Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Borel's conjecture and the transcendence of the Iwasawa power series


Author: Hae-Sang Sun
Journal: Proc. Amer. Math. Soc. 138 (2010), 1955-1963
MSC (2010): Primary 11K16, 11R23, 11R42
DOI: https://doi.org/10.1090/S0002-9939-10-10287-1
Published electronically: February 4, 2010
MathSciNet review: 2596029
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We deduce the transcendence of the Iwasawa power series from Borel's conjecture, namely, the normality of the irrational algebraic $ p$-adic integers.


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

  • [AS] J.-P. Allouche and J. Shallit, Automatic sequences, Cambridge Univ. Press, Cambridge, 2003. MR 1997038 (2004k:11028)
  • [A] B. Anglès, On some $ p$-adic power series attached to the arithmetic of $ \mathbb{Q}(\zeta_p)$, J. Number Theory 122 (2007), no. 1, 221-246. MR 2287121 (2008g:11182)
  • [A2] B. Anglès, On the $ p$-adic Leopoldt transform of a power series, Acta Arith. 134 (2008), no. 4, 349-367. MR 2449158
  • [B] É. Borel, Sur les chiffres décimaux de $ \sqrt{2}$ et divers problèmes de probabilités en chaîne. C. R. Acad. Sci. Paris 230 (1950), 591-593. MR 0034544 (11:605d)
  • [FW] B. Ferrero and L. C. Washington, The Iwasawa invariant $ \mu_{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377-395. MR 528968 (81a:12005)
  • [H] H. Hida, Elementary Theory of $ L$-Functions and Eisenstein Series, London Math. Soc. Student Texts, 26, Cambridge Univ. Press, Cambridge, UK, 1993. MR 1216135 (94j:11044)
  • [Iw] K. Iwasawa, Lectures on $ p$-adic $ L$-functions, Ann. of Math. Studies, 74. Princeton Univ. Press, 1972. MR 0360526 (50:12974)
  • [KN] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Pure and Applied Mathematics, John Wiley $ \&$ Sons, New York-London-Sydney, 1974. MR 0419394 (54:7415)
  • [Si] W. Sinnott, On the $ \mu$-invariant of the $ \Gamma$-transform of a rational function, Invent. Math. 75 (1984), 273-282. MR 732547 (85g:11112)
  • [Si2] W. Sinnott, On the power series attached to $ p$-adic $ L$-functions, J. Reine Angew. Math. 382 (1987), 22-34. MR 921164 (88m:11102)
  • [Su] H.-S. Sun, Derivative of power series attached to $ \Gamma$-transform of $ p$-adic measures, J. Number Theory (2009), in press, doi:10.1016/j.jnt.2009.07.001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11K16, 11R23, 11R42

Retrieve articles in all journals with MSC (2010): 11K16, 11R23, 11R42


Additional Information

Hae-Sang Sun
Affiliation: Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul 130-722, Republic of Korea
Email: haesang@kias.re.kr

DOI: https://doi.org/10.1090/S0002-9939-10-10287-1
Received by editor(s): August 17, 2009
Received by editor(s) in revised form: September 15, 2009, and October 5, 2009
Published electronically: February 4, 2010
Additional Notes: The manuscript was prepared while the author was visiting Department of the Mathematics, Université de Caen. He thanks the department for their support and hospitality. He also thanks the referee for the valuable suggestions and corrections.
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society