Half-Ordinary Rankin–Selberg Deformations

• Half-ordinary Rankin–Selberg deformations are a framework that generalizes classical p-adic L-function constructions by relaxing ordinarity conditions to include finite-slope phenomena.
• The theory leverages universal deformation rings and Panchishkin filtrations to construct four-variable eigenvarieties, expanding the scope of Hida theory.
• Advanced techniques, such as Beilinson–Flach classes and functional equations, establish essential links between Selmer groups and p-adic L-functions in this setting.

The theory of half-ordinary Rankin–Selberg universal deformations is a central development in the study of $p$-adic Galois representations and arithmetic $L$-functions in non-ordinary and finite-slope settings. It combines Hida theory, universal deformation rings, Panchishkin conditions, and advanced Iwasawa-theoretic constructions to address the algebraic and analytic properties of $p$-adic Rankin–Selberg convolutions in families. This framework naturally generalizes classical constructions, introducing four-variable eigenvarieties and universal $L$-functions with precise interpolation and functional equation properties.

1. Half-Ordinary Condition and Panchishkin Filtration

The notion of half-ordinary (nearly ordinary) arises by relaxing the ordinarity condition at $p$ for one of the modular forms. For a $G_{\Q,\Sigma}$-representation $T$ of rank $d$ over a regular local domain $R\cong\O[[\Gamma_1\times\cdots\times\Gamma_r]]$, the Panchishkin condition prescribes a $G_{\Q_p}$-stable filtration

$0\subset F^+_pT\subset\cdots\subset T,$

with graded pieces $\Gr^m_pT$ of rank one and character structure

$\widetilde\chi_1^{e_{m,1}}\cdots\widetilde\chi_r^{e_{m,r}}\;\omega^{1-a_m}\;\chi_{\cyc}^{1-b_m}\;\alpha_m.$

In the Rankin–Selberg convolution of two Hida families $f_1, f_2$, this yields the "half-ordinary" subquotient

$F^{-+}T = \Gr^1_pT = (T_{f_1}/F^+_pT_{f_1}) \otimes F^+_pT_{f_2} \otimes \Lambda_{\cyc}^\iota.$

This structure enables the construction of families where only one factor is required to be ordinary at $p$, substantially enlarging the space of families for deformation, and accommodates finite-slope phenomena (Büyükboduk et al., 2017, Gu, 1 Dec 2025, Hao et al., 2024).

2. Universal Deformation Rings and Family Structure

The half-ordinary universal deformation functor $D^{ho}$ is defined on the category of complete Noetherian local $O$-algebras with residue field $F$, associating isomorphism classes of pairs $(\rho_1, \rho_2)$ where $\rho_1$ is ordinary at $p$ and $\rho_2$ is unrestricted. This functor is pro-representable by

$$R^\star = R^{ord}(\bar\rho_1) \widehat\otimes_O R(\bar\rho_2) \cong O \llbracket X, T_1, T_2, T_3 \rrbracket,$$

flat over $O$ of relative dimension $4$, parametrizing deformations in a four-dimensional weight space. In particular, the standard three-variable eigencurve arises when both factors are ordinary; removing this constraint enlarges the eigenvariety to four dimensions (Gu, 1 Dec 2025, Hao et al., 2024).

The associated universal deformation $\rho^\star$ gives a $G_{\Q,\{p\}}$-representation on $V^\star = V_1^{ord}\,\widehat\otimes_O\,V_2$. At $p$, this decomposes as

$0 \to V^{\star,+} \to V^\star|_{G_{\Q_p}} \to V^{\star,-} \to 0,$

with the half-ordinary Panchishkin submodule $V^{\star,+} = V_1^{ord,+} \otimes_O V_2$.

3. Construction and Properties of Universal $p$-adic Rankin–Selberg $L$-Functions

The $p$-adic $L$-function in the half-ordinary setting is constructed via Hida’s ordinary projector $e$, a dual Hida functional $\lambda_{F^c}$, and the multiplication of families and Eisenstein series in the universal setting. Specifically,

$L_p := \lambda_{F^c}\bigl(e(\cG \cdot \Eis_{m,\chi})\bigr) \in \Frac(T_{\frak a}\,\widehat\otimes R_{\rm univ}),$

where $\cG$ is the universal $pN$-depleted modular eigenform and $\Eis_{m,\chi}$ is a two-variable Eisenstein family. This analytic construction yields a meromorphic function over the four-dimensional base algebra $A = T_{\frak a}\widehat\otimes_{\cO} R_{\rm univ}$ (Hao et al., 2024).

