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Lang-Trotter revisited

Author(s): Nicholas M. Katz
Journal: Bull. Amer. Math. Soc. 46 (2009), 413-457.
MSC (2000): Primary 11F80, 11G05, 14G35
Posted: March 27, 2009
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Abstract: The Lang-Trotter Conjecture(s) concern elliptic curves over the field $ \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.

References:

[Ba]
Baier, Stephan, The Lang-Trotter conjecture on average. J. Ramanujan Math. Soc. 22 (2007), no. 4, 299-314. MR 2376806 (2008j:11065)

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

[Cl]
Clozel, Laurent, Nombre de points des variétés de Shimura sur un corps fini (d'après R. Kottwitz). Séminaire Bourbaki, Vol. 1992/93. Astérisque No. 216 (1993), Exp. No. 766, 4, 121-149. MR 1246396 (95c:11075)

[Co-Shp]
Cojocaru, Alina Carmen and Shparlinski, Igor E., Distribution of Farey fractions in residue classes and Lang-Trotter conjectures on average, Proc. Amer. Math. Soc. 136, Number 6 (2008), 1977-1986. MR 2383504 (2009a:11035)

[Cox]
Cox, David, Primes of the Form $ x^2 +ny^2$, John Wiley and Sons, New York, 1989. MR 1028322 (90m:11016)

[Da-Pa]
David, Chantal and Pappalardi, Francesco, Average Frobenius distributions of elliptic curves, Internat. Math. Res. Notices 1999 (1999), no. 4, 165-183. MR 1677267 (2000g:11045)

[De-VA]
Deligne, Pierre, Variétés abéliennes ordinaires sur un corps fini. (French) Invent. Math. 8 (1969) 238-243. MR 0254059 (40:7270)

[De-WeilII]
Deligne, Pierre, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. No. 52 (1980), 137-252. MR 601520 (83c:14017)

[Deu]
Deuring, M., Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272. MR 0005125 (3:104f)

[Deu-CM]
Deuring, M., Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt. (1953), 85-94. MR 0061133 (15:779d)

[Elkies-Real]
Elkies, Noam D. Supersingular primes for elliptic curves over real number fields. Compositio Math. 72 (1989), no. 2, 165-172. MR 1030140 (90i:11058)

[Elkies-SS]
Elkies, Noam D. The existence of infinitely many supersingular primes for every elliptic curve over $ Q$. Invent. Math. 89 (1987), no. 3, 561-567. MR 903384 (88i:11034)

[Ge]
Gekeler, Ernst-Ulrich, Frobenius distributions of elliptic curves over finite prime fields, IMRN 2003, no. 37, 1999-2018. MR 1995144 (2004d:11048)

[H-SB-T]
Harris, 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.

[He]
Hecke, Erich, Über eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilungen der Primzahlen. Math. Z. 1 (1918), 357-376;4 (1920), 11-21. Reprinted in Mathematishe Werke, Vandenhoeck and Ruprecht, Göttingen, 1983. MR 1544302

[Honda]
Honda, Taira, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83-95. MR 0229642 (37:5216)

[Howe]
Howe, Everett W., Principally polarized ordinary abelian varieties over finite fields. Trans. Amer. Math. Soc. 347 (1995), no. 7, 2361-2401. MR 1297531 (96i:11065)

[Ig]
Igusa, Jun-ichi, Class number of a definite quaternion with prime discriminant. Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 312-314. MR 0098728 (20:5183)

[Ir-Ros]
Ireland, Kenneth F. and Rosen, Michael I., A Classical Introduction to Modern Number Theory. Revised edition of Elements of Number Theory. Graduate Texts in Mathematics, 84. Springer-Verlag, New York-Berlin, 1982. MR 661047 (83g:12001)

[Ito]
Ito, Kyosi, An Introduction to Probability Theory, The Clarendon Press, Oxford University Press, New York, 1984. MR 753828 (85f:60001)

[Ka-ESDE]
Katz, Nicholas M., Exponential sums and differential equations, Annals of Mathematics Studies, 124. Princeton University Press, Princeton, NJ, 1990. MR 1081536 (93a:14009)

[Ka-TLFM]
Katz, Nicholas M., Twisted $ L$-Functions and Monodromy, Annals of Mathematics Studies, 150. Princeton University Press, Princeton, NJ, 2002. MR 1875130 (2003i:11087)

[K-M]
Katz, Nicholas M. and Mazur, Barry, Arithmetic moduli of elliptic curves. Annals of Mathematics Studies, 108. Princeton University Press, Princeton, NJ, 1985. MR 772569 (86i:11024)

[Ko1]
Kottwitz, Robert E., Points on some Shimura varieties over finite fields. J. Amer. Math. Soc. 5 (1992), no. 2, 373-444. MR 1124982 (93a:11053)

[Ko2]
Kottwitz, Robert E., Shimura varieties and $ \lambda$-adic representations. Automorphic forms, Shimura varieties, and $ L$-functions, Vol. I (Ann Arbor, MI, 1988), 161-209, Perspect. Math., 10, Academic Press, Boston, MA, 1990. MR 1044820 (92b:11038)

[L-T]
Lang, Serge and Trotter, Hale, Frobenius distributions in $ \mathrm{GL}_2$-extensions, Springer Lecture Notes in Mathematics 504, 1976. MR 0568299 (58:27900)

[Maz]
Mazur, Barry, Rational points of abelian varieties with values in towers of number fields. Invent. Math. 18 (1972), 183-266. MR 0444670 (56:3020)

[Mes]
Messing, William, 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 (50:337)

[Pa]
Pacheco, Amícar, Distribution of the traces of Frobenius on elliptic curves over function fields. Acta Arith. 106 (2003), no. 3, 255-263. MR 1957108 (2004a:11046)

[Sch]
Schoof, René, Nonsingular plane cubic curves over finite fields. J. Combin. Theory Ser. A 46 (1987), no. 2, 183-211. MR 914657 (88k:14013)

[Se-Cheb]
Serre, Jean-Pierre, Quelques applications du théorème de densité de Chebotarev, Publ. Math. IHES 54 (1981), 323-401.MR 0644559 (83k:12011)

[Se-Mot]
Serre, Jean-Pierre, Propriétés conjecturales des groupes de Galois motiviques et des représentations $ \ell$-adiques, Proceedings of Symposia in Pure Mathematics 55 (1994) Part I, 377-400. MR 1265537 (95m:11059)

[Sh]
Shimura, Goro, Introduction to the arithmetic theory of automorphic functions. Kanô Memorial Lectures, No. 1. Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. MR 0314766 (47:3318)

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

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

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

[Wat]
Waterhouse, William C., Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 (1969) 521-560. MR 0265369 (42:279)

[Z]
Zarhin, Yuri G., Very simple 2-adic representations and hyperelliptic Jacobians, Mosc. Math. J. 2 (2002), no. 2, 403-431. MR 1944511 (2003k:11098)

[Yo]
Yoshida, Hiroyuki, On an analogue of the Sato conjecture. Invent. Math. 19 (1973), 261-277. MR 0337977 (49:2746)

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Additional Information:

Nicholas M. Katz
Affiliation: Department of Mathematics, Fine Hall, Princeton University, Princeton, New Jersey 08544-1000
Email: nmk@math.princeton.edu

DOI: 10.1090/S0273-0979-09-01257-9
PII: S 0273-0979(09)01257-9
Received by editor(s): December 21, 2008,
Received by editor(s) in revised form: February 23, 2009
Posted: March 27, 2009
Dedicated: Dedicated to the memory of Serge Lang
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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