On the Distribution of Supersingular Primes

Fouvry, Etienne ; Murty, M. Ram

Canadian journal of mathematics, Tome 48 (1996) no. 1, pp. 81-104

Résumé

Let E be a fixed elliptic curve defined over the rational numbers. We prove that the number of primes p ≤ x such that E has supersingular reduction mod p is greater than for any positive δ and x sufficiently large. Here logkx is defined recursively as log(logk-1 x) and log1x = logx. We also establish several results related to the Lang-Trotter conjecture.

Détail

DOI : 10.4153/CJM-1996-004-7

Mots-clés : 11G05, 11N36

Bibliographie

[Bi] Birch, B., How the number of points of an elliptic curve over a fixed prime field varies, J. London Math. Soc. 43(1968), 57–60. Google Scholar

[Bo] Bombieri, E., Le grand crible en theorie analytique des nombres, Asterisque, Soc. Math, de France 18 (1974/1987). Google Scholar

[Brl] Brown, M.L., Note on supersingular primes of elliptic curves over Q, Bull. London Math. Soc. 20(1988), 293–296. Google Scholar

[Br2] Brown, M.L., Singular Moduli and Supersingular Moduli of Drinfeld Modules, Inven. Math. 110(1992), 419–439. Google Scholar

[Dave] Davenport, H., Multiplicative Number Theory (Second Edition), Graduate Texts in Math. 74, Springer Verlag, 1980. Google Scholar

[Davi] David, C., Supersingular Reduction of Drinfeld Modules, Ph.D. Thesis, McGill University Montreal, 1993. Google Scholar

[De] Deuring, M., Die Typen der Multiplikatorenringe elliptischer Funktionenkorper, Abh. Math. Sem. Univ. Hamburg 14(1941), 197–272. Google Scholar

[Do] Dorman, D., Special values of the elliptic modular function and factorisation formulae, J. Reine Angew. Math. 383(1988), 207–220. Google Scholar

[Ell] Elkies, N., The existence of infinitely many supersingular primes for every elliptic curve over Q, Invent. Math. 89(1987), 561–567. Google Scholar

[E12] Elkies, N., Supersingular primes of a given elliptic curve over a number field, Ph.D. Thesis, Harvard Univ., 1987. Google Scholar

[EI3] Elkies, N., Distribution of Supersingular Primes, Astérisque, J. Arithmetiques de Luminy 198. 198–200.(1991), 127–132. Google Scholar

[Ga] Gallagher, P.X., The large sieve and probalistic Galois theory, Proc. Symp. Pure Math. XXIV(1973), 91–101. Google Scholar

[G-Z] Gross, B.H. and Zagier, D., On singular moduli, J. Reine Angew. Math. 335(1985), 191–220. Google Scholar

[He] Heilbronn, H., Zeta functions and L-functions. Algebraic Number Theory, (eds. Cassels, J.W.S. and Fröhlich), Academic Press, 1967. 204–230. Google Scholar

[HI] Hlawka, M.E., Bemerkungen zum grossen Sieb von Linnik, Österreich Akad. Wiss. Math.-Natur. S.B. II 178(1970), 13–18. Google Scholar

[Ho] Hooley, C., Applications of Sieve Methods to the Theory of Numbers. Cambridge Tracts in Math. 70, 1976. Google Scholar

[Hu] Huxley, M., The large sieve inequality for algebraic number fields, Mathematika 15(1968), 178—187. Google Scholar

[Ju] Jutila, M., On the mean-value of L(1\2, χ), Analysis 1(1981), 149–161. Google Scholar

[Ka] Kaneko, M., Super singular j-invariants mod p, Osaka J. Math. 26(1989), 849–855. Google Scholar

[L-T] Lang, S. and Trotter, H., Frobenius in GL2 extensions. Lecture Notes in Math. 504, Springer Verlag, 1976. Google Scholar

[MuJ Murty, R., Recent developments in the theory of elliptic curves, Proc. of the Ramanujan Centennial International Conf., 1987. 45–54. Google Scholar

[MMS] Murty, R., Murty, K. and Saradha, N., Modular forms and the Chebotarev density theorem, Amer. J. Math. 110(1988), 253–281. Google Scholar

[Sc] Schmidt, W.M., Equations over finite fields. An elementary approach. Lecture Notes in Math. 536, Springer Verlag, 1976. Google Scholar

[St] Stark, H.M., Some effective Cases of the Brauer-Siegel Theorem, Invent. Math. 23(1974), 135–152. Google Scholar

[Se] Serre, J-P., Quelques applications du theoreme de densite de Chebotarev, Inst. Hautes Etudes Sci. Publ. Math. 54(1982), 123–201. Google Scholar