---
title: Triple Product of Hida Families
url: https://www.emergentmind.com/topics/triple-product-of-hida-families
type: topic
---

# Triple Product of Hida Families

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 [1906.10474][1705.02717].

## 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 [1906.10474]. 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 [1906.10474][1705.02717].

**(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 [2401.13230][1708.02785][2403.18620].

**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 [1906.10474][1705.02717].

## 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 [1901.09301]. 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 [1906.10474]:
$$
\left. \frac{d^3}{ds^3} L_p(E, s) \right|_{s=2} = -p \mathscr{L}_p(E) \frac{L(E,2)}{2^4 \pi^5 \Omega(E)}
$$
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 [1705.02717][2409.08645][2202.06566].

## 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 [2207.01314].

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 [1906.10474]
- Hida families and $p$-adic triple product $L$-functions [1705.02717]
- Codimension two cycles in Iwasawa theory and tensor product of Hida families [1901.09301]
- Triple product p-adic L-function attached to p-adic families of modular forms [1909.03165]
- $p$-adic families of diagonal cycles [2207.01314]
- On the Artin formalism for triple product $p$-adic $L$-functions: Chow--Heegner points vs. Heegner points [2409.08645]
- On $p$-adic $L$-functions for $\mathrm{GL}(2)\times\mathrm{GL}(3)$ via pullbacks of Saito–Kurokawa lifts [2202.06566]

Source: https://www.emergentmind.com/topics/triple-product-of-hida-families