Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Zeroes of zeta functions and symmetry

Authors: Nicholas M. Katz and Peter Sarnak
Journal: Bull. Amer. Math. Soc. 36 (1999), 1-26
MSC (1991): Primary 11G, 11M, 11R, 11Y; Secondary 60B, 81Q
MathSciNet review: 1640151
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the ``function field'' analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the low-lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and $L$-functions.

References [Enhancements On Off] (What's this?)

  • [A-Z] A. Altland, M. Zirnbauer, ``Non-Standard Symmetry Classes in Mesoscopic Normal-Super-Conducting Hybrid Structures,'' Cond-Mat/9602137.
  • [AR] E. Artin, ``Quadratische Körper in Geibiet der Höheren Kongruzzen I and II,'' Math. Zeit., 19, 153-296, (1924).
  • [BO] E. Bombieri, ``Hilbert's 8-th Problem an Analogue,'' Proc. Sym. Pure Math. AMS XXVIII, 269-275, (1976). MR 55:2913
  • [BR] A. Brumer, ``The Rank of $J_0(N)$,'' Asterisque, 228, 41-68, (1995). MR 96f:11083
  • [B-HB] A. Brumer, R. Heath-Brown, ``The Average Rank of Elliptic Curves II,'' (preprint), (1992).
  • [B-K] E. Bogomolny, J. Keating, ``Random matrix theory and the Riemann zeros. I. Three- and four-point correlations,'' J. Non-linearity, 8, 1115, (1995). MR 97d:11132a
  • [B-M] A. Brumer, O. McGuinness, ``The Behavior of the Mordell-Weil Group of Elliptic Curves,'' Bull. AMS, 23, 375-382, (1990). MR 91b:11076
  • [B-S] B. Birch, P. Swinnerton-Dyer, ``Notes on Elliptic Curves (I) and (II),'' J. Reine Angew Math., 212, 7-25, (1963), 28, 79-108, (1965). MR 26:3669; MR 31:3419
  • [CO] A. Connes, ``Formule de Trace en G'eometrie Non-Commutative et Hypothesis de Riemann,'' CR. Acad. Sci., Paris, 323, I, 1231-1236, (1996). MR 97k:11124
  • [DA] H. Davenport, ``Multiplicative Number Theory,'' Springer Verlag, G.T.M., (1974). MR 82m:10001
  • [DE] P. Deligne, ``La Conjecture de Weil I and II,'' Publ. I.H.E.S., 43, 273-307, (1974), 52, 313-428, (1981). MR 49:5013; MR 83c:14017
  • [DEN] C. Denninger, ``Evidence for a Cohomological Approach to Analytic Number Theory,'' First European Math Congress, Vol. 1, (1992), 491-510, Birkhauser, (1994).
  • [D-F-I] W. Duke, J. Friedlander, H. Iwaniec, ``Representations by the Determinant and Mean Values of $L$-Functions,'' in Sieve Methods, Exponential Sums and their Applications in Number Theory, Cambridge University Press, 109-115, (1997). CMP 98:15
  • [DU] W. Duke, ``The Critical Order of Vanishing of Automorphic $L$-Functions with Large Level,'' Invent. Math., 119, 165-174, (1995). MR 95k:11075
  • [DY] F. Dyson, ``Statistical Theory of Energy Levels III,'' J. Math. Phys., 3, 166-175, (1962). MR 26:1113
  • [FE] W. Feller, ``Introduction to Probability Theory and its Applications,'' John Wiley, Vol. II, (1966). MR 35:1048
  • [GA] M. Gaudin, ``Sur la loi Limite de L'espacement de Valuers Propres D'une Matrics Aleatiore,'' Nucl. Phys., 25, 447-458, (1961).
  • [GO] D. Goldfeld, ``Conjectures on Elliptic Curves over Quadratic Number Fields,'' in Number Theory Conference, Carbondale & Spanger, L-N-M, 751, 108-118, (1979). MR 81i:12014
  • [GO-MA] F. Gouvea, B. Mazur ``The square free sieve and the rank of elliptic curves'', JAMS 4, (1991), No. 1, 1-23. MR 92b:11039
  • [GO-MO] D. Goldston, H. Montgomery, ``Pair Correlation of Zeros and Primes in Short Intervals,'' Progress in Math., Vol. 70, Birkhauser, 183-203, (1987). MR 90h:11084
