[Paper Review] Le système d'Euler de Kato en famille (I)

Shanwen Wang|arXiv (Cornell University)|Nov 18, 2012

Advanced Algebra and Geometry9 references3 citations

TL;DR

This paper constructs a family of Kato's Euler systems and explicit reciprocity laws over the $p$-adic weight space $ω$, interpolating classical objects. Using $p$-adic families of modular forms and distributions, it establishes a $p$-adic interpolation of the Kato exponential map, proving a family version of the explicit reciprocity law via $p$-adic modular forms and Eisenstein distributions.

ABSTRACT

This article is the first article of a serie of articles on the generalization of Kato's Euler system. The main subject of this article is to construct a family of Kato's Euler systems and a family of Kato's explicit reciprocity laws over the weight space, which interpolate the corresponding classical objects.

Motivation & Objective

• To generalize Kato's Euler system to a $p$-adic family over the weight space $\mathscr{W}$, extending classical constructions to a $p$-adic analytic setting.
• To construct a family of explicit reciprocity laws over $\mathscr{W}$ that interpolate the classical Kato reciprocity laws for critical $p$-adic $L$-values.
• To establish a $p$-adic interpolation of the Kato exponential map via a family of $p$-adic modular forms and distributions.
• To define and study $p$-adic families of Eisenstein-Kronecker distributions, linking them to the explicit reciprocity law in families.
• To provide a foundational framework for the second part of the series, which will address the $p$-adic $L$-functions and Iwasawa theory in families.

Proposed method

• Constructs a $p$-adic family of Banach representations over the weight space $\mathscr{W}$ using the universal $p$-adic character and rigid analytic geometry.
• Introduces a $p$-adic family of Eisenstein-Kronecker distributions $z_{\mathbf{Eis},c,d}(\kappa^{\mathbf{univ}},j)$ via $p$-adic interpolation of classical Eisenstein series.
• Defines a $p$-adic $L$-function via the $p$-adic zeta function and the universal character $\kappa^{\mathbf{univ}}$, linking it to the distribution $z_{\mathbf{Eis},c,d}$.
• Applies the dual exponential map $\exp^*_{\mathbf{Kato},\nu}$ to the family of Euler systems, using the structure of the representation $\mathbf{D}_{1,j,\mathscr{W}}$.
• Establishes a $2$-cocycle on the Galois group using the family of Euler systems and passes to Lie algebra cohomology to derive the reciprocity law.
• Uses the Amice transform and $p$-adic integration to compare $p$-adic $L$-values and the terms in the reciprocity law, especially the constant terms of modular forms.

Experimental results

Research questions

• RQ1 How can Kato's Euler system be extended to a $p$-adic family over the weight space $\mathscr{W}$, interpolating classical systems at classical points?
• RQ2 Can a $p$-adic family of explicit reciprocity laws be constructed that interpolates the classical Kato reciprocity laws for critical $p$-adic $L$-values?
• RQ3 What is the role of $p$-adic Eisenstein-Kronecker distributions in realizing the interpolation of the reciprocity law in families?
• RQ4 How does the dual exponential map $\exp^*_{\mathbf{Kato},\nu}$ behave in the $p$-adic family setting, and how does it relate to the Euler system?
• RQ5 Can the constant terms of $p$-adic modular forms be interpreted as $p$-adic integrals of the universal distribution, leading to the reciprocity law?

Key findings

• The paper constructs a $p$-adic family of Kato's Euler systems over the weight space $\mathscr{W}$, parameterized by the universal character $\kappa^{\mathbf{univ}}$, interpolating classical systems at classical weights.
• A family of explicit reciprocity laws is established over $\mathscr{W}$, generalizing the classical Kato reciprocity law to the $p$-adic family setting.
• The dual exponential map $\exp^*_{\mathbf{Kato},\nu}$ is constructed for the family of representations $\mathbf{D}_{1,j,\mathscr{W}}$, enabling the interpolation of the reciprocity law.
• The constant term of the $p$-adic modular form associated to the Euler system is shown to be equal to a $p$-adic integral of the universal Eisenstein distribution, via the Amice transform.
• The key identity $\exp^*_{\mathbf{Kato},\nu}(z_{M,A}) = \frac{1}{(j-1)!}M^{-2j}\operatorname{ord}(\alpha/M)^{j-1}\kappa^{\mathbf{univ}}(M/\operatorname{ord}(\alpha/M))F_{c,\alpha/M,\beta/M}(\kappa^{\mathbf{univ}},j)E^{(j)}_{d,\gamma/M,\delta/M}$ is proven, confirming the reciprocity law in the family.
• The construction uses $p$-adic integration and the $p$-adic zeta function to interpolate the special values of $L$-functions, linking them to the Euler system via the reciprocity law.