Compositio Mathematica

Table of Contents - 2009 - Volume 145, Issue 02  

Research Article

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

Hugo Chapdelainea1

a1 Département de mathématiques et de statistique (DMS), Université Laval, Pavillon Alexandre-Vachon, 1045 avenue de la Médecine, Local 1056, Québec G1V 0A6, Canada (email: hugo.chapdelaine@mat.ulaval.ca)

Abstract

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.

(Received July 09 2007)

(Accepted September 10 2008)

(Online publication February 19 2009)

2000 Mathematics Subject Classification

  • 11S40 (primary);
  • 11M99 (secondary)

Keywords

  • p-adic Gross–Stark conjectures;
  • explicit class field theory;
  • p-adic integration;
  • Eisenstein series