  • [G-J] S. Gelbart, H. Jacquet, ``A Relation Between Automorphic Representations of $GL$(2) and $GL$(3),'' Ann. Sci. Ecole Norm. Sup 4, 11, 471-542, (1978). MR 81e:10025
  • [G-M] M. Gaudin, M. Mehta, ``On the Density of Eigenvalues of a Random Matrix,'' Nucl. Phys., 18, 420-427, (1960). MR 22:3741
  • [G-Z] B. Gross, D. Zagier, ``Heegner Points and Derivatives of $L$-Series,'' Invent. Math., 84, 225-320, (1986). MR 87j:11057
  • [HB] D. R. Heath-Brown, ``The Size of Selmer Groups for the Congruent Number Problem II,'' Invent. Math., 118, 331-370, (1994). MR 95h:11064
  • [HE] D. Hejhal, ``On the Triple Correlation of the Zeroes of the Zeta Function,'' IMRN, 293-302, (1994). MR 96d:11093
  • [HEL] S. Helgason, ``Differential Geometry, Lie Groups and Symmetric Spaces,'' Academic Press, New York, (1978). MR 80k:53081
  • [IW] H. Iwaniec, ``Topics in Analytic Number Theory,'' Rutgers University Course, (1988).
  • [I-L-S] H. Iwaniec, W. Luo, P. Sarnak, ``Low Lying Zeroes of Families of $L$-Functions,'' (preprint), (1998).
  • [I-S] H. Iwaniec, P. Sarnak, ``The Non-Vanishing of Central Values of Automorphic $L$-Functions and Siegel's Zero,'' (preprint), (1997).
  • [JA] H. Jacquet, ``Principal $L$-Functions of the Linear Group,'' Proc. Sym. Pure Math., Vol. 33, A.M.S., Part 2, 63-86, (1979). MR 81f:22029
  • [KA1] N. Katz, ``An Overview of Deligne's Work on Hilbert's Twenty-First Problem,'' Proc. Pure Math. AMS XXVIII, 537-557, (1976). MR 55:5627
  • [KA2] N. Katz, ``Big Twists Have Big Monodromy'' (in preparation), (1998).
  • [KA3] N. Katz, ``Affine Cohomological Transforms, Perversity, and Monodromy'', JAMS 6, (1993), No. 1, 149-222. MR 94b:14013
  • [K-L] V. Kolyvagin and D. Lugachev, ``Finiteness of the Shafarevich-Tate Group and the Group of Rational Points for some Modular Abelian Varieties,'' Leningrad Math. J., 1, No. 5, 1229-1253, (1990). MR 91c:11032
  • [K-M1] E. Kowalski, P. Michel, ``Sur de Rang de $J_0 (N)$,'' (preprint), (1997).
  • [K-M2] E. Kowalski, P. Michel, ``Sur les Zeros de Fonctions l Automorphes de Grand Niveau,'' (preprint), (1997).
  • [K-S1] N. Katz, P. Sarnak, ``Random Matrices, Frobenius Eigenvalue and Monodromy,'' AMS Colloq. series (to appear), (1999).
  • [K-S2] N. Katz, P. Sarnak, ``Zeroes of Zeta Functions, their Spaces and their Spectral Nature,'' (1997 preprint version of the present paper).
  • [K-Z] G. Kramarz, D. Zagier, ``Numerical Investigations Related to $L$-Series of Certain Elliptic Curves,'' J. Indian Math. Soc., 52, 51-69, (1987). MR 90d:11072
  • [LA] R. Langlands, ``Problems in the Theory of Automorphic Forms,'' Springer, L.N.M., 170, 18-86, (1970). MR 46:1758
  • [MA] B. Mazur, ``Modular Curves and the Eisenstein Ideal,'' Publ. Math. I.H.E.S., 47, 33-186, (1977). MR 80c:14015
  • [MEH] M. Mehta, ``Random Matrices,'' Second Edition, Academic Press, Boston (1991). MR 92f:82002
  • [MER] L. Merel, ``Bornes pour la Torsion de Courbes Elliptiques sur les Corps de Nombres,'' Invent. Math., 124, 437-449, (1996). MR 96i:11057
  • [MO1] H. Montgomery, ``Topics in Multiplicative Number Theory,'' L.N.M., 227, Springer, (1971). MR 49:2616
  • [MO2] H. Montgomery, ``The Pair Correlation of Zeroes of the Zeta Function,'' Proc. Sym. Pure Math., 24, AMS, 181-193, (1973). MR 49:2590
  • [MU] M. Murty, ``The Analytic Rank of $J_0(N)/Q$,'' Number Theory, (Halifax, NS), CMS Conf. Proc., AMS, 15, 263-277, (1995). MR 96i:11054
