Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T07:22:51.870Z Has data issue: false hasContentIssue false
  • English
  • Français

p-adic class invariants

Part of: Arithmetic algebraic geometry Computational number theory

Published online by Cambridge University Press:  01 April 2011

Reinier Bröker
Reinier Bröker*
Affiliation:
Brown University, Box 1917, 151 Thayer Street Providence, RI, USA (email: reinier@math.brown.edu)

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 develop a new p-adic algorithm to compute the minimal polynomial of a class invariant. Our approach works for virtually any modular function yielding class invariants. The main algorithmic tool is modular polynomials, a concept which we generalize to functions of higher level.

MSC classification

Secondary: 11G15: Complex multiplication and moduli of abelian varieties 11Y40: Algebraic number theory computations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 108 - 126
Copyright
Copyright © London Mathematical Society 2011

References

[1]Agashe, A., Lauter, K. and Venkatesan, R., ‘Constructing elliptic curves with a known number of points over a prime field’, High primes and misdemeanours: lectures in honour of the 60th birthday of H. C. Williams, Fields Institute Communications Series 41 (American Mathematical Society, Providence, RI, 2004) 1–17.CrossRefGoogle Scholar
[2]Belding, J., Bröker, R., Enge, A. and Lauter, K., ‘Computing Hilbert class polynomials’, Algorithmic number theory symposium VIII, Lecture Notes in Computer Science 5011 (Springer, Berlin, 2008).Google Scholar
[3]Bröker, R., ‘Constructing elliptic curves of prescribed order’, PhD Thesis, Universiteit Leiden, 2006.Google Scholar
[4]Bröker, R, ‘A p-adic algorithm to compute the Hilbert class polynomial’, Math. Comp. 77 (2008) 2417–2435.CrossRefGoogle Scholar
[5]Bröker, R. and Stevenhagen, P., ‘Constructing elliptic curves of prime order’, Contemp. Math. 463 (2008) 17–28.CrossRefGoogle Scholar
[6]Cohen, H., Frey, G.et al., Handbook of elliptic and hyperelliptic curve cryptography (Chapman & Hall, 2006).Google Scholar
[7]Couveignes, J.-M. and Henocq, T., ‘Action of modular correspondences around CM-points’, Algorithmic number theory symposium V, Lecture Notes in Computer Science 2369 (Springer, Berlin, 2002) 234–243.CrossRefGoogle Scholar
[8]Enge, A., ‘The complexity of class polynomial computation via floating point approximations’, Math. Comp. 78 (2009) 1089–1107.CrossRefGoogle Scholar
[9]Enge, A. and Morain, F., ‘Comparing invariants for class fields of imaginary quadratic fields’, Algorithmic number theory symposium V, Lecture Notes in Computer Science 2369 (Springer, Berlin, 2002) 252–266.CrossRefGoogle Scholar
[10]Enge, A. and Schertz, R., ‘Constructing elliptic curves over finite fields using double eta-quotients’, J. Théor. Nombres Bordeaux 16 (2004) 555–568.CrossRefGoogle Scholar
[11]Enge, A. and Schertz, R., ‘Modular curves of composite level’, Acta Arith. 118 (2005) 129–141.CrossRefGoogle Scholar
[12]Franke, J., Kleinjung, T., Morain, F. and Wirth, T., ‘Proving the primality of very large numbers with fast ECPP’, Algorithmic number theory symposium VI, Lecture Notes in Computer Science 3076 (Springer, Berlin, 2004) 194–207.CrossRefGoogle Scholar
[13]Gee, A. and Stevenhagen, P., ‘Generating class fields using Shimura reciprocity’, Algorithmic number theory, Lecture Notes in Computer Science 1423 (Springer, Berlin, 1998) 441–453.CrossRefGoogle Scholar
[14]Hindry, M. and Silverman, J. H., Diophantine geometry, an introduction, Graduate Texts in Mathematics 201 (Springer, Berlin, 2000).CrossRefGoogle Scholar
[15]de Jong, A. J., ‘Families of curves and alterations’, Ann. Inst. Fourier (Grenoble) 47 (1997) 599–621.CrossRefGoogle Scholar
[16]Kohel, D., ‘Endomorphism rings of elliptic curves over finite fields’, PhD Thesis, University of California, Berkeley, 1996.Google Scholar
[17]Lang, S., Elliptic functions, Graduate Texts in Mathematics 112 (Springer, Berlin, 1987).CrossRefGoogle Scholar
[18]Neukirch, J., Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322 (Springer, Berlin, 1999).CrossRefGoogle Scholar
[19]Schoof, R., ‘Counting points on elliptic curves over finite fields’, J. Théor. Nombres Bordeaux 7 (1993) 219–254.CrossRefGoogle Scholar
[20]Shimura, G., Introduction to the arithmetic theory of automorphic forms (Princeton University Press, Princeton, NJ, 1971).Google Scholar
[21]Stevenhagen, P., ‘Hilbert’s 12th problem, complex multiplication and Shimura reciprocity’, Class field theory—its centenary and prospect, Advanced Studies in Pure Mathematics 30 (ed. K. Miyake; Mathematical Society of Japan, Tokyo, 2001) 161–176.Google Scholar
[22]Sutherland, A. V., ‘Computing Hilbert class polynomials with the Chinese remainder theorem’ Math. Comp., to appear, available at http://arxiv.org/abs/0903.2785.Google Scholar
[23]Vélu, J., ‘Isogénies entre courbes elliptiques’, C. R. Acad. Sci. Paris Sér. A–B 273 (1971) A238–A241.Google Scholar
[24]Weber, H., Lehrbuch der Algebra (Friedrich Vieweg und Sohn, 1908).Google Scholar