$\rho(T)_{\alpha,\beta} = \delta_{\alpha,\beta} e(Q(\alpha)),\qquad \rho(S)_{\alpha,\beta} = e(n/8)/p^{n/2} e(-\alpha^T\beta/p).$$

Such formulas unify the representation theory across prime and composite moduli, avoiding the necessity of local data or theta-series limits (Zhu, 17 Dec 2025).

7. Geometric and Computational Methods

The vector bundle aspects of the theory are formulated on the stacky curve $\mathbb{P}(8,12)$. The free module structure is proved via geometric arguments, employing Riemann–Roch for Deligne–Mumford stacks and Serre duality to determine vanishing and non-vanishing of the relevant cohomology groups. The critical weights are handled by results of Skoruppa–Serre–Stark. Closed-form counting of generating weights ultimately depends on explicit evaluation of Gauss sum traces and class number formulae (Candelori et al., 2016).

Algorithmically, invariants and bases of the Weil representation for arbitrary finite quadratic modules can be computed via linear algebra methods. Efficiency improvements stem from symmetrizations and splitting into $p$-parts, with the global invariant space recovered as a tensor product of local ones. Under integrality and suitable reduction, the dimensions and bases remain unchanged modulo suitable primes (Ehlen et al., 2017).

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The finite-prime Weil quadratic form thus exhibits a highly explicit and computable structure, both algebraically and geometrically, with direct consequences for the arithmetic of vector-valued modular forms and the representation theory of the modular and metaplectic groups (Candelori et al., 2016, Ehlen et al., 2017, Zhu, 17 Dec 2025).