Finite Shimura Correspondence

• Finite Shimura correspondence is a collection of explicit isomorphisms and kernel constructions that generalize the classical relation between half-integral and integral weight modular forms.
• The framework employs analytic kernels, rigid analytic cocycles, and theta lifts to connect Fourier coefficients with central L-values and structural arithmetic properties.
• It also establishes functorial equivalences in representation theory and Hecke algebra isomorphisms for finite and p-adic groups, enhancing harmonic analysis in non-classical settings.

The finite Shimura correspondence refers to a collection of explicit isomorphisms and kernel constructions that generalize the classical Shimura correspondence—linking modular forms of half-integral and integral weight—to settings with additional arithmetic constraints, such as rigid analytic cocycles, wild metaplectic covers, eta-multiplier spaces, and certain non-squarefree levels, as well as finite reductive or double-cover groups over local fields. This concept synthesizes structural, spectral, and arithmetic correspondences that arise when the usual modular forms theory is extended to finite or non-classical situations.

1. Classical Shimura Correspondence and Its Generalizations

The classical Shimura correspondence provides a Hecke module isomorphism between spaces of cusp forms of half-integral weight $k+\frac12$ (e.g., on $\Gamma_0(4)$, plus space) and integral weight $2k$ forms, realized via kernel integrals involving sums over binary quadratic forms. Kohnen and Zagier constructed a theta kernel whose Fourier coefficients indexed by discriminants $D$ establish this correspondence. A key arithmetic property is the relation of Fourier coefficients of half-integral weight forms to central $L$-values of twists of the corresponding integral weight form (Waldspurger’s formula).

Finite generalizations arise in multiple contexts:

• Rigid analytic cocycles, which replace modular symbols and classical spaces with $p$-adic or rigid analytic objects, as in the work of Darmon, Vonk, and Negrini (Negrini, 2022).
• Modular forms with eta-multiplier or more general multiplier systems (Ahlgren et al., 2023, Boylan et al., 2 May 2025).
• Hecke action and representations of finite groups or metaplectic covers, especially over local fields (Karasiewicz et al., 27 Oct 2025).

2. Rigid Analytic Cocycle Correspondence

A significant finite realization is the correspondence between modular forms of weight $k+\frac12$, level $4p^2$ (plus space), and rigid analytic cocycles of weight $2k$ for the Ihara group $\mathrm{SL}_2(\mathbb{Z}[1/p])$ (Negrini, 2022). The central kernel is

$$\mathcal{J}_{k,D}\{r,s\}(z) := \sum_{Q\in F_D(\mathbb{Z}[1/p])}\, (\mathcal{Y}_Q\cdot (r,s))\cdot Q(z,1)^{-k}$$

where $F_D(\mathbb{Z}[1/p])$ denotes binary quadratic forms of discriminant $D$ over the $p$-restricted integers, and $(\mathcal{Y}_Q\cdot (r,s))$ is a $p$-adic geodesic intersection number. The correspondence is realized by generating series

$$\Omega_{2k}(q) = \sum_{D>0,\, (\frac{D}{p})=1} D^{k-1/2}\,\mathcal{J}_{k,D}\,q^{D}$$

with coefficients in the space of rigid analytic cocycles. This construction mirrors the theta kernel in the classical case and connects the Fourier expansion of half-integral weight forms to analytic cocycles valued in $p$-adic functions.

3. Spectral Correspondences and Metaplectic Covers

The finite Shimura correspondence extends to spectral and error-term results in the prime geodesic theorem for arithmetic Fuchsian groups and metaplectic covers (Kaneko, 25 Feb 2025). When considering vector-valued multiplier systems (e.g., 3-fold theta), an explicit spectral correspondence is established: $s_f - 1/2 = 2(s_g - 1/2)$ between Laplace eigenparameters of half-integral weight (metaplectic) forms and weight zero (integral) forms, inducing a quantitative relation between error exponents in the prime geodesic theorem. For cubic metaplectic settings, a generalized Shimura-type correspondence

$s_f - 1 = \pm 3(s_g - 1)$

matches spectral data and error terms for the Kubota character, providing strong evidence toward the optimal conjectural exponent in such error terms.

4. Explicit Lifts for Eta-Multiplier and Theta-Multiplier Forms

Analytic constructions for forms with eta-multiplier and theta-multiplier produce explicit formulas for Shimura lifts, particularly for products of Hecke eigenforms and theta-function eta-quotients (Boylan et al., 2 May 2025, Ahlgren et al., 2023). For example,

$$\mathcal{S}_1(\eta(z) g(z)) = g(z)g(6z) - g(2z)g(3z)$$

and

$$\mathcal{S}_r(\eta(z)^r f(z)) = \sum_i \alpha_i \Big( g_i(z) g_i(6z) - g_i(2z) g_i(3z) \Big)$$

where the coefficients $\alpha_i$ arise in the decomposition of $\eta(z)^r f(z)$ into Hecke eigenforms $g_i$. These lifts land in explicit spaces of newforms with prescribed Atkin-Lehner eigenvalues. Rankin-Cohen bracket lifts are also covered, exhibiting multiplicity-one properties in certain finite dimensional modular spaces.

5. Hecke Algebra, Representation Theory, and Double Covers

In the setting of representation theory for finite or $p$-adic groups, the finite Shimura correspondence manifests as functorial equivalences between representation categories of double covers and their linear analogues, using explicit types and Hecke algebra isomorphisms (Karasiewicz et al., 27 Oct 2025). For wild covers of Chevalley groups over 2-adic fields, Iwahori types are constructed and extended to hyperspecial maximal compact subgroups. The main equivalence is

$$\mathcal{M}(G_\kappa) \longrightarrow \mathcal{M}_{((1),\sigma)}(K) \quad \pi \mapsto \pi \otimes \sigma$$

mapping linear representations of the finite group $G_\kappa$ to genuine $K$-representations via minimal depth types $\sigma$. The associated Hecke algebras admit Iwahori-Matsumoto presentations for several Cartan types, supporting explicit harmonic analysis for double covers.

6. Arithmetic Applications and Gross–Zagier Extensions

The construction enables arithmetic applications—such as explicit formulas for central $L$-values and congruences of Fourier coefficients—by relating the special values in $L$-functions to geometric data indexed by ideals or special points in quaternionic orders (Pacetti et al., 2010). For modular forms of weight $2$ and level $p^2$, two weight $3/2$ forms are constructed for each eigenform $f$, via theta series associated to splits of special points by norm $p$ bilateral ideals: $$L(f,1) L(f, D, 1) = 4\pi^2 \frac{ \langle f, f \rangle }{ \langle \vec{e}_f, \vec{e}_f \rangle } \, \frac{c_d^2}{\sqrt{p d}}$$ where $c_d$ are Fourier coefficients of the associated theta series. The method extends the Gross–Zagier formula to cases where the level and discriminant are not coprime, using explicit quantum and ideal-theoretic splittings.

7. Significance, Structure, and Outlook

The finite Shimura correspondence unifies disparate generalizations of the classical correspondence, providing analytic, algebraic, and arithmetic machinery for explicit lifts, Hecke algebraic structures, and error term analyses in automorphic and modular settings. It offers transfer principles between metaplectic, twisted, and classical objects, with consequences for explicit arithmetic formulas, harmonic analysis of double covers, and further development of $p$-adic or non-archimedean automorphic theory. Continued research aims to resolve technical obstacles in Hecke presentations for certain types and extend these correspondences to broader classes of groups, forms, and local-global settings.