Triple Product of Hida Families

• Triple Product of Hida Families is a framework linking three p-adic families through tensor product Galois representations and motives.
• The methodology employs integral representations and nearly overconvergent cohomology to interpolate p-adic L-functions across varying weights.
• Advanced techniques in p-adic analysis and Selmer groups illuminate exceptional zero phenomena and factorization properties within Iwasawa theory.

The triple product of Hida families and its associated $p$-adic $L$-functions form a deep and highly structured area at the intersection of arithmetic geometry, automorphic forms, and Iwasawa theory. The theory centers on the construction and properties of multi-variable $p$-adic $L$-functions interpolating critical values of complex-valued triple product $L$-functions attached to three Hida families and extends to questions of special value formulas, exceptional zero phenomena, Selmer groups, and codimension-two cycles in Iwasawa theory. The subject integrates techniques from representation theory, $p$-adic deformation spaces, arithmetic geometry, and the analytic theory of modular and Hilbert modular forms.

1. Hida Families and Galois Representations

Given a rational prime $p$, the Iwasawa algebra $$\Lambda = \mathcal{O}\llbracket 1 + p\mathbb{Z}_p\rrbracket$$ parametrizes $p$-adic weights. A primitive ordinary $\Lambda$-adic cusp form (Hida family) of tame level $N$ and Dirichlet character $\chi$ is a formal $q$-expansion

$$\mathbf{f}(q) = \sum_{n=1}^\infty a_n(\mathbf{f}) q^n \in \Lambda\llbracket q\rrbracket$$

such that for each arithmetic specialization $Q$ (of weight $k_Q \ge 2$ and character $\psi_Q$),

$$\mathbf{f}_Q(q) = \sum a_n(\mathbf{f})_Q q^n \in S_{k_Q}(Np, \chi_Q; \overline{\mathbb{Q}}_p)$$

is a $p$-ordinary eigenform. The associated "big" Galois representation

$$\rho_{\mathbf{f}}: G_\mathbb{Q} \longrightarrow \mathrm{GL}_2(\operatorname{Frac}(\Lambda))$$

specializes at arithmetic points to Deligne's two-dimensional $p$-adic representations of the corresponding modular forms.

Given three such families $\mathbf{f}_1, \mathbf{f}_2, \mathbf{f}_3$, the associated tensor product Galois representation $V_1 \otimes V_2 \otimes V_3$ (optionally cyclotomically twisted and self-dualized) plays a central role in the construction of triple product motives and $p$-adic $L$-functions (Hsieh et al., 2019, Hsieh, 2017).

2. Triple Product Motives and $L$-Functions

For arithmetic points $(Q_1, Q_2, Q_3, P)$ of weights $(k_1, k_2, k_3, k_P)$, the rank-8 geometric motive

$$\mathbf{V}_{(Q_1, Q_2, Q_3, P)} = V_{1,Q_1} \otimes V_{2,Q_2} \otimes V_{3,Q_3} \otimes \omega_\mathrm{cyc}^{a-k_P}$$

has as its $L$-function (in the automorphic normalization)

$$L(\mathbf{V}_{(Q_1, Q_2, Q_3, P)}, s) = L(s + \tfrac{1}{2}, \pi_{f_1,Q_1} \times \pi_{f_2,Q_2} \times \pi_{f_3,Q_3} \otimes \omega_\mathrm{cyc}^{a-k_P})$$

where $\pi_{f_i,Q_i}$ denote the automorphic representations attached to the given specializations (Hsieh et al., 2019). These $L$-functions satisfy deep conjectural and proven relationships with cohomological cycles, special values, and Galois representations.

3. Construction of Triple Product $p$-adic $L$-Functions

The major constructions interpolate critical values of the complex triple product $L$-functions as the weights of the Hida families and cyclotomic variable vary $p$-adically. There are two principal methodologies:

(a) Garrett’s Integral and Rankin–Selberg–Hida Theory:

Garrett’s zeta integrals provide a representation of the triple product $L$-function via integrals involving Eisenstein series on $\mathrm{GSp}_6$ and cusp forms on $\mathrm{GL}_2^3$. The $p$-adic interpolation replaces classical input by Hida families and constructs $p$-adic families of cohomology classes, Eisenstein sections, and Hecke operators. The resulting $p$-adic $L$-function

$$L_p(\mathbf{f}_1, \mathbf{f}_2, \mathbf{f}_3; X_1, X_2, X_3, T) \in \Lambda\llbracket X_1, X_2, X_3, T \rrbracket$$

enjoys explicit interpolation at arithmetic specializations in the balanced critical region, matching archimedean factors, Euler factors, and Petersson norms—precisely as conjectured by Perrin-Riou for Panchishkin-ordinary self-dual motives (Hsieh et al., 2019, Hsieh, 2017).

(b) Nearly Overconvergent Cohomology Approach:

For ordinary as well as finite-slope families, vector bundles with marked sections and $p$-adically interpolated Gauss–Manin connections give a cohomological construction of $p$-adic triple product $L$-functions. These methods extend from modular curves to Hilbert and quaternionic settings and allow specializations at classical and non-classical points. Projectors (ordinary, slope-bounded) and connections generalizing the Serre $\theta$-operator control the construction (Kazi, 2024, Andreatta et al., 2017, Huang, 2024).

