Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of p-adic L-functions

Liu, Zheng; Rosso, Giovanni

doi:10.1007/s00208-020-01966-x

Cited by 7 publications

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“…Since we are often faced with representations of L-functions not as a finite sum but rather as an integral of an Eisenstein series against cusp form(s), we now briefly explain the key ideas for adapting such a representation to the p-adic setting. We discuss this strategy in the context of the Rankin-Selberg zeta function, where it was first developed (by Hida in [37]), but it has also since been extended to various settings, including, among others, in [23,38,53,54,58].…”

Section: Working With Pairings and Pullback Methodsmentioning

confidence: 99%

“…Shimura and Rankin proved in [60,67] that < m < k. In [37], Hida constructed p-adic Rankin-Selberg zeta functions by building on Shimura's approach to studying algebraicity. In particular, the idea to interpret the Rankin-Selberg zeta function in terms of the Peterssen pairing plays a key role, and this remains true in extensions to higher rank groups (including in the discussions of algebraicity in [35] and in extensions to the p-adic case in PEL settings involving the doubling method in [23,53,54]). The idea is to reinterpret the linear Petersson pairing h 1 , h 2 as a functional h 1 (h 2 ).…”

Section: Working With Pairings and Pullback Methodsmentioning

confidence: 99%

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An introduction to Eisenstein measures

Eischen¹

2022

Journal De Théorie Des Nombres De Bordeaux

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L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Journal de Théorie des Nombres de Bordeaux 33 (2021), 779-808 An introduction to Eisenstein measures par Ellen EISCHEN Résumé. Cet article fournit une introduction aux mesures d'Eisenstein, un outil puissant pour construire certaines fonctions L p-adiques. Vues pour la première fois dans la réalisation par Serre des fonctions zêta de Dedekind p-adiques associées aux corps totalement réels, les mesures d'Eisenstein fournissent un moyen d'étendre les congruences de style kummerien, observées par Kummer pour les valeurs de la fonction zêta de Riemann (dites congruences de Kummer) à certaines autres fonctions L. En plus de retracer les développements clés, nous discutons certains défis qui se posent dans des contextes plus généraux, en concluant par certains qui restent ouverts.

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Section: Working With Pairings and Pullback Methodsmentioning

confidence: 99%

Section: Working With Pairings and Pullback Methodsmentioning

confidence: 99%

An introduction to Eisenstein measures

Eischen¹

2022

Journal De Théorie Des Nombres De Bordeaux

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“…Nevertheless, one can hope to develop Hida theory after restricting to weights for which there is no a priori obstruction to the existence of Eisenstein series. The interested reader is referred to [LR17] where this strategy has been exploited to construct non-cuspidal Hida families of Siegel modular forms. Proposition 4.3.…”

Section: Hida Theorymentioning

confidence: 99%

Hida theory over some unitary Shimura varieties without ordinary locus

Brasca¹,

Rosso²

2021

American Journal of Mathematics

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“…We apply the approach in [LR20] to prove the vertical control theorem for semi-ordinary forms on GU(3, 1) by analyzing the quotient V /V 0 and using the vertical control theorem for cuspidal semi-ordinary forms on GU(3, 1). When studying V /V 0 , we introduce an auxiliary space V ♭ .…”

Section: Shimura Varietymentioning

confidence: 99%

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

Castella¹,

Liu²,

Wan³

2021

Preprint

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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

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Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of p-adic L-functions

Cited by 7 publications

References 40 publications

An introduction to Eisenstein measures

An introduction to Eisenstein measures

Hida theory over some unitary Shimura varieties without ordinary locus

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

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