  • [OD] A. Odlyzko, ``The $10^{20}$-th Zero of the Riemann Zeta Function and 70 Million of its Neighbors,'' (preprint), A.T.T., (1989).
  • [O-S] A. Ozluk, C. Snyder, ``Small Zeroes of Quadratic $L$-Functions,'' Bull. Aust. Math. Soc., 47, 307-319, (1993). MR 94c:11080
  • [P-P] A. Perelli, J. Pomykala, ``Averages Over Twisted Elliptic $L$-Functions,'' Acta Arith., 80, 149-163, (1997). MR 98c:11044
  • [P-S] R. Phillips, P. Sarnak, ``On Cusp Forms for Cofinite Subgroups of $PSL(2, \Bbb{R})$,'' Invent. Math., 80, 339-364, (1985). MR 86m:11037
  • [RI] B. Riemann, ``Über die Anzahl der Primzahlen uter Einer Gegebenen Gröbe,'' Montasb. der Berliner Akad., (1858160), 671-680.
  • [RUB] M. Rubinstein, ``Evidence for a Spectral Interpretation of Zeros of $L$-Functions,'' Thesis, Princeton University, (1998).
  • [RUM] R. Rumely, ``Numerical Computations Concerning ERH,'' Math. Comp., 61, 415-440, (1993). MR 94b:11085
  • [R-S] Z. Rudnick, P. Sarnak, ``Zeros of Principal $L$-Functions and Random Matrix Theory,'' Duke Math. Jnl., 81, 2, 269-322, (1996). MR 97f:11074
  • [SA] P. Sarnak, ``$L$-Functions,'' ICM Talk, Berlin, 1998.
  • [SCH] F.K. Schmidt, ``Analytische Zahlen Theorie in Körpen der Charakteristik $p$,'' Math. Zeit., 33, 1-32, (1931).
  • [SH1] G. Shimura, ``On the Holomorphy of Certain Dirichlet Series,'' Proc. London Math. Soc. (3), 31, 79-98, (1975). MR 52:3064
  • [SH2] G. Shimura, ``On Modular Forms of Half-Integral Weight,'' Ann. Math., 97, 440-481, (1973). MR 48:10989
  • [SI1] J. Silverman, ``The Arithmetic of Elliptic Curves,'' G.T.M., Springer, (1986). MR 87g:11070
  • [SI2] J. Silverman, ``The Average Rank of a Family of Elliptic Curves,'' (preprint), (1997).
  • [ST] S.A. Stepanov, ``The Number of Points of Hyperelliptic Curve Over a Prime Field,'' Izv. Akad. Nauk. Sssr. Mat., 33, 1171-1181, (1969). MR 40:5620
  • [TA] J. Tate, ``On the Conjecture of Birch Swinnerton-Dyer and a Geometric Analogue,'' Seminar Bourbaki, Fev, (1966, Exp. 306). CMP 98:09
  • [TI] E. Titchmarsh, ``The Theory of the Riemann Zeta Function,'' Oxford University Press, Second Edition, (1986). MR 88c:11049
  • [V] J. Vanderkam, ``The Rank of Quotients of $J_0 (N)$,'' (preprint), (1997).
  • [WE1] A. Weil, ``Über die Bestimmung Dirichletscher Reihen Durch Funktion Algleichungen,'' Math. Ann., 168, 149-156, (1967). MR 34:7473
  • [WE2] A. Weil, ``Basic Number Theory,'' Third Edition, Springer, Berlin, (1974). MR 55:302
  • [WE3] A. Weil, ``Sur les Functions Algebriques à corps de Constantes Fini,'' C.R.Acad. Sci., Paris, 210, 592-594, (1940). MR 2:123d
  • [WE4] A. Weil, ``On the Riemann Hypothesis in Function Fields,'' Proc. Nat. Acad. Sci., U.S.A., 27, 345-349, (1941). MR 2:345b
  • [WI] E. Wigner, ``Random Matrices in Physics,'' Siam Review, 9, 1-23, (1967).
  • [WO] S. Wong, ``Rank Zero Twists of Elliptic Curves,'' (preprint), (1996).

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 11G, 11M, 11R, 11Y, 60B, 81Q

Retrieve articles in all journals with MSC (1991): 11G, 11M, 11R, 11Y, 60B, 81Q

Additional Information

Nicholas M. Katz
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544

Peter Sarnak
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544

Received by editor(s): October 15, 1997
Received by editor(s) in revised form: August 28, 1998
Additional Notes: Research partially supported by NSF grants DMS 9506412 and DMS 9401571.
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society