Arithmetic of critical $p$-adic $L$-functions

Technical Summary: Arithmetic of Critical p-adic L-functions

1. Problem Statement and Context
The paper addresses the arithmetic theory of $p$-adic $L$-functions associated with modular forms at $\theta$-critical points on the cuspidal Coleman–Mazur–Buzzard eigencurve. A modular form $f_\alpha$ (a $p$-stabilization of a newform $f$) is $\theta$-critical if it lies in the image of the $p$-adic $\theta$-operator, $\theta^{k-1}$.

In the classical (non-$\theta$-critical) setting, the eigencurve is étale over the weight space, and $p$-adic $L$-functions are constructed via modular symbols or Beilinson–Kato elements, satisfying standard interpolation properties. However, at a $\theta$-critical point $x_0$, the weight map is ramified (with ramification index $e=2$), and the eigencurve is no longer étale. This geometric degeneracy leads to an "extreme exceptional zero" phenomenon: the specialized $p$-adic $L$-function $L^{[0]}$ vanishes on the entire critical strip ($1 \le j \le k-1$), while the "secondary" $p$-adic $L$-function $L^{[1]}$ (the derivative with respect to the weight parameter) carries the non-trivial interpolation data.

The central problem is to develop a complete arithmetic framework for these critical $L$-functions, including:

• An "étale" construction of Bellaïche's $p$-adic $L$-functions using Galois cohomology.
• The formulation of Iwasawa Main Conjectures in this degenerate setting.
• The definition of algebraic counterparts (Selmer complexes) and $L$-invariants that explain the vanishing and leading terms.
• The interpretation of the derivative of the secondary $L$-function in terms of $p$-adic heights and regulators.

2. Methodology
The authors employ a synthesis of Iwasawa theory, $p$-adic Hodge theory, and the formalism of Selmer complexes (à la Nekovář and Perrin-Riou).

• Étale Construction: Instead of relying on modular symbols (as in Bellaïche's original work), the authors construct the $p$-adic $L$-functions by interpolating Beilinson–Kato elements over the eigencurve. They utilize the overconvergent étale cohomology of Andreatta–Iovita–Stevens to define a "big" Beilinson–Kato class $z(X, \xi)$ over an affinoid neighborhood $X$ of the $\theta$-critical point.
• Eigenspace-Transition Principle: A core technical innovation is the "eigenspace-transition via differentiation" principle. This relates the exponential maps of the infinitesimal deformation of the Galois representation to the exponential maps of its crystalline sub-quotients. It allows the authors to deduce the interpolation properties of the secondary $L$-function ($L^{[1]}$) from the properties of the slope-zero $L$-function.
• Thick Selmer Complexes: To handle the non-semisimple nature of the infinitesimal deformation $V_k$ (where $V_k$ is the deformation of the Galois representation along the eigencurve), the authors introduce thick Selmer complexes. These are defined over the ring $\tilde{E} = E[X]/(X^2)$ and capture the arithmetic information lost in the standard Selmer groups due to the degeneracy at $x_0$.
• Iwasawa Theoretic $L$-invariants: The authors define a new invariant, $L^{cr}_{Iw}$, as the slope of a localization map in Iwasawa cohomology. This invariant serves as the bridge between the critical $p$-adic $L$-function and the classical slope-zero $L$-function.
• Regulators and Heights: The paper establishes leading term formulae relating the derivatives of $p$-adic $L$-functions to $p$-adic height pairings and regulators. Specifically, they define an $\mathcal{O}_X$-adic regulator and show that the derivative of the secondary $L$-function is computed by the second-order derivative of this regulator.

3. Key Contributions and Results

• Étale Construction of Critical $L$-functions: The authors provide an alternative construction of Bellaïche's $p$-adic $L$-functions ($L^{[0]}$ and $L^{[1]}$) using Beilinson–Kato elements and the triangulation of the eigencurve. They prove that their construction coincides with Bellaïche's modular symbol construction (Theorem C).
• The "Eigenspace-Transition" Principle: They establish a general principle (Proposition 2.6) linking the cohomology of the infinitesimal deformation to the cohomology of the crystalline sub-representations via differentiation. This is crucial for proving the interpolation properties of the secondary $L$-function.
• Iwasawa Theoretic $L$-invariant ($L^{cr}_{Iw}$): They define an Iwasawa theoretic $L$-invariant $L^{cr}_{Iw}(V_f)$ and prove that the cyclotomic improvement of the critical $L$-function is equal to the slope-zero $L$-function multiplied by this invariant (Proposition F). Consequently, the non-vanishing of the critical $L$-function is equivalent to the non-vanishing of $L^{cr}_{Iw}$.
• Main Conjectures:
  • Punctual Main Conjecture: They formulate a Main Conjecture for the critical point, relating the module of algebraic $L$-functions to the critical $L$-function. They show this is equivalent to the classical slope-zero Main Conjecture, provided $L^{cr}_{Iw} \neq 0$.
  • Infinitesimal Thickening: They formulate a stronger "infinitesimal thickening" of the Main Conjecture ($\tilde{MC}(f)$) involving the thick Selmer complex. This conjecture implies both the punctual and slope-zero conjectures.
• Leading Term Formulae:
  • They prove an $\mathcal{O}_X$-adic leading term formula for the two-variable $p$-adic $L$-function (Theorem M).
  • Crucially, they show that when the Hecke $L$-function vanishes to order one at the central critical point, the derivative of the secondary $p$-adic $L$-function is computed by the second-order derivative of an $\mathcal{O}_X$-adic regulator, rather than the regulator itself. This contrasts with standard leading term formulae involving first-order regulators.
• Non-vanishing Results: In the case where $f$ has Complex Multiplication (CM), they prove unconditionally that the critical $L$-function and the Iwasawa theoretic $L$-invariant are non-zero (Corollary 3.20), providing the first concrete examples where the Main Conjectures hold in the $\theta$-critical setting.

4. Significance and Claims
The authors claim that this work develops a "fairly complete arithmetic theory" for critical $p$-adic $L$-functions, filling a gap left by previous works which focused on non-critical or non-$\theta$-critical scenarios.

• Resolution of Degeneracy: The paper successfully interprets the "extreme exceptional zero" phenomenon (where the $L$-function vanishes on the critical strip) as a consequence of the non-semisimplicity of the Galois representation's infinitesimal deformation.
• New Invariants: The introduction of the Iwasawa theoretic $L$-invariant $L^{cr}_{Iw}$ and the thick Selmer complex provides new tools for studying the arithmetic of the eigencurve at ramified points.
• Connection to Heights: The derivation of the leading term formula involving the second-order derivative of a regulator is presented as a novel result, offering a new perspective on the relationship between $L$-values and $p$-adic heights in degenerate cases.
• Generalizability: The authors suggest that their abstract framework (particularly the eigenspace-transition principle and the thick Selmer formalism) can be generalized to other contexts, such as $p$-adic Rankin–Selberg $L$-functions in the $\theta$-critical case, though they do not construct these explicitly in this paper.

The paper dedicates its work to the memories of Joël Bellaïche and Jan Nekovář, whose foundational work on $p$-adic $L$-functions and Selmer complexes underpins the constructions presented here.