Skip to main content Accessibility help
Hostname: page-component-65d66dc8c9-vx887 Total loading time: 0.827 Render date: 2021-09-29T00:24:09.928Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }


Published online by Cambridge University Press:  13 August 2019

Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada;
Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden;
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA;


HTML view is 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.

Research Article
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 (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
© The Author(s) 2019


Armitage, J. V., ‘Zeta functions with a zero at ’, Invent. Math. 15 (1972), 199205.CrossRefGoogle Scholar
Ash, A., Brakenhoff, J. and Zarrabi, T., ‘Equality of polynomial and field discriminants’, Exp. Math. 16(3) (2007), 367374.CrossRefGoogle Scholar
Baily, A. M., ‘On the density of discriminants of quartic fields’, J. Reine Angew. Math. 315 (1980), 190210.Google Scholar
Belabas, K., Bhargava, M. and Pomerance, C., ‘Error estimates for the Davenport-Heilbronn theorems’, Duke Math. J. 153(1) (2010), 173210.CrossRefGoogle Scholar
Bhargava, M., ‘Higher composition laws III: the parametrization of quartic rings’, Ann. of Math. (2) 159(3) (2004), 13291360.CrossRefGoogle Scholar
Bhargava, M., ‘Higher composition laws IV: the parametrization of quintic rings’, Ann. of Math. (2) 167(1) (2008), 5394.CrossRefGoogle Scholar
Bhargava, M., ‘The density of discriminants of quartic rings and fields’, Ann. of Math. (2) 162(2) (2005), 10311063.CrossRefGoogle Scholar
Bhargava, M., ‘The density of discriminants of quintic rings and fields’, Ann. of Math. (2) 172(3) (2010), 15591591.CrossRefGoogle 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.CrossRefGoogle Scholar
Bhargava, M., Shankar, A. and Tsimerman, J., ‘On the Davenport-Heilbronn theorems and second order terms’, Invent. Math. 193(2) (2013), 439499.CrossRefGoogle 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), 385394.CrossRefGoogle Scholar
Booker, A. R. and Strömbergsson, A., ‘Numerical computations with the trace formula and the Selberg eigenvalue conjecture’, J. Reine Angew. Math. 607 (2007), 113161.CrossRefGoogle 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), 447485.CrossRefGoogle Scholar
Calegari, F., ‘The Artin conjecture for some S 5 -extensions’, Math. Ann. 356(1) (2013), 191207.CrossRefGoogle 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), 669688.CrossRefGoogle Scholar
Cho, P. J. and Kim, H. H., ‘ n-level densities of Artin L-functions’, Int. Math. Res. Not. IMRN (17) (2015), 78617883.CrossRefGoogle Scholar
Cohen, H., Diaz y Diaz, F. and Olivier, M., ‘Enumerating Quartic Dihedral Extensions of ℚ’, Compos. Math. 133(1) (2002), 6593.CrossRefGoogle Scholar
Conrey, J. B. and Soundararajan, K., ‘Real zeros of quadratic Dirichlet L-functions’, Invent. Math. 150(1) (2002), 144.CrossRefGoogle Scholar
Curtis, C. W., Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, 15 (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
Davenport, H. and Heilbronn, H., ‘On the density of discriminants of cubic fields II’, Proc. R. Soc. Lond. Ser. A 322(1551) (1971), 405420.CrossRefGoogle Scholar
David, C., Fearnley, J. and Kisilevsky, H., ‘On the vanishing of twisted L-functions of elliptic curves’, Experiment. Math. 13(2) (2004), 185198.CrossRefGoogle Scholar
Dedekind, R., ‘Konstruktion von Quaternionkörpern’, inGesammelte mathematische Werke, Bd. 2 (Vieweg & Sohn, Braunschweig, 1931), 376384.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), 3544.CrossRefGoogle Scholar
Ellenberg, J., Pierce, L. B. and Wood, M. M., ‘On -torsion in class groups of number fields’, Algebra Number Theory 11(8) (2017), 17391778.CrossRefGoogle 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), 12301261.CrossRefGoogle 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), 315351.CrossRefGoogle Scholar
Fouvry, É., Luca, F., Pappalardi, F. and Shparlinski, I. E., ‘Counting dihedral and quaternionic extensions’, Trans. Amer. Math. Soc. 363(6) (2011), 32333253.CrossRefGoogle Scholar
Frobenius, G. and Schur, I., ‘Über die reellen Darstellungen der endlichen Gruppen’, Sitzungsber, Preuss, Akad. d (1906), 186208.Google Scholar
Fröhlich, A., ‘Artin root numbers and normal integral bases for quaternion fields’, Invent. Math. 17(2) (1972), 143166.CrossRefGoogle Scholar
