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BookDOI

Non-commutative q-expansions

01 Aug 2016-arXiv: Number Theory (World Scientific Publishing)-pp 317-331
TL;DR: In this article, the authors partially answer a question of Fukaya and Kato by constructing a $q$-expansion with coefficients in a non-commutative Iwasawa algebra whose constant term is a non commutative p-adic zeta function.
Abstract: In this short note we partially answer a question of Fukaya and Kato by constructing a $q$-expansion with coefficients in a non-commutative Iwasawa algebra whose constant term is a non-commutative p-adic zeta function.
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Journal ArticleDOI
01 Jan 2011
TL;DR: The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that: • a full bibliographic reference is made to the original source • a link is made in DRO • the fulltext is not changed in any way as mentioned in this paper.
Abstract: The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details.

4 citations

Journal ArticleDOI
TL;DR: In this article , a new strategy for studying low weight specializations of p-adic families of ordinary modular forms was developed, which is designed to explicitly avoid use of the related facts that the Galois representation attached to a classical weight one eigenform has finite image.
Abstract: We develop a new strategy for studying low weight specializations of p-adic families of ordinary modular forms. In the elliptic case, we give a new proof of a result of Ghate–Vatsal which states that a Hida family contains infinitely many classical eigenforms of weight one if and only if it has complex multiplication. Our strategy is designed to explicitly avoid use of the related facts that the Galois representation attached to a classical weight one eigenform has finite image, and that classical weight one eigenforms satisfy the Ramanujan conjecture. We indicate how this strategy might be used to prove similar statements in the case of partial weight one Hilbert modular forms, given a suitable development of Hida theory in that setting.
References
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Book ChapterDOI
01 Jan 1973
TL;DR: In this article, Cela permet de demontrer des proprietes de congruence reliant les ζK(1−k) for diverses valeurs de k, and deduire par interpolation une fonction zeta p-adique for le corps k, au sens de Kubota-Leopoldt (cf.
Abstract: Soient K un corps de nombres algebriques totalement reel, et ζK sa fonction zeta. D’apres un theoreme de Siegel [24], ζK(1 − k) est un nombre rationnel si k est entier ⩾ 1; il est ≠ 0 si k est pair. Lorsque K est abelien sur Q, on peut ecrire ce nombre comme produit de «nombres de Bernoulli generalises»: $$ \zeta _K (1 - k) = \mathop \prod \limits_X L(X,1 - k) = \mathop \prod \limits_X ({{ - b_k (X)} \mathord{\left/ {\vphantom {{ - b_k (X)} k}} \right. \kern- ulldelimiterspace} k}), cf. [18], $$ , ou χ parcourt l’ensemble des caracteres de Q attaches a K. Cela permet de demontrer des proprietes de congruence reliant les ζK(1−k) pour diverses valeurs de k, et d’en deduire par interpolation une fonction zeta p-adique pour le corps k, au sens de Kubota-Leopoldt (cf. [7], [10], [11], [16]).

383 citations


"Non-commutative q-expansions" refers background in this paper

  • ...Introduction The theory of p-adic modular forms essentially began with the paper of Serre [15]....

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  • ...The theory of p-adic modular forms essentially began with the paper of Serre [15]....

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Journal ArticleDOI
TL;DR: In this paper, it was shown that a p-adic L-function associated to a one-dimensional Artin character 4 with F is continuous for s e Zp{ 1, and even at s = 1 if 4, is not trivial.
Abstract: Let F be a totally real number field. Let p be a prime number and for any integer n let Fun denote the group of nth roots of unity. Let 41 be a p-adic valued Artin character for F and let F,, be the extension of F attached to 4, i.e., so that 4 is the character of a faithful representation of Gal(F,,/F). We will assume that F,, is also totally real. For a number field K let K., denote the cyclotomic Zp-extension of K. Following Greenberg we say that 4 is of type S if F., n Fc, = F and of type W if 4 is one-dimensional with F.,p c Fcc. Deligne and Ribet (in [DR], following Kubota and Leopoldt for the case F= Q) have proved the existence of a p-adic L-function associated to a one-dimensional Artin character 4 with F,, totally real. This function Lp(s, 4) is continuous for s e Zp{ 1}, and even at s = 1 if 4, is not trivial, and satisfies the following interpolation property:

360 citations


"Non-commutative q-expansions" refers methods in this paper

  • ...Since then they have formed a central tool in number theory and have most notably been used to prove main conjectures of commutative Iwasawa theory (Wiles [18], Skinner-Urban [17] etc....

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Journal ArticleDOI

355 citations


"Non-commutative q-expansions" refers methods in this paper

  • ...Theorem 1 (Deligne-Ribet [4], theorem 6....

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  • ...It was generalised by Katz [11] and Deligne-Ribet [4] and used to construct p-adic Lfunctions for CM and totally real number fields respectively....

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  • ...Theorem 1 (Deligne-Ribet [4], theorem 6.1)....

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Journal ArticleDOI

302 citations


"Non-commutative q-expansions" refers background in this paper

  • ...As F1 is an abelian extension of Q and G1 is pro-p we know by the theorem of Ferrero-Washington [5] that Ei ∈ A(G ab i ) × (it is enough to show that the constant term, i....

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  • ...As F1 is an abelian extension of Q and G1 is pro-p we know by the theorem of Ferrero-Washington [5] that Ei ∈ A(G ab i ) × (it is enough to show that the constant term, i.e. the p-adic zeta functions ζ(Ki/Fi), of Ei are units in Λ(G ab i )S ....

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Journal ArticleDOI

284 citations


"Non-commutative q-expansions" refers methods in this paper

  • ...It was generalised by Katz [11] and Deligne-Ribet [4] and used to construct p-adic Lfunctions for CM and totally real number fields respectively....

    [...]