[Paper Review] Anticyclotomic p-ordinary Iwasawa Theory of Elliptic Modular Forms

Kâzım Büyükboduk, Antonio Lei|arXiv (Cornell University)|Feb 24, 2016

Algebraic Geometry and Number Theory31 references5 citations

TL;DR

This paper establishes the anticyclotomic $p$-ordinary Iwasawa main conjecture for elliptic modular forms of weight $k \geq 2$ over an imaginary quadratic field where $p$ splits, using Beilinson-Flach elements and explicit reciprocity laws. It proves that the $p$-adic $L$-function lies in the characteristic ideal of the dual Selmer group, extending classical results to non-CM, $p$-ordinary forms via Euler system techniques and $p$-adic height pairings.

ABSTRACT

This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic Zp-tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson-Flach elements.

Motivation & Objective

• To establish the Iwasawa main conjecture for $p$-ordinary elliptic modular forms along the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field where $p$ splits.
• To extend previous results on CM forms and elliptic curves to non-CM, $p$-ordinary modular forms using Euler systems of Beilinson-Flach elements.
• To unify the definite and indefinite cases in the anticyclotomic Iwasawa theory via a single framework based on Rankin-Selberg $L$-functions and $p$-adic $L$-functions.
• To prove a Rubin-style formula for the anticyclotomic $p$-adic height pairing using the cyclotomic derivative of Beilinson-Flach elements.

Proposed method

• Constructs an Euler system of Beilinson-Flach classes over $K$ from Rankin-Selberg convolutions using results from [KLZ15a].
• Applies Coleman maps and explicit reciprocity laws to relate Beilinson-Flach elements to $p$-adic $L$-functions.
• Uses the $p$-adic height pairing on the anticyclotomic tower to relate the derivative of the Beilinson-Flach element to the $p$-adic $L$-function.
• Employs the Coleman-Perrin-Riou map to relate the cyclotomic derivative of the Euler system to the $p$-adic $L$-function $\mathfrak{L}_{f,1}^{(\alpha)}$.
• Applies Nekovář’s theory of $p$-adic height pairings and Tate duality to derive a Rubin-style formula in the anticyclotomic setting.
• Establishes divisibility results between the $p$-adic $L$-function and the characteristic ideal of the Selmer group, upgrading to equality under additional hypotheses.

Experimental results

Research questions

• RQ1 Does the anticyclotomic Iwasawa main conjecture hold for $p$-ordinary, non-CM modular forms over imaginary quadratic fields where $p$ splits?
• RQ2 Can Beilinson-Flach elements be used to construct an Euler system that controls the Selmer group in the anticyclotomic $\mathbb{Z}_p$-extension?
• RQ3 Is there a $p$-adic height pairing formula that relates the derivative of the Beilinson-Flach element to the $p$-adic $L$-function in the anticyclotomic setting?
• RQ4 Under what conditions does the containment of the $p$-adic $L$-function in the characteristic ideal of the Selmer group become an equality?
• RQ5 How do the local conditions at $\mathfrak{p}^c$ influence the structure of the Selmer group and the $p$-adic $L$-function?

Key findings

• The $p$-adic $L$-function $\mathfrak{L}_{f,1}^{(\alpha)}$ lies in the ideal $\mathfrak{Reg}_{\textup{ac}} \cdot \mathrm{char}(\mathfrak{X}(f\otimes\alpha/D_{\infty})_{\textup{tor}}) \otimes_{\Lambda_{\textup{ac}}^{\mathfrak{o}}} \mathcal{R}_{L}^{\textup{ac}}$, proving the main conjecture up to a regulator factor.
• Under additional hypotheses—$N$ square-free, $\rho_f$ ramified at a prime dividing $N^-$, and the weight-2 condition—the containment becomes an equality: $\mathfrak{L}_{f,1}^{(\alpha)}$ generates the characteristic ideal of the Selmer group.
• The cyclotomic derivative $\mathfrak{d}\textup{BF}_{D_{\infty}}$ of the Beilinson-Flach element is shown to map to $\mathfrak{L}_{f,1}^{(\alpha)}$ under the Coleman-Perrin-Riou map.
• A Rubin-style formula is established: the anticyclotomic $p$-adic height pairing satisfies $\mathfrak{h}^{\textup{ac}}_p(\textup{BF}_{D_{\infty}}, \mathfrak{y}) = -\langle \mathfrak{d}\textup{BF}_{D_{\infty}}, \textup{res}_{\mathfrak{p}^c}(\mathfrak{y}) \rangle_{\textup{Tate}}$, linking the height pairing to the derivative of the Euler system.
• The construction of the Euler system of Beilinson-Flach elements over $K$ is achieved via a generalization of [KLZ15a], extending the method to higher weights and non-CM forms.
• The proof relies on the non-vanishing of the anticyclotomic Beilinson-Flach Euler system, which is established via the non-triviality of the $p$-adic $L$-function and the explicit reciprocity law.