The finite-slope universal $p$-adic $L$-function $L_p^{\rm fs}$ is defined by pairing the globalized Beilinson–Flach class $\BF_U$ with the family Perrin–Riou regulator:

$L_p^{\rm fs} := \cL_{V_U, V_U^+}(\BF_U) \in \sH(\Gamma)\,\widehat\otimes\,\O(U) \cong \O(U \times \scrW),$

with analytic interpolation in both cyclotomic and modular variables (Gu, 1 Dec 2025).

4. Main Conjectures, Selmer Groups, and Interpolation Theorems

The main conjecture for nearly ordinary families, as proved in (Büyükboduk et al., 2017), asserts

$\charc\bigl(H^1_{F_{\Gr}^*}(\Q,T^\vee(1))^\vee\bigr) \supset H \bigl(L_p^{\rm Hida}(f_1,f_2;j)\bigr)$

for the three-variable $L$-function, relating the characteristic ideal of the dual Selmer group to the $p$-adic $L$-function and congruence divisor.

In the half-ordinary universal deformation context, the interpolation formula at classical points $(f_{k_1}, g_{k_2}; j)$ is

$$L_p(f_{k_1}, g_{k_2}; j) = E_p(k_1, k_2, j) \times \frac{L(f_{k_1} \times g_{k_2}, j)}{\Omega^\pm_{f_{k_1}, g_{k_2}}},$$

with $E_p$ an explicit $p$-Euler factor dependent on Atkin–Lehner pseudo-eigenvalues, local Euler polynomials, and adjoint Euler factors from Panchishkin subrepresentations (Hao et al., 2024). In the finite-slope theory,

$$L_p^{\rm fs}(x, \kappa) = \frac{E_p(f_x, g_x, s)}{\Omega_p(f_x, g_x, \pm)} \cdot \frac{L^{(p)}(f_x \otimes g_x, s)}{(2\pi i)^{2s}},$$

refining the interpolation in terms of analytic periods for each arithmetic point (Gu, 1 Dec 2025).

The universal Beilinson–Flach class and Coleman maps play a critical role, establishing divisibility relations between Selmer groups and $p$-adic $L$-functions via reciprocity laws and dimension reduction for Euler systems (Büyükboduk et al., 2017).

5. Functional Equations and Reflection Symmetries

A universal functional equation is established for the four-variable $p$-adic $L$-function:

$$L_p(\kappa, \lambda; s) = N^{2(\kappa - \lambda - 1)} \gamma(\kappa, \lambda) L_p(\kappa^c, \lambda^c; \kappa + \lambda - 2 - s),$$

where $\gamma(\kappa, \lambda)$ interpolates local gamma-factors and involutions act on universal weight-characters and cyclotomic parameters. This generalizes the classical functional equation for Rankin–Selberg $L$-functions, ensuring the $p$-adic object retains the expected symmetry about the center of the critical strip (Hao et al., 2024).

6. Auxiliary Constructions: Projectors, Measures, and Dimension Reduction

The half-ordinary projector

$$e^{\rm ho} := \lim_{n\to\infty} \alpha_{f_0}^{-n} U_p^n,$$

acts on overconvergent cohomology, enforcing ordinarity in one factor (Gu, 1 Dec 2025). Associated $p$-adic $L$-functions can also be realized via two-variable $p$-adic measures

$$L_p^{\rm fs}(x) = \int_{\Z_p^\times \times \Z_p^\times} \chi_1(t_1) \chi_2(t_2) \, d\mu_{f_x, g_x}(t_1, t_2),$$

reflecting cyclotomic and modular variation.

Dimension reduction procedures for Euler systems and Selmer groups, utilizing specialization at linear primes and control lemmas, reduce higher-rank problems to dimension 1, enabling the application of Kolyvagin and Euler-system techniques (Büyükboduk et al., 2017).

7. Corollaries, Mod $p$ Congruences, and Open Directions

The reduced structure of deformation rings ensures that mod $p$ congruences among specializations of $L_p^{\rm fs}$ detect congruences in global Galois representations (Gu, 1 Dec 2025). The Iwasawa main conjecture is expected to equate the characteristic ideal of universal Selmer modules with the ideal generated by the universal $L$-function. Analysis of exceptional zeroes and functional equations via local Euler factors illuminates derivative formulas and offers prospects for further generalization.

A plausible implication is that the theory provides a robust framework for comparing arithmetic invariants across families with varying local conditions, bridging ordinary, finite-slope, and non-ordinary regimes.

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Principal technical references: (Büyükboduk et al., 2017, Gu, 1 Dec 2025, Hao et al., 2024).