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Averages and moments associated to class numbers of imaginary quadratic fields

  • D. R. Heath-Brown (a1) and L. B. Pierce (a2)

Abstract

For any odd prime $\ell$ , let $h_{\ell }(-d)$ denote the $\ell$ -part of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$ . Nontrivial pointwise upper bounds are known only for $\ell =3$ ; nontrivial upper bounds for averages of $h_{\ell }(-d)$ have previously been known only for $\ell =3,5$ . In this paper we prove nontrivial upper bounds for the average of $h_{\ell }(-d)$ for all primes $\ell \geqslant 7$ , as well as nontrivial upper bounds for certain higher moments for all primes $\ell \geqslant 3$ .

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References

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[BST13] Bhargava, M., Shankar, A. and Tsimerman, J., On the Davenport–Heilbronn theorem and second order terms , Invent. Math. 193 (2013), 439499.
[BS96] Brumer, A. and Silverman, J. H., The number of elliptic curves over Q with conductor N , Manuscripta Math. 91 (1996), 95102.
[CL84] Cohen, H. and Lenstra, H. W. Jr., Heuristics on class groups of number fields , in Number theory, Noordwijkerhout 1983, Lecture Notes in Mathematics, vol. 1068 (Springer, Berlin, 1984), 3362.
[Dav58] Davenport, H., Indefinite quadratic forms in many variables II , Proc. Lond. Math. Soc. (3) 8 (1958), 109126.
[Dav00] Davenport, H., Multiplicative number theory, Graduate Texts in Mathematics, vol. 74, third edition (Springer, New York, 2000).
[DH71] Davenport, H. and Heilbronn, H., On the density of discriminants of cubic fields II , Proc. R. Soc. Lond. A 322 (1971), 405420.
[Duk98] Duke, W., Bounds for arithmetic multiplicities , in Proc. Int. Congress of Mathematicians, Berlin, 1998, Doc. Math., Extra Volume II (1998), 163172.
[EV07] Ellenberg, J. S. and Venkatesh, A., Reflection principles and bounds for class group torsion , Int. Math. Res. Not. IMRN 2007 (2007), rnm002.
[HB07] Heath-Brown, D. R., Quadratic class numbers divisible by 3 , Funct. Approx. Comment. Math. 37 (2007), 203211.
[HV06] Helfgott, H. A. and Venkatesh, A., Integral points on elliptic curves and 3-torsion in class groups , J. Amer. Math. Soc. 19 (2006), 527550.
[Hou10] Hough, R., Average equidistribution of Heegner points associated to the 3-part of the class group of imaginary quadratic fields, Preprint (2010), arXiv:1005.1458v2.
[Sch32] Scholz, A., Über die Beziehung der Klassenzahlen quadratischer Körper , J. Reine Angew. Math. 166 (1932), 201203.
[Sou00] Soundararajan, K., Divisibility of class numbers of imaginary quadratic fields , J. Lond. Math. Soc. (2) 61 (2000), 681690.
[TT13] Taniguchi, T. and Thorne, F., The secondary term in the counting function for cubic fields , Duke Math. J. 162 (2013), 24512508.
[Zha05] Zhang, S.-W., Equidistribution of CM-points on quaternion Shimura varieties , Int. Math. Res. Not. IMRN 2005 (2005), 36573689.
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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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