Fröhlich, A. and Queyrut, J., ‘On the functional equation of the Artin L-function for characters of real representations’, Invent. Math. 20 (1973), 125138.CrossRefGoogle Scholar
Gan, W. T., Gross, B. and Savin, G., ‘Fourier coefficients of modular forms on G 2 ’, Duke Math. J. 115(1) (2002), 105169.CrossRefGoogle Scholar
Heilbronn, H., ‘On the 2-classgroup of cubic fields’, inStudies in Pure Mathematics (Presented to Richard Rado) (Academic Press, London, 1971), 117119.Google Scholar
Iwaniec, H., ‘Conversations on the exceptional character’, inAnalytic Number Theory, Lecture Notes in Mathematics, 1891 (Springer, Berlin, 2006), 97132.CrossRefGoogle Scholar
Jensen, C. U. and Yui, N., ‘Quaternion extensions’, inAlgebraic Geometry and Commutative Algebra, Vol. I (Kinokuniya, Tokyo, 1988), 155182.CrossRefGoogle 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), 359377.CrossRefGoogle Scholar
Katz, N. M. and Sarnak, P., ‘Zeroes of zeta functions and symmetry’, Bull. Amer. Math. Soc. (N.S.) 36(1) (1999), 126.CrossRefGoogle 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), 825855.CrossRefGoogle 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), 540.Google Scholar
Lagarias, J. C. and Weiss, B. L., ‘Splitting behavior of S n-polynomials’, Res. Number Theory 1 (2015), Art. 7, 30 pp.CrossRefGoogle 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), 32013210.CrossRefGoogle Scholar
Levi, F., ‘Kubische Zahlkörper und binäre kubische Formenklassen’, Ber. Sächs. Akad. Wiss. Leipzig, Math.-Naturwiss 66 (1914), 2637.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), 555563.CrossRefGoogle Scholar
Nakagawa, J., ‘Binary forms and orders of algebraic number fields’, Invent. Math. 97(2) (1989), 219235.CrossRefGoogle Scholar
Neukirch, J., Class Field Theory, Grundlehren der Mathematischen Wissenschaften, 280 (Springer, Berlin, 1986).CrossRefGoogle Scholar
Perlis, R., ‘On the equation 𝜁K(s) =𝜁 K (s)’, J. Number Theory 9(3) (1977), 342360.CrossRefGoogle Scholar
Peyre, E., ‘Hauteurs et mesures de Tamagawa sur les variétés de Fano’, Duke Math. J. 79(1) (1995), 101218.CrossRefGoogle Scholar
Reichardt, H., ‘Über Normalkörper mit Quaternionengruppe’, Math. Z. 41(1) (1936), 218221.CrossRefGoogle Scholar
Rubinstein, M., ‘Low-lying zeros of L-functions and random matrix theory’, Duke Math. J. 109(1) (2001), 147181.CrossRefGoogle Scholar
Rudnick, Z. and Sarnak, P., ‘Zeros of principal L-functions and random matrix theory’, Duke Math. J. 81(2) (1996), 269322.CrossRefGoogle 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), 531578.CrossRefGoogle Scholar
Sato, M. and Kimura, T., ‘A classification of irreducible prehomogeneous vector spaces and their relative invariants’, Nagoya Math. J. 65 (1977), 1155.CrossRefGoogle 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).CrossRefGoogle 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), 1177.CrossRefGoogle Scholar
Siegel, C. L., ‘The average measure of quadratic forms with given determinant and signature’, Ann. of Math. (2) 45 (1944), 667685.CrossRefGoogle Scholar
Soundararajan, K., ‘Nonvanishing of quadratic Dirichlet L-functions at ’, Ann. of Math. (2) 152(2) (2000), 447488.CrossRefGoogle Scholar
Taniguchi, T. and Thorne, F., ‘Secondary terms in counting functions for cubic fields’, Duke Math. J. 162(13) (2013), 24512508.CrossRefGoogle 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), 326.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), 2936.CrossRefGoogle Scholar
Vignéras, M.-F., Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, 800 (Springer, Berlin, 1980).CrossRefGoogle 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), 237245.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), 391405.CrossRefGoogle Scholar
Wood, M. M., ‘Rings and ideals parameterized by binary n-ic forms’, J. Lond. Math. Soc. (2) 83(1) (2011), 208231.CrossRefGoogle Scholar
Wood, M. M., ‘How to determine the splitting type of a prime, unpublished note’, available at∼mmwood/Splitting.pdf.Google Scholar
Wright, D. J. and Yukie, A., ‘Prehomogeneous vector spaces and field extensions’, Invent. Math. 110(2) (1992), 283314.CrossRefGoogle Scholar
Yang, A., ‘Distribution problems associated to zeta functions and invariant theory’, PhD Thesis, Princeton University, 2009.Google Scholar
You have Access
Open access
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *