Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T16:43:22.844Z Has data issue: false hasContentIssue false
  • English
  • Français

SATO–TATE EQUIDISTRIBUTION OF CERTAIN FAMILIES OF ARTIN $L$-FUNCTIONS

Part of: Zeta and $L$-functions: analytic theory Algebraic number theory: global fields

Published online by Cambridge University Press:  13 August 2019

ARUL SHANKAR ,
ANDERS SÖDERGREN  and
NICOLAS TEMPLIER
ARUL SHANKAR
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada; arul.shnkr@gmail.com
ANDERS SÖDERGREN
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden; andesod@chalmers.se
NICOLAS TEMPLIER
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA; templier@math.cornell.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study various families of Artin $L$-functions attached to geometric parametrizations of number fields. In each case we find the Sato–Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros.

MSC classification

Primary: 11R42: Zeta functions and $L$-functions of number fields 11M41: Other Dirichlet series and zeta functions
Secondary: 11M50: Relations with random matrices
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e23
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

References

Armitage, J. V., ‘Zeta functions with a zero at ’, Invent. Math. 15 (1972), 199–205.Google Scholar
Ash, A., Brakenhoff, J. and Zarrabi, T., ‘Equality of polynomial and field discriminants’, Exp. Math. 16(3) (2007), 367–374.Google Scholar
Baily, A. M., ‘On the density of discriminants of quartic fields’, J. Reine Angew. Math. 315 (1980), 190–210.Google Scholar
Belabas, K., Bhargava, M. and Pomerance, C., ‘Error estimates for the Davenport-Heilbronn theorems’, Duke Math. J. 153(1) (2010), 173–210.Google Scholar
Bhargava, M., ‘Higher composition laws III: the parametrization of quartic rings’, Ann. of Math. (2) 159(3) (2004), 1329–1360.Google Scholar
Bhargava, M., ‘Higher composition laws IV: the parametrization of quintic rings’, Ann. of Math. (2) 167(1) (2008), 53–94.Google Scholar
Bhargava, M., ‘The density of discriminants of quartic rings and fields’, Ann. of Math. (2) 162(2) (2005), 1031–1063.Google Scholar
Bhargava, M., ‘The density of discriminants of quintic rings and fields’, Ann. of Math. (2) 172(3) (2010), 1559–1591.Google Scholar
Bhargava, M., ‘Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants’, Int. Math. Res. Not. IMRN (2007), no. 17, Art. ID rnm052, 20 pp.Google Scholar
Bhargava, M., Shankar, A. and Tsimerman, J., ‘On the Davenport-Heilbronn theorems and second order terms’, Invent. Math. 193(2) (2013), 439–499.Google Scholar
Bhargava, M., Shankar, A. and Wang, X., ‘Squarefree values of polynomial discriminants I’, Preprint, 2016, arXiv:1611.09806.Google Scholar
Bhargava, M., Shankar, A. and Wang, X., ‘Geometry-of-numbers methods over global fields I: prehomogeneous vector spaces’, Preprint, 2015, arXiv:1512.03035.Google Scholar
Birch, B. J. and Merriman, J. R., ‘Finiteness theorems for binary forms with given discriminant’, Proc. Lond. Math. Soc. (3) 24 (1972), 385–394.Google Scholar
Booker, A. R. and Strömbergsson, A., ‘Numerical computations with the trace formula and the Selberg eigenvalue conjecture’, J. Reine Angew. Math. 607 (2007), 113–161.Google Scholar
Buchsbaum, D. and Eisenbud, D., ‘Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3’, Amer. J. Math. 99(3) (1977), 447–485.Google Scholar
Calegari, F., ‘The Artin conjecture for some S 5 -extensions’, Math. Ann. 356(1) (2013), 191–207.Google Scholar
Cassels, J. W. S., Rational Quadratic Forms, London Mathematical Society Monographs, 13 (Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London–New York, 1978), pp. xvi+413.Google Scholar
Cassels, J. W. S. and Fröhlich, A., Algebraic Number Theory, (Academic Press, London–New York, 1967).Google Scholar
Cho, P. J. and Kim, H. H., ‘Low lying zeros of Artin L-functions’, Math. Z. 279(3–4) (2015), 669–688.Google Scholar
Cho, P. J. and Kim, H. H., ‘ n-level densities of Artin L-functions’, Int. Math. Res. Not. IMRN (17) (2015), 7861–7883.Google Scholar
Cohen, H., Diaz y Diaz, F. and Olivier, M., ‘Enumerating Quartic Dihedral Extensions of ℚ’, Compos. Math. 133(1) (2002), 65–93.Google Scholar
Conrey, J. B. and Soundararajan, K., ‘Real zeros of quadratic Dirichlet L-functions’, Invent. Math. 150(1) (2002), 1–44.Google Scholar
