Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Sato-Tate conjecture on average for small angles


Authors: Stephan Baier and Liangyi Zhao
Journal: Trans. Amer. Math. Soc. 361 (2009), 1811-1832
MSC (2000): Primary 11G05
DOI: https://doi.org/10.1090/S0002-9947-08-04498-X
Published electronically: October 31, 2008
MathSciNet review: 2465818
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain average results on the Sato-Tate conjecture for elliptic curves for small angles.


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

  • [1] S. Baier, The Lang-Trotter conjecture on average, J. Ramanujan Math. Soc. 22 (2007), 299-314. MR 2376806 (2008j:11065)
  • [2] W. D. Banks, I. E. Shparlinski, Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height, Israel J. Math. (to appear), preprint available at ArXiv:math.NT/0609144.
  • [3] L. Clozel, M. Harris, and R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations, preprint, available at www.math.harvard.edu/$ \sim$rtaylor.
  • [4] C. David, F. Pappalardi, Average Frobenius Distributions of Elliptic Curves, Int. Math. Res. Not. (1999) 165-183. MR 1677267 (2000g:11045)
  • [5] H. Davenport, Multiplicative Number Theory, Third Edition, Graduate Texts in Mathematics, Springer-Verlag, Barcelona, 2000. MR 1790423 (2001f:11001)
  • [6] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941) 197-272. MR 0005125 (3:104f)
  • [7] J. Friedlander, H. Iwaniec, The divisor problem for arithmetic progressions, Acta Arith. 45 (1985) 273-277. MR 808026 (87b:11087)
  • [8] E. Fouvry, M.R. Murty, On the distribution of supersingular primes, Canad. J. Math. 48 (1996) 81-104. MR 1382477 (97a:11084)
  • [9] M. Harris, N. Shepherd-Barron and R. Taylor, Ihara's lemma and potential automorphy, preprint, available at www.math.harvard.edu/$ \sim$rtaylor.
  • [10] M. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, I, Math. Z. 1 (1918) 357-376. MR 1544302
  • [11] M. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, II, Math. Z. 6 (1920) 11-51. MR 1544392
  • [12] A. Ivic, The Riemann Zeta-Function, Wiley-Interscience, New York, 1985. MR 792089 (87d:11062)
  • [13] H. Iwaniec, E. Kowalski, Analytic Number Theory, American Mathematical Society, Colloquium Publications, Volume 53, American Mathematical Society, Providence, 2004. MR 2061214 (2005h:11005)
  • [14] K. James, G. Yu, Average Frobenius Distribution of Elliptic Curves, Acta Arith. 124 (2006), 79-100. MR 2262142
  • [15] S. Lang, H. Trotter, Frobenius Distributions in GL$ _2$ extensions, Lecture Notes in Math. 504 (1976) Springer-Verlag, Berlin. MR 0568299 (58:27900)
  • [16] R.P. Langlands, Problems in the theory of automorphic forms, Lectures Modern Analysis Appl. 3, Lect. Notes Math. 170 (1970) 18-61. MR 0302614 (46:1758)
  • [17] V. Kumar Murty, On the Sato-Tate conjecture, Number theory related to Fermat's last theorem, Proc. Conf., Prog. Math. 26, (1982) 195-205. MR 685296 (84e:14021)
  • [18] M. Ram Murty, Recent developments in the Langlands program, C. R. Math. Acad. Sci., Soc. R. Can. 24 (2002) 33-54. MR 1902021 (2003j:11055)
  • [19] F. Shahidi, Symmetric power $ L$-functions for $ GL(2)$, Elliptic Curves and Related Topics, CRM Proc. Lecture Notes 4, Amer. Math. Soc. (1994) 159-182. MR 1260961 (95c:11066)
  • [20] J.T. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geom., Harper and Row, New York, 1965. MR 0225778 (37:1371)
  • [21] R. Taylor, Automorphy for some $ l$-adic lifts of automorphic mod $ l$ representations II, preprint, available at www.math.harvard.edu/$ \sim$rtaylor.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11G05

Retrieve articles in all journals with MSC (2000): 11G05


Additional Information

Stephan Baier
Affiliation: Department of Mathematics and Statistics, Queen’s University, University Ave., Kingston, Ontario, Canada K7L 3N6
Address at time of publication: School of Engineering and Sciences, Jacobs University, P.O. Box 750 561, Bremen 28725 Germany
Email: sbaier@mast.queensu.ca

Liangyi Zhao
Affiliation: Department of Mathematics, University of Toronto, 40 Saint George Street, Toronto, Ontario, Canada M5S 2E4
Address at time of publication: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371
Email: lzhao@math.toronto.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04498-X
Keywords: Sato-Tate conjecture, average Frobenius distribution
Received by editor(s): August 15, 2006
Received by editor(s) in revised form: February 12, 2007
Published electronically: October 31, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society