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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
DOI: https://doi.org/10.1090/S0025-5718-09-02215-7
Published electronically: January 29, 2009
MathSciNet review: 2521291
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Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [Cha] H. Chapdelaine.
    Elliptic units in ray class fields of real quadratic number fields, version with a few corrections and supplements.
    available at http://www.mat.ulaval.ca/fileadmin/Pages_personnelles_des_profs/hchapd/thesis_final.pdf.
  • [Cha07a] H. Chapdelaine.
    $ p$-units in ray class fields of real quadratic number fields.
    accepted for publication in Compositio, 1:1-34, 2007.
  • [Cha07b] H. Chapdelaine.
    Zeta functions twisted by additive characters, $ p$-units and Gauss sums.
    International J. Number Theory, 1:1-40, 2007.
  • [Dar04] Henri Darmon, Rational points on modular elliptic curves, CBMS Regional Conference Series in Mathematics, vol. 101, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020572
  • [Das07a] Samit Dasgupta, Computations of elliptic units for real quadratic fields, Canad. J. Math. 59 (2007), no. 3, 553–574. MR 2319158, https://doi.org/10.4153/CJM-2007-023-0
  • [Das07b] Samit Dasgupta, Shintani zeta functions and Gross-Stark units for totally real fields, Duke Math. J. 143 (2008), no. 2, 225–279. MR 2420508, https://doi.org/10.1215/00127094-2008-019
  • [DD06] Henri Darmon and Samit Dasgupta, Elliptic units for real quadratic fields, Ann. of Math. (2) 163 (2006), no. 1, 301–346. MR 2195136, https://doi.org/10.4007/annals.2006.163.301
  • [Gro81] Benedict H. Gross, 𝑝-adic 𝐿-series at 𝑠=0, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 979–994 (1982). MR 656068
  • [Gro88] Benedict H. Gross, On the values of abelian 𝐿-functions at 𝑠=0, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 (1988), no. 1, 177–197. MR 931448
  • [Tat84] John Tate, Les conjectures de Stark sur les fonctions 𝐿 d’Artin en 𝑠=0, Progress in Mathematics, vol. 47, Birkhäuser Boston, Inc., Boston, MA, 1984 (French). Lecture notes edited by Dominique Bernardi and Norbert Schappacher. MR 782485

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Additional Information

Hugo Chapdelaine
Affiliation: Département de Mathématiques et de Statistique, Université Laval, Québec, Canada G1K 7P4
Email: hugo.chapdelaine@mat.ulaval.ca

DOI: https://doi.org/10.1090/S0025-5718-09-02215-7
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.