Interpolation Formula (balanced critical range):

$$L_p(\mathbf{f}_1, \mathbf{f}_2, \mathbf{f}_3; k_1, k_2, k_3, \chi) = \mathcal{E}_p(\mathbf{V}_{(k_i, s)}) \frac{ (2\pi i)^{-s} \Gamma_{\mathbf{V}(s)} L(f_{1,k_1}, f_{2,k_2}, f_{3,k_3}, s) }{ \langle f_{1,k_1}, f_{1,k_1} \rangle \langle f_{2,k_2}, f_{2,k_2} \rangle \langle f_{3,k_3}, f_{3,k_3} \rangle }$$

where $s$ is in the balanced range, and $\mathcal{E}_p$ matches the local factor prescribed by the Panchishkin condition and Coates–Perrin-Riou’s conjecture (Hsieh et al., 2019, Hsieh, 2017).

4. Selmer Groups, Codimension-Two Cycles, and Main Conjectures

Recent developments have examined the interaction between pairs of $p$-adic $L$-functions (balanced and unbalanced) and the algebraic structure of Selmer-type modules in Iwasawa theory. Given the Galois representation $L_{8,3}$ associated to the tensor product of three Hida families $F, G, H$, one constructs both a balanced and an (often conjectural) unbalanced four-variable $p$-adic $L$-function. These generate a height-two ideal in the relevant Iwasawa algebra.

The main theorem (under standard hypotheses) asserts that the sum of the codimension-two characteristic cycles of two pseudo-null Selmer intersection modules equals the ideal generated by these two $p$-adic $L$-functions:

$$\operatorname{Char}_R(M_1) + \operatorname{Char}_R(M_2) = (L_p^\mathrm{bal}, L_p^\mathrm{unb}) \in Z^2(R)$$

giving a higher-codimension analog of the classical main conjecture (Lei et al., 2019). This result realizes a Greenberg-style program for higher codimension phenomena.

5. Exceptional Zero Phenomena and the Trivial Zero Conjecture

The four-variable $p$-adic $L$-function constructed from three ordinary Hida families of elliptic curves restricts to a cyclotomic $p$-adic $L$-function for the associated 8-dimensional motive. In cases of split multiplicative reduction or mixed reduction types, the interpolation formula produces trivial zeros at critical points. Differentiating the $p$-adic $L$-function yields formulas involving Greenberg–Benois $\mathscr{L}$-invariants and special values of the complex $L$-function, as conjectured and proved in (Hsieh et al., 2019):

$$\left. \frac{d^3}{ds^3} L_p(E, s) \right|_{s=2} = -p \mathscr{L}_p(E) \frac{L(E,2)}{2^4 \pi^5 \Omega(E)}$$

This type of result is pivotal in the understanding of $p$-adic analogs of the Birch–Swinnerton-Dyer conjecture for higher rank motives.

6. Factorization and Artin Formalism in Triple Product $p$-adic $L$-Functions

Artin formalism predicts that in the presence of CM or adjoint-type structures, triple product $p$-adic $L$-functions factor into products of lower-rank $p$-adic $L$-functions. Explicitly, in certain settings,

$$L_p(\mathbf{f}_1, \mathbf{f}_2, \mathbf{f}_3) \sim \Theta_{\mathbf{f}_1/K}(X_2) \cdot \Theta_{\mathbf{f}_1/K,\chi}(X_3)$$

where the right-hand side consists of anticyclotomic $p$-adic $L$-functions for the base form $\mathbf{f}_1$ and varying Hecke characters, and in certain "diagonal" coordinates, factorizations into adjoint and cyclotomic factors—mirroring classical Artin formalism—are established via explicit reciprocity laws and comparison of diagonal cycles with Heegner-type cycles (Hsieh, 2017, Büyükboduk et al., 2024, Casazza et al., 2022).

7. Cohomology, Diagonal Cycles, and Explicit Reciprocity Laws

A core aspect of the theory is the relationship between the triple product $p$-adic $L$-function and the class of diagonal cycles in the product of three towers of modular curves. A three-variable family of cohomology classes, constructed via Abel–Jacobi images of algebraic cycles (typically, "Gross–Kudla–Schoen cycles"), specializes to the natural motivic cycles at arithmetic points. Perrin–Riou’s $\Lambda$-adic regulator identifies the special value of the triple product $p$-adic $L$-function with the image of the diagonal cycle, establishing a deep explicit reciprocity law:

$$R(\kappa_p(f_1, f_2, f_3)) = L_p(\tilde{f}_1, \tilde{f}_2, \tilde{f}_3)$$

for appropriate test vectors, weights, and regulators (Darmon et al., 2022).

The existence of this law is crucial for applications to the study of Selmer groups, Euler systems, and the main conjecture in multi-variable Iwasawa theory.

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References:

• Four-variable $p$-adic triple product $L$-functions and the trivial zero conjecture (Hsieh et al., 2019)
• Hida families and $p$-adic triple product $L$-functions (Hsieh, 2017)
• Codimension two cycles in Iwasawa theory and tensor product of Hida families (Lei et al., 2019)
• Triple product p-adic L-function attached to p-adic families of modular forms (Fukunaga, 2019)
• $p$-adic families of diagonal cycles (Darmon et al., 2022)
• On the Artin formalism for triple product $p$-adic $L$-functions: Chow--Heegner points vs. Heegner points (Büyükboduk et al., 2024)
• On $p$-adic $L$-functions for $\mathrm{GL}(2)\times\mathrm{GL}(3)$ via pullbacks of Saito–Kurokawa lifts (Casazza et al., 2022)