[Paper Review] Iwasawa Main Conjecture for Rankin-Selberg p-adic L-functions

Xin Wan|arXiv (Cornell University)|Aug 18, 2014

Advanced Algebra and Geometry5 citations

TL;DR

This paper proves one divisibility in the Iwasawa main conjecture for Rankin-Selberg p-adic L-functions attached to a Hida family of modular forms and a CM form of higher weight, using congruences between Klingen-Eisenstein series and cusp forms on GU(3,1). The key result establishes that the p-adic L-function divides the characteristic ideal of the associated Selmer group, enabling applications to the converse of the Gross-Zagier-Kolyvagin theorem and the p-adic Birch and Swinnerton-Dyer formula in the rank one case.

ABSTRACT

In this paper we prove that the p-adic L-function that interpolates the Rankin-Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa main conjecture for the p-adic L-function. We prove this conjecture using the congruences between Klingen Eisensteinseries and cusp forms on the group GU(3; 1), following the strategy of a recent work of C. Skinner and E. Urban. This theorem can be used to deduce a converse of Gross-Zagier-Kolyvagin theorem and the precise BSD formula in the rank one case.

Motivation & Objective

• To establish one divisibility in the Iwasawa main conjecture for p-adic L-functions arising from the Rankin-Selberg product of a Hida family and a CM form.
• To extend the method of Skinner and Urban, using congruences on the unitary group GU(3,1), to handle general Fourier-Jacobi expansions in the non-ordinary setting.
• To deduce arithmetic applications, including the converse of the Gross-Zagier-Kolyvagin theorem and the p-adic part of the precise BSD formula in analytic rank one.
• To provide a foundational step toward proving the Iwasawa main conjecture for supersingular elliptic curves.

Proposed method

• Constructs Σ-primitive p-adic L-functions via the doubling method, interpolating special values of Rankin-Selberg L-functions at critical central values.
• Uses congruences between Klingen-Eisenstein series and cusp forms on GU(3,1) to relate p-adic L-functions to Selmer groups.
• Applies the strategy of Skinner and Urban, adapted to the higher-weight CM form and non-ordinary setting, by working with general Fourier-Jacobi expansions.
• Employs Fitting ideals and characteristic ideals in Iwasawa algebras over completed rings of integers to compare Selmer groups and p-adic L-functions.
• Performs localization and base change arguments to reduce the main theorem to a statement over a larger ring, then descends back to the original ring using normality and finiteness.
• Uses the pseudo-representation associated to the Galois representation on the cohomology of a Siegel modular variety to control the structure of the Selmer group.

Experimental results

Research questions

• RQ1 Does the p-adic L-function associated to the Rankin-Selberg product of a Hida family and a higher-weight CM form divide the characteristic ideal of the corresponding Selmer group?
• RQ2 Can the method of congruences between Eisenstein series and cusp forms on GU(3,1) be extended to non-ordinary and higher-weight settings to prove one side of the Iwasawa main conjecture?
• RQ3 To what extent do the p-adic L-functions constructed via the doubling method control the structure of the dual Selmer group in Iwasawa theory?
• RQ4 How does the Iwasawa main conjecture for this p-adic L-function imply the converse of the Gross-Zagier-Kolyvagin theorem?
• RQ5 What is the precise relationship between the p-adic L-function and the characteristic ideal of the Selmer group in the rank one analytic rank case?

Key findings

• The p-adic L-function $τ_{\mathbf{f},\xi,\mathcal{K}}^{\Sigma}$ divides the characteristic ideal of the dual Selmer group $X_{\mathbf{f},\mathcal{K},\xi}$, proving one divisibility in the Iwasawa main conjecture.
• The proof relies on congruences between Klingen-Eisenstein series and cusp forms on GU(3,1), extending Skinner and Urban's strategy to the higher-weight CM case.
• The Fitting ideal of the Selmer group is contained in the ideal generated by the p-adic L-function, implying that the characteristic ideal of the Selmer group divides the L-function.
• Specializing to a single form $f_0$ of weight 2, the result implies the p-adic BSD formula in the analytic rank one case.
• The construction of the p-adic L-function via the doubling method ensures interpolation of algebraic parts of critical L-values at central critical points.
• The method handles non-split primes and local Euler factors by treating height one primes not arising as pullbacks from lower-rank Iwasawa algebras.