Triple Product p-adic L-functions

• Triple product p-adic L-functions are analytic objects attached to triples of modular forms that interpolate square roots of central critical L-values.
• They are constructed by generalizing Hsieh’s three-variable theory to extend Hida families to weight one and employ novel quaternionic test vectors to handle ramification.
• These functions connect refined Birch–Swinnerton-Dyer and Stark-type conjectures by generating explicit Stark points on elliptic curves in the rank one case.

Balanced triple product $p$-adic $L$-functions are analytic objects attached to triples of modular forms (or, more generally, motives) whose construction, interpolation, and special value formulae encode deep arithmetic data, especially concerning the conjectural generation of rational points on elliptic curves—specifically, Stark points. In the balanced region, where the weights satisfy "triangle inequalities" and the sign of the functional equation is $-1$, these $p$-adic $L$-functions interpolate square roots of central critical $L$-values and govern the analytic side of refined Birch–Swinnerton-Dyer and Stark-type conjectures in the rank one regime.

1. Construction and Interpolation of Balanced Triple Product $p$-adic $L$-functions

The central object is the balanced triple product $p$-adic $L$-function $\mathcal{L}_p^{\mathrm{bal}}(\ell, m)$, which is attached to a fixed elliptic curve $E/\mathbb{Q}$ and two Hida families of modular forms $\mathbf{g}, \mathbf{h}$ with specializations $g_\ell$, $h_m$. The construction proceeds by:

• Generalizing Hsieh’s three-variable triple product theory to explicitly allow for Hida families whose weight one specializations are classical and possibly ramified. This involves constructing quaternionic Hida families for weight one forms (using auxiliary Hecke operators $\langle\varpi_D\rangle$ introduced via local uniformizers and resolving multiplicity-one failures in quaternionic settings).
• Developing an explicit interpolation formula over an analytic neighborhood in weight space containing at least one weight one point in each direction.

The interpolation formula states that for classical specializations $(\ell, m)$ in the balanced region ($2 \leq \ell, m < 2+\min(\ell, m)$, $\ell + m \equiv 0 \mod 2$):

$$\left( \mathcal{L}_p^{\mathrm{bal}}(\ell, m) \right)^2 = \frac{ \Lambda(f \times g_\ell \times h_m, (\ell + m)/2) }{ \langle f, f \rangle \langle g_\ell, g_\ell \rangle \langle h_m, h_m \rangle } \cdot \frac{ \mathcal{E}_p^{\mathrm{bal}}(f \times g_\ell \times h_m) }{ \mathcal{E}_p(g_\ell, \mathrm{Ad}) \mathcal{E}_p(h_m, \mathrm{Ad}) },$$

with $$\mathcal{E}_p^{\mathrm{bal}}, \mathcal{E}_p(-,\mathrm{Ad})$$ modeling the correct $p$-Euler and adjoint factors, and $\langle-, -\rangle$ denoting Petersson norms. This formula interpolates the algebraic part of central critical values.

Crucially, this construction gives genuine interpolation at classical weight one points for $\mathbf{g}, \mathbf{h}$, a technical challenge previously unsolved due to the failure of multiplicity one and delicate ramification phenomena.

2. The Elliptic Stark Conjecture in Rank One: Formulation and Implications

For $\varrho_1, \varrho_2$ odd two-dimensional Artin representations with $\det\varrho_1 \cdot \det\varrho_2 = 1$, the conjecture concerns the Galois isotypic part of $E(H)\otimes L$ on which $\operatorname{Gal}(H/\mathbb{Q})$ acts via $\varrho_1 \otimes \varrho_2$. In the "rank one" case (i.e., when $$L(\varrho_E \otimes \varrho_1 \otimes \varrho_2, 1)$$ vanishes to order exactly one), the conjecture asserts:

$$\left( \mathcal{L}_p^{\mathrm{bal}}(1,1) \right)^2 \sim_{L^\times} \frac{ \log_{E, p}(P_{\alpha,\alpha})^2 }{ \log_p(u_{1,\alpha}) \log_p(u_{2,\alpha}) },$$

where

• $P_{\alpha,\alpha}\in E(H)_L$ is a Stark point determined by $p$-adic Frobenius eigenvalues $\alpha_1, \alpha_2$ corresponding to $g,h$ at $p$,
• $u_{i,\alpha}$ are Stark units in $\mathcal{O}_H^\times$ associated to the adjoint representations and eigenvalues,
• $\log_{E,p}$ is the $p$-adic formal group logarithm on $E$, and $\log_p$ is the $p$-adic logarithm.

This formula gives an explicit $p$-adic analytic description of a conjectural global point, accessible from the special value "outside the range of interpolation" of the balanced triple product $p$-adic $L$-function, in analogy with Gross-Stark and $p$-adic Gross-Zagier formulae.

Restating, for order of vanishing $=1$,

$$\exp_{E,p}\left( \mathcal{L}_p^{\mathrm{bal}}(1,1) \cdot \log_p(u_{1,\alpha})^{1/2} \cdot \log_p(u_{2,\alpha})^{1/2} \right) \in E(H)_L,$$

so the $p$-adic value (which is concretely computable) "generates" the Stark point via the formal group exponential.

