Iwasawa-Greenberg main conjecture for non-ordinary modular forms and Eisenstein congruences on $\mathrm{GU}(3,1)$

Abstract

In this paper we prove one side divisibility of the Iwasawa-Greenberg main conjecture for Rankin-Selberg product of a weight two cusp form and an ordinary CM form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on GU(3, 1), generalizing earlier result of the third-named author to allow non-ordinary cusp forms. The main result is a key input in the third author's proof for Kobayashi's ±-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semi-ordinary Hida theory along an appropriate smaller weight space, and a study of the semi-ordinary Eisenstein family. Contents 1. 2.1. Some notation 2.2. Shimura variety 2.3. Hasse invariant 2.4. Some groups 2.5. Igusa towers 2.6. p-adic forms forms 2.7. Classical automorphic forms 2.8. The U p -operator 2.9. Statement of main theorem 3.The proof of Theorem 2.9.1 for the cuspidal part 4.The proof of Theorem 2.9.1 for the non-cuspidal part 4.1. Cusp labels 4.2. The formal completion along boundary strata 4.3. The subset) ord and the subspace V ♭ of V 4.4. The exact sequence for V ♭ n,m 4.5. Three propositions on the U p -action on V and V ♭ . 4.6. The ordinary projection on V and the fundamental exact sequence 4.7. The Fourier-Jacobi expansion 5.The construction of the Klingen family 5.1. Some notation 5.2. Our setup 5.3. The weight space 5.4. The groups 5.5. The Klingen Eisenstein series and the doubling method 5.6. The auxiliary data for the Klingen family 5.7. The choice of the local sections for the Siegel Eisenstein series on GU(3, 3) 5.8. p-adic measures and p-adic families of automorphic forms 5.9. The p-adic family of Siegel Eisenstein series on GU(3, 3) 1 5.10. The semi-ordinary family of Klingen Eisenstein series 44 6.The degenerate Fourier-Jacobi coefficients of the Klingen family 46 6.1. The divisibility of the degenerate Fourier-Jacobi coefficients by p-adic L-functions 46 6.2. The computation of local zeta integrals at p 49 7.The non-degenerate Fourier-Jacobi coefficients of the Klingen family 51 7.1. The Schrödinger model of Weil representation and the intertwining maps for different polarizations 52 7.2. The Heisenberg group and Jacobi forms 54 7.3. Theta series 56 7.4. The unfolding 56 7.5. Our strategy of analyzing the non-degenerate Fourier-Jacobi coefficients of the Klingen Eisenstein family 58 7.6. The auxiliary Jacobi form θ J 1 61 7.7. The construction of the auxiliary CM families h, h3 , θ, θ3 63 7.8. p-adic Petersson inner product on U(2) 66 7.9. The Rallis inner product formulas for θ, θ3 p-adic and h, h3 p-adic . 67 7.10. Extending CM forms on U(2) to GU(2) 67 7.11. The nonvanishing property of the degenerate Fourier-Jacobi coefficients of the Klingen Eisenstein family 68 7.12. The local triple product integrals at non-split places 73 8. Proof of Greenberg-Iwasawa Main Conjecture 77 8.1. The Klingen Eisenstein ideal 77 8.2. The main theorem 78 Index 81 References 81

On Fine Selmer Groups and Signed Selmer Groups of Elliptic Modular Forms

“…The reverse inclusion of (2.2) has been established in the monumental work of Skinner and Urban [26] for a p-ordinary modular form (under certain hypotheses; see [26,Theorem 1]). In the nonordinary form, there have been several recent breakthroughs in this reverse direction (see [5,6,9]). Finally, the work [16] has supplied many sufficient conditions for the nonvanishing of L p and L p (see [16,Corollary 3.29 and Proposition 3.39]).…”