Lang-Trotter revisited

Katz, Nicholas

doi:10.1090/S0273-0979-09-01257-9

Lang-Trotter revisited

Author: Nicholas M. Katz
Journal: Bull. Amer. Math. Soc. 46 (2009), 413-457
MSC (2000): Primary 11F80, 11G05, 14G35
DOI: https://doi.org/10.1090/S0273-0979-09-01257-9
Published electronically: March 27, 2009
MathSciNet review: 2507277
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Lang–Trotter Conjecture(s) concern elliptic curves over the field $\mathbb {Q}$ of rational numbers. We first explain the broader number-theoretic context into which they fit. Then we turn to formulating their “function field” analogues. We explain how these analogues can be proven in some very special cases, and we speculate about what might be true in the general function field case.

Stephan Baier, The Lang-Trotter conjecture on average, J. Ramanujan Math. Soc. 22 (2007), no. 4, 299–314. MR 2376806

[B-K]B-KBombieri, E. and Katz, N., A Note on Lower bounds for Frobenius traces, 2008 preprint available at www.math.princeton.edu/~nmk.

Laurent Clozel, Nombre de points des variétés de Shimura sur un corps fini (d’après R. Kottwitz), Astérisque 216 (1993), Exp. No. 766, 4, 121–149 (French, with French summary). Séminaire Bourbaki, Vol. 1992/93. MR 1246396
Alina Carmen Cojocaru and Igor E. Shparlinski, Distribution of Farey fractions in residue classes and Lang-Trotter conjectures on average, Proc. Amer. Math. Soc. 136 (2008), no. 6, 1977–1986. MR 2383504, DOI https://doi.org/10.1090/S0002-9939-08-09324-6
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
Chantal David and Francesco Pappalardi, Average Frobenius distributions of elliptic curves, Internat. Math. Res. Notices 4 (1999), 165–183. MR 1677267, DOI https://doi.org/10.1155/S1073792899000082
Pierre Deligne, Variétés abéliennes ordinaires sur un corps fini, Invent. Math. 8 (1969), 238–243 (French). MR 254059, DOI https://doi.org/10.1007/BF01406076
Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272 (German). MR 5125, DOI https://doi.org/10.1007/BF02940746
Max Deuring, Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt. 1953 (1953), 85–94 (German). MR 61133
Noam D. Elkies, Supersingular primes for elliptic curves over real number fields, Compositio Math. 72 (1989), no. 2, 165–172. MR 1030140
Noam D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over ${\bf Q}$, Invent. Math. 89 (1987), no. 3, 561–567. MR 903384, DOI https://doi.org/10.1007/BF01388985
Ernst-Ulrich Gekeler, Frobenius distributions of elliptic curves over finite prime fields, Int. Math. Res. Not. 37 (2003), 1999–2018. MR 1995144, DOI https://doi.org/10.1155/S1073792803211272

[H-SB-T]H-SB-THarris, Michael, Shepherd-Barron, Nicholas, and Taylor, Richard, A family of Calabi-Yau varieties and potential automorphy, preprint available at www.math. harvard.edu/~rtaylor/cy3.pdf.

E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z. 1 (1918), no. 4, 357–376 (German). MR 1544302, DOI https://doi.org/10.1007/BF01465095
Taira Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83–95. MR 229642, DOI https://doi.org/10.2969/jmsj/02010083
Everett W. Howe, Principally polarized ordinary abelian varieties over finite fields, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2361–2401. MR 1297531, DOI https://doi.org/10.1090/S0002-9947-1995-1297531-4
Jun-ichi Igusa, Class number of a definite quaternion with prime discriminant, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 312–314. MR 98728, DOI https://doi.org/10.1073/pnas.44.4.312
Kenneth F. Ireland and Michael I. Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York-Berlin, 1982. Revised edition of Elements of number theory. MR 661047
P. A. P. Moran, An introduction to probability theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1984. Corrected reprint of the 1968 original. MR 753828
Nicholas M. Katz, Exponential sums and differential equations, Annals of Mathematics Studies, vol. 124, Princeton University Press, Princeton, NJ, 1990. MR 1081536
Nicholas M. Katz, Twisted $L$-functions and monodromy, Annals of Mathematics Studies, vol. 150, Princeton University Press, Princeton, NJ, 2002. MR 1875130
Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569
Robert E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), no. 2, 373–444. MR 1124982, DOI https://doi.org/10.1090/S0894-0347-1992-1124982-1
Robert E. Kottwitz, Shimura varieties and $\lambda $-adic representations, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 161–209. MR 1044820
Serge Lang and Hale Trotter, Frobenius distributions in ${\rm GL}_{2}$-extensions, Lecture Notes in Mathematics, Vol. 504, Springer-Verlag, Berlin-New York, 1976. Distribution of Frobenius automorphisms in ${\rm GL}_{2}$-extensions of the rational numbers. MR 0568299
Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266. MR 444670, DOI https://doi.org/10.1007/BF01389815
William Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, Vol. 264, Springer-Verlag, Berlin-New York, 1972. MR 0347836
Amílcar Pacheco, Distribution of the traces of Frobenius on elliptic curves over function fields, Acta Arith. 106 (2003), no. 3, 255–263. MR 1957108, DOI https://doi.org/10.4064/aa106-3-4
René Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A 46 (1987), no. 2, 183–211. MR 914657, DOI https://doi.org/10.1016/0097-3165%2887%2990003-3
Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
Jean-Pierre Serre, Propriétés conjecturales des groupes de Galois motiviques et des représentations $l$-adiques, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 377–400 (French). MR 1265537
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1. MR 0314766

[Sie]SieSiegel, Carl Ludwig, Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1 (1935), 83-86.

[St]StStein, William, The Modular Forms Explorer, database available at http://modular. fas.harvard.edu.

[Tate]TateTate, J., Classes d’isogénie des variétés abéliennes sur un corps fini, Séminaire Bourbaki 1968-69, exp. 352, 95-110.

William C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. (4) 2 (1969), 521–560. MR 265369
Yuri G. Zarhin, Very simple 2-adic representations and hyperelliptic Jacobians, Mosc. Math. J. 2 (2002), no. 2, 403–431. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1944511, DOI https://doi.org/10.17323/1609-4514-2002-2-2-403-431
Hiroyuki Yoshida, On an analogue of the Sato conjecture, Invent. Math. 19 (1973), 261–277. MR 337977, DOI https://doi.org/10.1007/BF01425416