THE IWASAWA MAIN CONJECTURE FOR HILBERT MODULAR FORMS

Xin Wan

doi:10.1017/fms.2015.16

Abstract

Following the ideas and methods of a recent work of Skinner and Urban, we prove the one divisibility of the Iwasawa main conjecture for nearly ordinary Hilbert modular forms under certain local hypotheses. As a consequence, we prove that for a Hilbert modular form of parallel weight, trivial character, and good ordinary reduction at all primes dividing$p$, if the central critical$L$-value is zero then the$p$-adic Selmer group of it has rank at least one. We also prove that one of the local assumptions in the main result of Skinner and Urban can be removed by a base-change trick.

Highlights

( ) K is not contained in the narrow Hilbert class field of F and all primes v ramified in F/Q are split in K
The Hida families are just components of the Hecke algebra h acting on the space of cuspidal ΛW -adic nearly ordinary forms
8. p-adic properties of Fourier coefficients of ED following [44, Section 13], using the theta correspondence between different unitary groups, we prove that certain Fourier coefficient of ED is not divisible by certain height-one prime P

Summary

Introduction

Let Mκ,w(U0(N ), ε; C) be the space of Hilbert modular forms of weight (κ, w) with level group U0(N ) and nebentypus ε. First of all let us define the weight space for Hilbert modular Hida families. H} with the property that for a Zariski dense set of primes φ ∈ SpecI which map to arithmetic points in Spec(ΛW ), the specializations φ(ci (ξ, I)) are the qexpansions a0(ti , fφ) or a(ti ξ, fφ) (see (2.1)) of nearly ordinary cusp forms fφ of weight (κφ, κφ/2), prime to p level M and nebentypus εφ at primes dividing p. Let A be a finite extension of Qp. One can define the space of Hilbert modular forms Mκ,w(U0(N ), ε, A) and the corresponding cuspidal spaces Sκ,w(U0(N ), ε, A). (We have only focused on the special case when the character is trivial in the main conjecture in the introduction.)

Hida theory for unitary Hilbert modular forms

Siegel Eisenstein series and their pullbacks

Fourier–Jacobi expansions

D KdMD2 pvrv

Proof of the main results

Let integral