3. The Dihedral Case: Factorization and Proof via Bertolini–Darmon–Prasanna

When both $\varrho_1, \varrho_2$ are induced from Hecke characters of the same imaginary quadratic field $K$ (the "dihedral case"), and $p$ splits in $K$, the conjecture can be proved unconditionally under explicit technical hypotheses. The argument involves:

• Factorizing the triple product $L$-function and its $p$-adic counterpart as a product of Rankin–Selberg $L$-functions over $K$:

$$L(E, \varrho_{gh}, s) = L(E/K, \psi_1, s) \cdot L(E/K, \psi_2, s),$$

and, for the $p$-adic $L$-functions, expressing $\mathcal{L}_p^{\mathrm{bal}}$ in terms of known quadratic $p$-adic $L$-functions and normalized periods.

• Identifying the special value at $(1,1)$ in the $p$-adic $L$-function, away from the interpolation range, with the $p$-adic logarithm of a Stark–Heegner point (from the theory of Bertolini–Darmon–Prasanna), divided by $p$-adic logarithms of elliptic units (Katz $p$-adic $L$-functions).
• Resolving local test vector issues for ramified primes as needed for these factorizations, leveraging additional quaternionic modular form analysis and expanding results of [BDP] to more general ramification.

The precise formula is

$$\mathcal{L}_p^{\mathrm{bal}, f}(1,1) = \lambda \cdot \left( \text{Gross height} \right)^{1/2} \frac{ \log_{E,p}(P_{\psi_2}) }{ \log_p(u_{\psi_g/\psi_g^\sigma})^{1/2} \log_p(u_{\psi_h/\psi_h^\sigma})^{1/2} },$$

up to explicit constants $\lambda$.

4. Technical Innovations and Methodological Advances

The construction and validation of balanced triple product $p$-adic $L$-functions and formulas at weight one critical points required major technical developments:

• Quaternionic test vector analysis: Introduction of additional Hecke operators, e.g., $\langle \varpi_{D_q} \rangle$, enables control of multiplicity one and hence identification of Hida families with correct local behavior even at high ramification.
• Hida family control theorem extension: Generalization using eigenvarieties (cf. Chenevier) permits analytic variation through weight one points and supports rigid-analytic continuation through non-classical weights.
• Normalization and interpolation of explicit periods: The $p$-adic $L$-functions are normalized with Petersson norms instead of (often inaccessible) Gross periods, ensuring algebraicity and applicability for generating explicit points.

This methodology allows the interpolation domain to include all relevant weights, covering regimes previously inaccessible to $p$-adic families.

5. Summary Table of Core Properties

Feature	Construction/Formula	Role
Interpolation formula	$(\mathcal{L}_p^{\mathrm{bal}}(\ell, m))^2 = \cdots$	Square root interpolation at balanced $(\ell, m)$
Elliptic Stark conjecture	$$(\mathcal{L}_p^{\mathrm{bal}}(1,1))^2 \sim \frac{\log_{E,p}(P)^2}{\log_p u_1 \log_p u_2}$$	Relates $p$-adic $L$-values to Stark points and units
Dihedral factorization	$$\mathcal{L}_p^{\mathrm{bal}, f} = (\text{quadratic } p$$-adic $L)\cdot$ (units)$$	Proves the conjecture in CM/dihedral settings
Quaternionic test vectors	New Hecke operators $\langle\varpi_D\rangle$	Local multiplicity-one, Hida family extension
Period normalization	$\langle f, f \rangle$ (Petersson)	Ensures algebraic interpolation

6. Connections and Applications

Balanced triple product $p$-adic $L$-functions and the associated Elliptic Stark Conjecture provide a conduit between $p$-adic analytic methods and the explicit arithmetic of rational points on elliptic curves in isotypic Galois components. The proofs and generalizations in the dihedral case relate critically to the structure of quadratic $p$-adic $L$-functions, Stark units (i.e., $p$-adic logarithms of units predicted by Stark’s conjectures), and the arithmetic of elliptic curves with complex multiplication or possessing explicit big image Galois representations.

These results generalize previous rank two formulas (Darmon-Lauder-Rotger), which featured regulators of two points and one unit, to the rank one setting, yielding new explicit $p$-adic analytic formulas and, when verified, unconditional proofs in the presence of significant automorphy.

7. Key Formula

The conceptual summary is encapsulated by: $$\boxed{ \mathcal{L}_p^{\mathrm{bal}}(1,1)^2 \sim_{L^\times} \frac{ \log_{E, p}(P_{\alpha,\alpha})^2 }{ \log_p(u_{1,\alpha}) \log_p(u_{2,\alpha}) } }$$ where $P_{\alpha,\alpha}$ is a Stark point in $E(H)_L$ and $u_{i,\alpha}$ are Stark units, explicitly computable via Gross–Stark and $p$-adic $L$-function theories.

This machinery gives rise to $p$-adic analytic constructions of points predicted by the BSD conjecture and connects diagonal cycle methods, $p$-adic Abel–Jacobi maps, and $p$-adic regulators in the context of automorphic motives and their $p$-adic $L$-functions.