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

Author: Hugo Chapdelaine
Journal: Math. Comp. 78 (2009), 2307-2345
MSC (2000): Primary 11S31
Published electronically: January 29, 2009
MathSciNet review: 2521291
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Abstract: Let $ K$ be a real quadratic field, let $ p$ be a prime number which is inert in $ K$ and let $ K_p$ be the completion of $ K$ at $ p$. As part of a Ph.D. thesis, we constructed a certain $ p$-adic invariant $ u\in K_p^{\times}$, and conjectured that $ u$ is, in fact, a $ p$-unit in a suitable narrow ray class field of $ K$. In this paper we give numerical evidence in support of that conjecture. Our method of computation is similar to the one developed by Dasgupta and relies on partial modular symbols attached to Eisenstein series.

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Additional Information

Hugo Chapdelaine
Affiliation: Département de Mathématiques et de Statistique, Université Laval, Québec, Canada G1K 7P4

Keywords: $p$-adic Gross-Stark conjectures, explicit Class field theory, $p$-adic integration, Eisenstein series
Received by editor(s): November 14, 2007
Received by editor(s) in revised form: August 27, 2008
Published electronically: January 29, 2009
Additional Notes: The author is grateful to the Max Planck Institut für Mathematik for the financial support during the writing of the paper.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.