On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

1. Problem Statement

The central problem addressed in this paper is the generalization of a known factorization formula for $p$-adic $L$-functions from the ordinary case to the supersingular case.

• Classical Context: For a modular form $f$, the Rankin–Selberg convolution $L(f \otimes f, s)$ factors into the product of the symmetric square $L$-function and a Dirichlet $L$-function:
  $$L(f \otimes f, s) = L(\text{Sym}^2 f, s) L(\psi, s - k - 1)$$
• Ordinary Case: Dasgupta (2016) proved a $p$-adic analogue of this factorization when $f$ is ordinary at $p$. The proof relied on the density of classical weights in Hida families and the use of Bloch–Kato exponential maps on unit groups (specifically cyclotomic units).
• The Supersingular Obstacle: When $f$ is supersingular at $p$ (i.e., the $p$-adic valuation of the Hecke eigenvalue $a_p(f)$ is positive), the standard $p$-adic $L$-functions are not measures but unbounded distributions. Furthermore, classical weights are not dense in Coleman families (the $p$-adic families passing through supersingular forms) in the same way they are in Hida families. Consequently, Dasgupta's method of interpolating from finite-order characters fails because the values at these characters do not uniquely determine the distribution, and the "unit group" approach does not yield sufficient density to fix the $p$-adic $L$-function.

2. Methodology

The authors overcome the non-ordinary obstacles by shifting the theoretical framework from unit groups to higher $K$-theory and motivic cohomology. Their strategy involves three main pillars:

A. Higher $K$-Theory and Motivic Classes

Instead of using cyclotomic units (which live in $K_1$), the authors construct Beilinson–Flach K-elements ($bf,j$) in higher motivic cohomology groups ($H^1_M$).

• These elements are constructed as pairings of compactified Beilinson–Flach classes (derived from Eisenstein classes on modular curves) with specific differential forms associated with cusp forms.
• The construction utilizes the Kuga–Sato varieties ($W_k$) and their cohomology to lift these classes to a geometric setting where they can be paired with cusp forms.

B. Analytic Continuation via Coleman Families

To address the lack of density in the weight variable:

• The authors work within a Coleman family $\mathcal{F}$ passing through the $p$-stabilization of the supersingular form $f$.
• They utilize the fact that while classical weights are not dense, the set of pairs $(k, j)$ (weight and twist) where the motivic classes are non-trivial and satisfy specific regulator formulas is sufficiently dense to uniquely determine the $p$-adic $L$-functions via analytic continuation.

C. Regulator Formulae

The core of the proof relies on computing the images of these motivic classes under two different regulators:

1. Complex Regulator ($r_D$): Links the motivic class to the derivative of the complex Rankin–Selberg $L$-function, $L'(f \otimes f, j+1)$.
2. Syntomic/Étale Regulator ($r_{syn}$): Links the motivic class to the geometric $p$-adic Rankin–Selberg $L$-function, $L_p^{geom}(f, f)$.

By establishing a relation between these two regulators via the motivic class, the authors bridge the gap between complex and $p$-adic $L$-values.

3. Key Contributions and Results

Theorem A: Construction of a 2-Variable $p$-adic Symmetric Square $L$-function

The authors construct a 2-variable meromorphic $p$-adic $L$-function, denoted $L_p^{imp}(\text{Sym}^2 \mathcal{F})$, defined on the product of the weight space and the character space ($U \times W$).

• Interpolation: This function interpolates the values of the imprimitive complex symmetric square $L$-functions for specializations of the Coleman family $\mathcal{F}$ at classical weights $t$ and twists $j$.
• Significance: This provides the first construction of a $p$-adic symmetric square $L$-function for Coleman families in the supersingular case, filling a gap in the literature where such a construction was previously missing.

Theorem B: $p$-adic Factorization Formula

The main result establishes the factorization of the geometric $p$-adic Rankin–Selberg $L$-function in the supersingular case:
$$L_p^{geom}(\mathcal{F}, \mathcal{F})(\kappa, \kappa, \sigma) = L_p^{imp}(\text{Sym}^2 \mathcal{F})(\kappa, \sigma) \cdot L_p(\psi, \sigma - \kappa - 1)$$

• This generalizes Dasgupta's result to the supersingular setting.
• The proof proceeds by showing that the motivic class $bf,j$ is proportional to the cyclotomic element $\phi_{\psi}$ in the motivic cohomology of cyclotomic fields. The constant of proportionality involves the critical value $L(\text{Sym}^2 f, j+1)$. Applying the regulators to this proportionality yields the factorization of the $p$-adic $L$-values.

Theorem C: Functional Equation

Under the assumption that the character $\psi$ and its square $\psi^2$ are primitive, the authors prove that the newly constructed $L_p^{imp}(\text{Sym}^2 \mathcal{F})$ satisfies a functional equation:
$$L_p^{imp}(\text{Sym}^2 \mathcal{F})(\kappa, 2\kappa + 3 - \sigma) = \varepsilon(\text{Sym}^2 \mathcal{F})(\kappa, \sigma) \cdot L_p^{imp}(\text{Sym}^2 \mathcal{F}^*)(\kappa, \sigma)$$
This allows the interpolation property to be extended from the "left half" of the critical strip (where the motivic construction is direct) to the "right half" via the functional equation.

4. Technical Innovations

• Use of Higher $K$-Theory: The shift from $K_1$ (units) to higher $K$-groups (motivic cohomology) is crucial. It allows the authors to access the $\wedge^2$ component of the tensor product of Galois representations, which corresponds to the symmetric square, and provides classes that are non-trivial even in the supersingular case.
• Compactified Beilinson–Flach Classes: The authors carefully handle the transition from open modular curves to compactified Kuga–Sato varieties. They define "compactified" Beilinson–Flach classes ($\Xi_{k,k,j}$) to ensure the pairing with differential forms is well-defined and to control the "cuspidal defect" (terms that vanish when paired with cusp forms).
• Analytic Continuation Strategy: Instead of relying on the density of classical weights in the weight variable (which fails for Coleman families), they rely on the density of the set of pairs $(k, j)$ in the product space, leveraging the rigidity of $p$-adic analytic functions.

5. Significance

• Resolution of a Major Gap: This work resolves the long-standing difficulty of defining and factorizing $p$-adic $L$-functions for supersingular modular forms in the context of Rankin–Selberg convolutions.
• Foundation for Iwasawa Theory: The construction of the 2-variable $p$-adic symmetric square $L$-function is a prerequisite for formulating and proving Iwasawa main conjectures for symmetric square motives in the supersingular case.
• Methodological Advancement: The paper demonstrates that motivic cohomology and higher $K$-theory provide a robust framework for $p$-adic $L$-function theory that transcends the limitations of the ordinary case, offering a blueprint for future work on other non-ordinary $p$-adic $L$-functions.
• Verification of Conjectures: It confirms the expected factorization structure of $p$-adic $L$-functions in the supersingular setting, aligning the $p$-adic theory with the classical complex factorization and the geometric properties of the underlying Galois representations.