CM liftings of supersingular elliptic curves
Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 635-663.

Sous GRH, nous présentons un algorithme qui, étant donné un nombre premier p, calcule l’ensemble des discriminants fondamentaux D<0, tels que l’application de réduction, modulo un premier aux dessus de p, des courbes elliptiques avec multiplication complexe par 𝒪 D vers les courbes elliptiques supersingulières en caractéristique p est surjective. Dans l’algorithme, nous déterminons d’abord une borne D p explicite telle que |D|>D p implique que l’application est nécessairement surjective et nous calculons ensuite explicitement les cas |D|<D p .

Assuming GRH, we present an algorithm which inputs a prime p and outputs the set of fundamental discriminants D<0 such that the reduction map modulo a prime above p from elliptic curves with CM by 𝒪 D to supersingular elliptic curves in characteristic p is surjective. In the algorithm we first determine an explicit constant D p so that |D|>D p implies that the map is necessarily surjective and then we compute explicitly the cases |D|<D p .

DOI : 10.5802/jtnb.692
Classification : 11G05, 11E20, 11E45, 11Y35, 11Y70
Mots clés : Quaternion Algebra, Elliptic Curves, Maximal Orders, Half Integer Weight Modular Forms, Kohnen’s Plus Space, Shimura Lifts
Kane, Ben 1

1 Department of Mathematics Radboud Universiteit Nijmegen Heijendaalseweg 135, 6525 AJ Nijmegen, Netherlands
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Kane, Ben. CM liftings of supersingular elliptic curves. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 635-663. doi : 10.5802/jtnb.692. http://www.numdam.org/articles/10.5802/jtnb.692/

[1] J. Cremona, Algorithms for elliptic curves. Cambridge Univ. Press, 1992. | MR | Zbl

[2] P. Deligne, La conjecture de weil i. Inst. Hautes Études Sci. Publ. Math 43 (1974), 273–307. | Numdam | MR | Zbl

[3] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörpen. Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272. | MR

[4] W. Duke, Hyperbolic distribution problems and half-integral weight maass forms. Invent. Math. 92 (1998), 73–90. | MR | Zbl

[5] W. Duke and R. Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent. Math. 99 (1990), no. 1, 49–57. | MR | Zbl

[6] A. Earnest, J. Hsia, and D. Hung, Primitive representations by spinor genera of ternary quadratic forms. J. London Math. Soc. (2) 50 (1994), no. 2, 222–230. | MR | Zbl

[7] N. Elkies, Supersingular primes for elliptic curves over real number fields. Compositio Mathematica 72 (1989), 165–172. | Numdam | MR | Zbl

[8] N. Elkies, K. Ono, and T. Yang, Reduction of CM elliptic curves and modular function congruences. Int. Math. Res. Not. 44 (2005), 2695–2707. | MR | Zbl

[9] U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Math. Comp. (1985), 463–471. | MR | Zbl

[10] W. Gautschi, A computational procedure for incomplete Gamma functions. ACM Transactions on Mathematical Software 5 (1979), 466–481. | MR | Zbl

[11] B. Gross, Heights and the special values of L-series. Number theory (Montreal, Que., 1985), CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115–187. | MR | Zbl

[12] B. Gross and D. Zagier, On singular moduli. J. Reine Angew. Math. 335 (1985), 191–220. | MR | Zbl

[13] T. Ibukiyama, On maximal order of division quaternion algebras over the rational number field with certain optimal embeddings. Nagoya Math J. 88 (1982), 181–195. | MR | Zbl

[14] H. Iwaniec, Fourier coefficients of modular forms of half-integral weight. Invent. Math. 87 (1987), 385–401. | MR | Zbl

[15] B. Jones, The arithmetic theory of quadratic forms. Carcus Monograph Series, no. 10, The Mathematical Association of America, Buffalo, Buffalo, NY, 1950. | MR | Zbl

[16] B. Kane, Representations of integers by ternary quadratic forms. preprint (2007).

[17] D. Kohel, Endomorphism rings of elliptic curves over finite fields. University of California, Berkeley, Ph.D. Thesis (1996), pp. 1–96.

[18] W. Kohnen, Newforms of half integral weight. J. reine angew. Math. 333 (1982), 32–72. | MR | Zbl

[19] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip. Invent. Math. 64 (1981), 175–198. | MR | Zbl

[20] J. Oesterlé, Nombres de classes des corps quadratiques imaginaires. Astérique 121-122 (1985), 309–323. | Numdam | MR | Zbl

[21] O. T. O’Meara, Introduction to quadratic forms. Classics in Mathematics, Springer-Verlag, Berlin, 2000, Reprint of the 1973 edition. | MR | Zbl

[22] K. Ono, Web of modularity: Arithmetic of the coefficients of modular forms and q-series. CBMS Regional Conference Series in Mathematics, no. 102, Amer. Math. Soc., Providence, RI, 2003. | MR | Zbl

[23] K. Ono and K. Soundarajan, Ramanujan’s ternary quadratic form. Invent. Math. 130 (1997), 415–454. | MR | Zbl

[24] A. Pizer, An algorithm for compute modular forms on Γ 0 (N). J. Algebra 64 (1980), no. 2, 340–390. | MR | Zbl

[25] R. Schulze-Pillot, Darstellungsmaße von Spinorgeschlechtern ternärer quadratischer Formen. J. Reine Angew. Math., 352 (1984), 114–132. | MR | Zbl

[26] G. Shimura, On modular forms of half integer weight. Ann. of Math. 97 (1973), 440–481. | MR | Zbl

[27] C. Siegel, Über die klassenzahl quadratischer zahlkorper. Acta Arith. 1 (1935), 83–86.

[28] J. Silverman, The arithmetic of elliptic curves. Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original. | MR | Zbl

[29] W. Stein, Explicit approaches to modular abelian varieties. Ph.D. thesis, University of California, Berkeley (2000), pp. 1–96.

[30] , Modular forms, a computational approach. Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, RI, 2007, Appendix by P. Gunnells. | MR | Zbl

[31] J. Sturm, On the congruence of modular forms. Number theory (New York, 1984–1985), Springer, Berlin, 1987, pp. 275–280. | MR | Zbl

[32] M.-F. Vignéras, Arithmétique des algèbres de quaternions. Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980. | MR | Zbl

[33] M. Watkins, Class numbers of imaginary quadratic fields. Math. Comp. 73 (2004), 907–938. | MR | Zbl

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