A proof of $p$-adic Gross--Zagier theorem via BDP formula

Abstract: This paper provides a new proof of the $p$-adic Gross--Zagier formula for the $p$-adic $L$-function associated with the base change of a normalised cuspidal eigen-newform $f$ of weight $k \geq 2$ (and families of such) to an imaginary quadratic field $K$. Our results encompass both the classical $p$-ordinary cases and non-ordinary scenarios, including new cases where $k > 2$ and $\mathrm{ord}_p(a_p(f)) > 0$. Unlike the traditional approach of comparing geometric and analytic kernels, we employ a ``wall-crossing'' strategy centred on the BDP formula and the theory of Beilinson--Flach elements.

• The paper introduces a new, unified proof of the p-adic Gross–Zagier theorem by exploiting the BDP formula and Beilinson–Flach elements.
• It bypasses traditional analytic-geometric kernel comparisons by utilizing wall-crossing, enabling explicit treatment of both ordinary and non-ordinary cases.
• The work establishes a precise identity linking the derivative of the p-adic L-function to p-adic height pairings, advancing Iwasawa theory and the study of Heegner cycles.

$p$-adic Gross--Zagier Theorems via the BDP Formula: A Uniform Proof

Introduction and Context

The $p$-adic Gross--Zagier formula provides a deep and precise explicit connection between the derivative of $p$-adic $L$-functions at their central critical point and the $p$-adic height pairings of Heegner points or cycles associated with modular forms. Originally established by Perrin-Riou, Nekovář, Kobayashi, and others in disparate cases, this formula underpins Iwasawa-theoretic investigations of the Birch and Swinnerton-Dyer conjecture and $p$-adic height theory. The paper "A proof of $p$-adic Gross–Zagier theorem via BDP formula" (2604.13854) presents a new, unified proof of the $p$-adic Gross–Zagier theorem for $p$-adic $L$-functions associated to (families of) newforms of weight $p$0. This approach is notable for its applicability to both $p$1-ordinary and non-ordinary settings (including instances where $p$2 and $p$3), and for avoiding the comparison of analytic and geometric kernels central to previous arguments. Instead, the proof exploits the structure of Beilinson–Flach elements, the wall-crossing formalism, and crucially the Bertolini–Darmon–Prasanna (BDP) formula.

Overview of Main Results

The central theorem establishes, for the $p$4-adic $p$5-function $p$6 attached to the base change of a $p$7-stabilized newform $p$8 to an imaginary quadratic field $p$9, under relevant Heegner conditions and big image assumptions, the following precise identity at the central point $p$0:

$p$1

where $p$2 is a Heegner cycle (defined classically or ad hoc for the non-ordinary case), $p$3 is an explicit algebraic factor depending only on $p$4, and $p$5 denotes the Nekovář $p$6-adic height pairing. In settings where the non-vanishing hypothesis (\textsf{NV}) holds, the factor $p$7, leading to a clean formula between heights and the analytic derivative.

This result encompasses all previously proven ranges—including higher weight and non-ordinary cases not covered by earlier methods, thus providing a uniform treatment.

Methodological Innovations

BDP Formula and Wall-Crossing

The crux of the proof lies in a series of key steps radically distinct from the traditional analytic-geometric kernel comparison:

1. Beilinson–Flach Euler System Machinery: Beilinson–Flach elements, constructed using $p$8-adic families of modular forms, are realized as classes in Iwasawa cohomology with explicit $p$9-adic interpolation and explicit control over local properties.
2. Rubin-Style Formula: Derivatives of $L$0-adic $L$1-functions associated to $L$2 can be related to $L$3-adic heights of Beilinson–Flach elements via explicit local and global duality arguments.
3. Comparison with Heegner Classes via BDP and Wall-Crossing: The BDP formula provides a reciprocity law relating the local images of Beilinson–Flach elements and Heegner classes. Leveraging the wall-crossing principle (the interpolation of $L$4-adic heights across families and GGP regions), the global Selmer classes associated to these cycles are shown to coincide (up to explicit constants), under big image or irreducibility conditions.
4. Explicit Height Computations and Specialization: By considering the $L$5-adic variations and specializing to crystalline points in the Hida/Coleman family, the authors transit to cases where direct calculation of heights and $L$6-function derivatives is possible.

This machinery is fundamentally more flexible than previous approaches and accommodates both ordinary and non-ordinary settings, including critical slope and supersingular primes.

Anticyclotomic Iwasawa Theory

A crucial technical input is the control over the anticyclotomic variation of Heegner classes along $L$7-extensions of $L$8. The authors show that, modulo algebraic factors, the $L$9-adic heights interpolate the corresponding specializations of the two-variable $p$0-adic $p$1-function, invoking results on the structure of anticyclotomic Selmer groups and the non-triviality of generalized Heegner cycles in families.

Numerical and Structural Highlights

• Uniformity Across $p$2 and Slope: The main theorem holds for any weight $p$3 and for both $p$4-ordinary and critical slope $p$5-stabilizations, subject to the established big image and Heegner-type hypotheses.
• Explicit Constants: The factors $p$6, $p$7, and the Euler–type factors appearing in the formulas are given in terms of $p$8, $p$9, and modular form constants. In key settings, these constants are shown to be $p$0.
• Unification and Generalization: Prior $p$1-adic Gross--Zagier formulas—due to Perrin-Riou [perrinriou87], Nekovář [nekovarGZ], Kobayashi [kobayashi13, kobayashi2014_GZ], Disegni [disegni17], and others—are recovered as specializations.
• Strong Contradictory to Previous Paradigms: The proof does not require vanishing of local heights at $p$2, nor the matching of Fourier expansions for analytic/geometric kernels, which had been essential previously and challenging in the non-ordinary setting.

Theoretical and Practical Implications

Advancements in Iwasawa Theory and Main Conjectures

This result significantly advances the study of the anticyclotomic Iwasawa main conjecture for modular forms over general congruence subgroups, especially in non-ordinary settings. The explicit computation of $p$3-adic height pairings enables new progress on:

• Iwasawa main conjectures for Selmer groups attached to higher weight (and higher rank) motives, especially those with non-ordinary local behavior at $p$4.
• Verification and generalization of Perrin-Riou's predictions for derivatives of $p$5-adic $p$6-functions and their relation to heights.
• Structure and freedom of Selmer groups in families, relevant to the study of exceptional zeros, $p$7-invariants, and the distribution of Heegner points in towers.

Potential for Further Generalizations

The methods are expected to carry over, with suitable modifications, to more general settings, including:

• Modular forms with non-trivial central character, non-split quaternionic forms, and self-dual twists.
• $p$8-adic triple product $p$9-functions, diagonal cycles, and the $p$0-adic Gross–Kudla–Schoen formula.
• Eigenvarieties and completed cohomology cases, supporting conjectures on the variation of heights in (larger) $p$1-adic families.

Computational and Algorithmic Prospects

Given the explicit nature of the formulas for $p$2-adic heights and derivatives, the results have applications to algorithms for computing (or bounding) Mordell–Weil ranks and for explicit verification of the Birch and Swinnerton-Dyer conjecture over quadratic fields in both ordinary and non-ordinary circumstances.

Conclusion

This paper establishes a uniform, conceptually novel proof of the $p$3-adic Gross--Zagier theorem via the BDP formula and Beilinson–Flach elements, overcoming the limitations of analytic-geometric kernel comparison and achieving explicit, flexible relations for $p$4-adic heights in a broad range of settings. The methods and structural advances made here are expected to have sustained impact on arithmetic geometry, Iwasawa theory, and computational aspects of the arithmetic of modular forms (2604.13854).