Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T04:49:40.526Z Has data issue: false hasContentIssue false
  • English
  • Français

p-units in ray class fields of real quadratic number fields

Part of: Zeta and $L$-functions: analytic theory Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  01 March 2009

Hugo Chapdelaine
Hugo Chapdelaine*
Affiliation:
Département de mathématiques et de statistique (DMS), Université Laval, Pavillon Alexandre-Vachon, 1045 avenue de la Médecine, Local 1056, Québec G1V 0A6, Canada (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K be a real quadratic number field and let p be a prime number which is inert in K. We denote the completion of K at the place p by Kp. We propose a p-adic construction of special elements in Kp× and formulate the conjecture that they should be p-units lying in narrow ray class fields of K. The truth of this conjecture would entail an explicit class field theory for real quadratic number fields. This construction can be viewed as a natural generalization of a construction obtained by Darmon and Dasgupta who proposed a p-adic construction of p-units lying in narrow ring class fields of K.

Keywords

p-adic Gross–Stark conjecturesexplicit class field theoryp-adic integrationEisenstein series

MSC classification

Secondary: 11S40: Zeta functions and $L$-functions 11M99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 2 , March 2009 , pp. 364 - 392
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Chapdelaine, H., Elliptic units in ray class fields of real quadratic number fields, version with a few corrections and supplements. available athttp://www.mat.ulaval.ca/fileadmin/Pages_personnelles_des_profs/hchapd/thesis_final.pdf.Google Scholar
[2]Chapdelaine, H., Computation of p-units in ray class fields or real quadratic number fields, J. Math. Comp. (accepted).Google Scholar
[3]Chapdelaine, H., Elliptic p-units and Gauss sums (submitted).Google Scholar
[4]Darmon, H., Rational points on modular elliptic curves (American Mathematical Society, Providence, RI, 2004).Google Scholar
[5]Darmon, H. and Dasgupta, S., Elliptic units for real quadratic fields, Ann. of Math. (2) 163 (2006), 301–346.CrossRefGoogle Scholar
[6]Dasgupta, S., Computations of elliptic units for real quadratic number fields, Canad. J. Math. 59 (2007), 553–574.CrossRefGoogle Scholar
[7]Dasgupta, S., Shintani zeta functions and Gross-Stark units for totally real fields, Duke Math. J. 143 (2008), 225–279.CrossRefGoogle Scholar
[8]Lang, S. and Kubert, D. S., Modular units (Springer, New York, 1981).Google Scholar
[9]Mennike, J., On Ihara’s modular group, Invent. Math. 4 (1967), 202–228.CrossRefGoogle Scholar
[10]Serre, J.-P., Le problème des groupes de congruences pour SL 2, Ann. of Math. (2) 92 (1970), 489–527.CrossRefGoogle Scholar
[11]Siegel, C. L., Bernoullische Polynome und quadratische Zahlkörper, Nach. Akad. Wiss. Göttingen Math.-Phys. K1. II (1968), 7–38.Google Scholar
[12]Young, P. T., Kummer congruences for values of Bernoulli and Euler polynomials, Acta Arith. 48 (2001), 277–288.CrossRefGoogle Scholar