Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T14:55:25.278Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE MERTENS CONJECTURE FOR ELLIPTIC CURVES OVER FINITE FIELDS

Published online by Cambridge University Press:  28 February 2013

PETER HUMPHRIES
PETER HUMPHRIES*
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544, USA email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for which this conjecture holds in terms of the size of the finite field and the trace of the Frobenius endomorphism acting on the curve.

Keywords

Mertens conjectureelliptic curvefinite fieldfunction field
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 1 , February 2014 , pp. 19 - 32
Copyright
©2013 Australian Mathematical Publishing Association Inc. 

References

Bateman, P. T., Brown, J. W., Hall, R. S., Kloss, K. E. and Stemmler, R. M., ‘Linear relations connecting the imaginary parts of the zeros of the zeta function’, in: Computers in Number Theory, (eds Atkin, A. O. L. and Birch, B. J.) (Academic Press, London, 1971), pp. 11–19.Google Scholar
Best, D. G. and Trudgian, T. S., ‘Linear relations of zeroes of the zeta-function’. arXiv:math.NT/1209.3843.Google Scholar
Cha, B., ‘The summatory function of the Möbius function in function fields’. arXiv:math.NT/1008.4711v2.Google Scholar
Howe, E. W., Nart, E. and Ritzenthaler, C., ‘Jacobians in isogeny classes of abelian surfaces over finite fields’, Ann. Inst. Fourier 59 (2009), 239–289.Google Scholar
Humphries, P., ‘On the Mertens conjecture for function fields’. arXiv:math.NT/1210.0945.Google Scholar
Ingham, A. E., ‘On two conjectures in the theory of numbers’, Amer. J. Math. 64 (1942), 313–319.Google Scholar
Kotnik, T. and te Riele, H., ‘The Mertens conjecture revisited’, in: Algorithmic Number Theory, Lecture Notes in Comput. Sci., 4076 (eds Hess, F., Pauli, S. and Pohst, M.) (Springer, Berlin, 2006), pp. 156–167.Google Scholar
Mertens, F., ‘Über eine zahlentheoretische Funktion’, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, Abteilung 2a 106 (1897), 761–830.Google Scholar
Odlyzko, A. M. and te Riele, H. J. J., ‘Disproof of the Mertens conjecture’, J. reine angew. Math. 357 (1985), 138–160.Google Scholar
Rosen, M., Number Theory in Function Fields, Graduate Texts in Mathematics, 210 (Springer, New York, 2002).Google Scholar
Waterhouse, W. C., ‘Abelian varieties over finite fields’, Ann. Sc. Éc. Norm. Supér., Série 4 2 (1969), 521–560.Google Scholar