Curtis, C. W., Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, 15 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Davenport, H. and Heilbronn, H., ‘On the density of discriminants of cubic fields II’, Proc. R. Soc. Lond. Ser. A 322(1551) (1971), 405–420.Google Scholar
David, C., Fearnley, J. and Kisilevsky, H., ‘On the vanishing of twisted L-functions of elliptic curves’, Experiment. Math. 13(2) (2004), 185–198.Google Scholar
Dedekind, R., ‘Konstruktion von Quaternionkörpern’, inGesammelte mathematische Werke, Bd. 2 (Vieweg & Sohn, Braunschweig, 1931), 376–384.Google Scholar
Deligne, P., SGA —Cohomologie étale, Lecture Notes in Mathematics, 569 (Springer, New York, 1977).Google Scholar
Delone, B. N. and Faddeev, D. K., The Theory of Irrationalities of the Third Degree, Translations of Mathematical Monographs, 10 (American Mathematical Society, Providence, RI, 1964).Google Scholar
Deng, A.-W., ‘Rational points on weighted projective spaces’, Preprint, 1998,arXiv:9812082.Google Scholar
Dietmann, R., ‘On the distribution of Galois groups’, Mathematika 58(1) (2012), 35–44.Google Scholar
Ellenberg, J., Pierce, L. B. and Wood, M. M., ‘On ℓ-torsion in class groups of number fields’, Algebra Number Theory 11(8) (2017), 1739–1778.Google Scholar
Entin, A., Roditty-Gershon, E. and Rudnick, Z., ‘Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and random matrix theory’, Geom. Funct. Anal. 23(4) (2013), 1230–1261.Google Scholar
Fiorilli, D., Parks, J. and Södergren, A., ‘Low-lying zeros of elliptic curve L-functions: beyond the ratios conjecture’, Math. Proc. Cambridge Philos. Soc. 160(2) (2016), 315–351.Google Scholar
Fouvry, É., Luca, F., Pappalardi, F. and Shparlinski, I. E., ‘Counting dihedral and quaternionic extensions’, Trans. Amer. Math. Soc. 363(6) (2011), 3233–3253.Google Scholar
Frobenius, G. and Schur, I., ‘Über die reellen Darstellungen der endlichen Gruppen’, Sitzungsber, Preuss, Akad. d (1906), 186–208.Google Scholar
Fröhlich, A., ‘Artin root numbers and normal integral bases for quaternion fields’, Invent. Math. 17(2) (1972), 143–166.Google Scholar
Fröhlich, A. and Queyrut, J., ‘On the functional equation of the Artin L-function for characters of real representations’, Invent. Math. 20 (1973), 125–138.Google Scholar
Gan, W. T., Gross, B. and Savin, G., ‘Fourier coefficients of modular forms on G 2 ’, Duke Math. J. 115(1) (2002), 105–169.Google Scholar
Heilbronn, H., ‘On the 2-classgroup of cubic fields’, inStudies in Pure Mathematics (Presented to Richard Rado) (Academic Press, London, 1971), 117–119.Google Scholar
Iwaniec, H., ‘Conversations on the exceptional character’, inAnalytic Number Theory, Lecture Notes in Mathematics, 1891 (Springer, Berlin, 2006), 97–132.Google Scholar
Jensen, C. U. and Yui, N., ‘Quaternion extensions’, inAlgebraic Geometry and Commutative Algebra, Vol. I (Kinokuniya, Tokyo, 1988), 155–182.Google Scholar
Katz, N. M., ‘Sato–Tate in the higher dimensional case: elaboration of 9. 5. 4 in Serre’s N X(p) book’, Enseign. Math. 59(3–4) (2013), 359–377.Google Scholar
Katz, N. M. and Sarnak, P., ‘Zeroes of zeta functions and symmetry’, Bull. Amer. Math. Soc. (N.S.) 36(1) (1999), 1–26.Google Scholar
Kedlaya, K. S., ‘Mass formulas for local Galois representations’, Int. Math. Res. Not. IMRN (17) (2007), Art. ID rnm021, 26 pp.Google Scholar
Kiming, I., ‘Explicit classification of some 2-extensions of a field of characteristic different from 2’, Canad. J. Math. 42(5) (1990), 825–855.Google Scholar
Klüners, J., ‘Über die Asymptotik von Zahlkörpern mit vorgegebener Galoisgruppe’, Habilitationsschrift, Universität Kassel, 2005.Google Scholar
Kowalski, E., ‘Families of cusp forms’, inActes de la Conférence ‘Théorie des Nombres et Applications’, Publ. Math. Besançon Algèbre Théorie Nr. (Presses Univ. Franche-Comté, Besançon, 2013), 5–40.Google Scholar
Lagarias, J. C. and Weiss, B. L., ‘Splitting behavior of S n-polynomials’, Res. Number Theory 1 (2015), Art. 7, 30 pp.Google Scholar
Lam, T. Y., The Algebraic Theory of Quadratic Forms, Mathematics Lecture Note Series (Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, MA, 1980).Google Scholar
Lemke Oliver, R. J. and Thorne, F., ‘The number of ramified primes in number fields of small degree’, Proc. Amer. Math. Soc. 145(8) (2017), 3201–3210.Google Scholar
