Minimal polynomials of singular moduli

Errthum, Eric

doi:10.1090/S0025-5718-2013-02709-3

Minimal polynomials of singular moduli
HTML articles powered by AMS MathViewer

by Eric Errthum PDF

Math. Comp. 83 (2014), 411-420 Request permission

Abstract:

Given a properly normalized parametrization of a genus-0 modular curve, the complex multiplication points map to algebraic numbers called singular moduli. In both cases there are known algorithms for algebraically computing the rational norms of the singular moduli without relying on the the recognition of a decimal or $p$-adic expansion as a rational number. We demonstrate a method of extending these norm algorithms to determine the minimal polynomial of the singular moduli below a discriminant threshold. We then use these minimal polynomials to compute the algebraic $abc$-ratios for the singular moduli.

References

Harald Baier, Efficient computation of singular moduli with application in cryptography, Fundamentals of computation theory (Riga, 2001) Lecture Notes in Comput. Sci., vol. 2138, Springer, Berlin, 2001, pp. 71–82. MR 1914098, DOI 10.1007/3-540-44669-9_{9}
Niklas Broberg, Some examples related to the $abc$-conjecture for algebraic number fields, Math. Comp. 69 (2000), no. 232, 1707–1710. MR 1659863, DOI 10.1090/S0025-5718-99-01153-9
David A. Cox, Primes of the form $x^2 + ny^2$, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR 1028322
Tim Dokchitser, LLL & ABC, J. Number Theory 107 (2004), no. 1, 161–167. MR 2059955, DOI 10.1016/j.jnt.2004.02.003
Noam D. Elkies, Shimura curve computations, Algorithmic number theory (Portland, OR, 1998) Lecture Notes in Comput. Sci., vol. 1423, Springer, Berlin, 1998, pp. 1–47. MR 1726059, DOI 10.1007/BFb0054850
Eric Errthum, Singular moduli of Shimura curves, Canad. J. Math. 63 (2011), no. 4, 826–861. MR 2848999, DOI 10.4153/CJM-2011-023-7
Gillien Geuze and Bart de Smit, Count along with ABC, Nieuw Arch. Wiskd. (5) 8 (2007), no. 1, 26–30 (Dutch, with Dutch summary). MR 2340133
Andrew Granville and H. M. Stark, $abc$ implies no “Siegel zeros” for $L$-functions of characters with negative discriminant, Invent. Math. 139 (2000), no. 3, 509–523. MR 1738058, DOI 10.1007/s002229900036
Matthew Greenberg, Computing Heegner points arising from Shimura curve parametrizations, Arithmetic geometry, Clay Math. Proc., vol. 8, Amer. Math. Soc., Providence, RI, 2009, pp. 115–124. MR 2498058
Benedict H. Gross and Don B. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220. MR 772491
Svetlana Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. MR 1177168
Magma Calculator, http://magma.maths.usyd.edu.au/calc/, 2010
D. W. Masser, Note on a conjecture of Szpiro, Astérisque 183 (1990), 19–23. Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). MR 1065152
Nitaj, A. The ABC Conjecture Homepage, http://www.math.unicaen.fr/ nitaj/abc.html, May 27, 2010.
Joseph Oesterlé, Nouvelles approches du “théorème” de Fermat, Astérisque 161-162 (1988), Exp. No. 694, 4, 165–186 (1989) (French). Séminaire Bourbaki, Vol. 1987/88. MR 992208
Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
David Peter Roberts, Shimura curves analogous to X(0)(N), ProQuest LLC, Ann Arbor, MI, 1989. Thesis (Ph.D.)–Harvard University. MR 2637583
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
Jarad Schofer, Borcherds forms and generalizations of singular moduli, J. Reine Angew. Math. 629 (2009), 1–36. MR 2527412, DOI 10.1515/CRELLE.2009.025
Jarad John Schofer, Borcherds forms and generalizations of singular moduli, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University of Maryland, College Park. MR 2707427
Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989