On a construction of $C^1(\mathbb{Z}_p)$ functionals from $\mathbb{Z}_p$-extensions of algebraic number fields

Timothy All; Bradley Waller

doi:10.5802/jtnb.968

On a construction of

C^{1} (ℤ_{p})

functionals from

ℤ_{p}

-extensions of algebraic number fields

Timothy All ¹ ; Bradley Waller ²

¹ 301 W. Wabash Ave Crawfordsville, IN 47933, USA
² 231 W. 18th Ave Columbus OH 43210, USA

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 29-50

Abstract (VO)
Résumé

Let $k$ be any number field, and let $k_{\infty} / k$ be any $ℤ_{p}$ -extension. We construct a natural $ℤ_{p} 〚 T - 1 〛$ -morphism from $lim_{\leftarrow} k_{n}^{\times} \otimes_{ℤ} ℤ_{p}$ into a special subset of $C^{1} {(ℤ_{p})}^{*}$ , the dual of the $ℂ_{p}$ -vector space of continuously differentiable functions from $ℤ_{p} \to ℂ_{p}$ . We apply the results to the problem of interpolating Gauss sums attached to Dirichlet characters.

Soit $k$ un corps de nombres et $k_{\infty} / k$ une $ℤ_{p}$ -extension. Nous construisons un $ℤ_{p} 〚 T - 1 〛$ -morphisme naturel de $lim_{\leftarrow} k_{n}^{\times} \otimes_{ℤ} ℤ_{p}$ dans un sous-ensemble particulier de $C^{1} {(ℤ_{p})}^{*}$ , le dual de l’espace vectoriel sur $ℂ_{p}$ des fonctions continûment dérivables de $ℤ_{p} \to ℂ_{p}$ . Nous appliquons les résultats au problème d’interpolation des sommes de Gauss attachées aux caractères de Dirichlet.

Reçu le : 2014-11-23
Révisé le : 2015-07-29
Accepté le : 2015-07-30
Publié le : 2017-01-26

DOI : 10.5802/jtnb.968

Classification : 11R23
Keywords: distributions, $L$-functions, Gauss sums, class group

Affiliations des auteurs :

Timothy All ¹ ; Bradley Waller ²

¹ 301 W. Wabash Ave Crawfordsville, IN 47933, USA
² 231 W. 18th Ave Columbus OH 43210, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On a construction of $C^1(\mathbb{Z}_p)$ functionals from $\mathbb{Z}_p$-extensions of algebraic number fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {29--50},
     year = {2017},
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Timothy All; Bradley Waller. On a construction of $C^1(\mathbb{Z}_p)$ functionals from $\mathbb{Z}_p$-extensions of algebraic number fields. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 29-50. doi: 10.5802/jtnb.968

Bibliographie
Cité par

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[8] Alain M. Robert A course in $p$ -adic analysis, Graduate Texts in Mathematics, 198, Springer-Verlag, New York, 2000

[9] W. Sinnott On the Stickelberger ideal and the circular units of an abelian field, Invent. Math., Volume 62 (1980/81) no. 2, pp. 181-234 | DOI

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