Levi, F., ‘Kubische Zahlkörper und binäre kubische Formenklassen’, Ber. Sächs. Akad. Wiss. Leipzig, Math.-Naturwiss 66 (1914), 26–37.Google Scholar
Macdonald, I. G., Symmetric Functions and Orthogonal Polynomials, University Lecture Series, 12 (American Mathematical Society, Providence, RI, 1998).Google Scholar
Munsch, M., ‘Character sums over squarefree and squarefull numbers’, Arch. Math. (Basel) 102(6) (2014), 555–563.Google Scholar
Nakagawa, J., ‘Binary forms and orders of algebraic number fields’, Invent. Math. 97(2) (1989), 219–235.Google Scholar
Neukirch, J., Class Field Theory, Grundlehren der Mathematischen Wissenschaften, 280 (Springer, Berlin, 1986).Google Scholar
Perlis, R., ‘On the equation 𝜁K(s) =𝜁 K ′ (s)’, J. Number Theory 9(3) (1977), 342–360.Google Scholar
Peyre, E., ‘Hauteurs et mesures de Tamagawa sur les variétés de Fano’, Duke Math. J. 79(1) (1995), 101–218.Google Scholar
Reichardt, H., ‘Über Normalkörper mit Quaternionengruppe’, Math. Z. 41(1) (1936), 218–221.Google Scholar
Rubinstein, M., ‘Low-lying zeros of L-functions and random matrix theory’, Duke Math. J. 109(1) (2001), 147–181.Google Scholar
Rudnick, Z. and Sarnak, P., ‘Zeros of principal L-functions and random matrix theory’, Duke Math. J. 81(2) (1996), 269–322.Google Scholar
Sarnak, P., Shin, S. W. and Templier, N., ‘Families of L-functions and their symmetry’, inProceedings of Simons Symposia, Families of Automorphic Forms and the Trace Formula (Springer Verlag, 2016), 531–578.Google Scholar
Sato, M. and Kimura, T., ‘A classification of irreducible prehomogeneous vector spaces and their relative invariants’, Nagoya Math. J. 65 (1977), 1–155.Google Scholar
Serre, J.-P., Lectures on N X(p), Chapman & Hall/CRC Research Notes in Mathematics, 11 (CRC Press, Boca Raton, FL, 2012).Google Scholar
Shankar, A. and Tsimerman, J., ‘Counting S 5 -fields with a power saving error term’, Forum Math. Sigma 2 (2014), e13 (8 pages).Google Scholar
Shin, S. W. and Templier, N., ‘Sato–Tate theorem for families and low-lying zeros of automorphic L-functions’, Invent. Math. 203(1) (2016), 1–177.Google Scholar
Siegel, C. L., ‘The average measure of quadratic forms with given determinant and signature’, Ann. of Math. (2) 45 (1944), 667–685.Google Scholar
Soundararajan, K., ‘Nonvanishing of quadratic Dirichlet L-functions at ’, Ann. of Math. (2) 152(2) (2000), 447–488.Google Scholar
Taniguchi, T. and Thorne, F., ‘Secondary terms in counting functions for cubic fields’, Duke Math. J. 162(13) (2013), 2451–2508.Google Scholar
Tate, J., ‘Number theoretic background’, inAutomorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proceedings of Symposia in Pure Mathematics, XXXIII (American Mathematical Society, Providence, RI, 1979), 3–26.Google Scholar
Taussky, O., ‘Pairs of sums of three squares of integers whose product has the same property’, inGeneral Inequalities 2 (Proc. Second Internat. Conf., Oberwolfach, 1978) (Birkhäuser, Basel–Boston, MA, 1980), 29–36.Google Scholar
Vignéras, M.-F., Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, 800 (Springer, Berlin, 1980).Google Scholar
Wilson, K. H., ‘Three perspectives on $n$ points in $\mathbb{P}^{n-2}$ ’, PhD Thesis, Princeton University, 2012.Google Scholar
Witt, E., ‘Konstruktion von galoisschen Körpern der Charakteristik p zu vorgegebener Gruppe der Ordung p f ’, J. Reine Angew. Math. 174 (1936), 237–245.Google Scholar
Wood, M. M., ‘Moduli spaces for rings and ideals’, PhD Thesis, Princeton University, June 2009.Google Scholar
Wood, M. M., ‘Mass formulas for local Galois representations to wreath products and cross products’, Algebra Number Theory 4 (2008), 391–405.Google Scholar
Wood, M. M., ‘Rings and ideals parameterized by binary n-ic forms’, J. Lond. Math. Soc. (2) 83(1) (2011), 208–231.Google Scholar
Wood, M. M., ‘How to determine the splitting type of a prime, unpublished note’, available at http://www.math.wisc.edu/∼mmwood/Splitting.pdf.Google Scholar
Wright, D. J. and Yukie, A., ‘Prehomogeneous vector spaces and field extensions’, Invent. Math. 110(2) (1992), 283–314.Google Scholar
Yang, A., ‘Distribution problems associated to zeta functions and invariant theory’, PhD Thesis, Princeton University, 2009.Google Scholar