---
title: Finite-Prime Weil Quadratic Form
url: https://www.emergentmind.com/topics/finite-prime-weil-quadratic-form
type: topic
---

# Finite-Prime Weil Quadratic Form

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 $B(x,y)=q(x+y)-q(x)-q(y)$ [1705.04572]. 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 [1606.07844].

## 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 [1705.04572, 2512.15334]. Notably, $\rho_A(S)^4 = i\cdot \mathrm{Id}$, reflecting the metaplectic double cover and parity structure [1606.07844].

The explicit computation of these operators reduces in particular cyclic and diagonal cases to elementary expressions involving Gauss sums (see below for generalizations) [2512.15334].

## 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 $0 \leq k_j \leq 23/2$ [1606.07844].

## 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 [1606.07844, 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 [1606.07844].

## 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 $p\equiv 1,3 \bmod 4$ [2512.15334].

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 [2512.15334].

## 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 [1606.07844].

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 [1705.04572].

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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 [1606.07844, 1705.04572, 2512.15334].

Source: https://www.emergentmind.com/topics/finite-prime-weil-quadratic-form