313
Views
4
CrossRef citations to date
0
Altmetric
Original Articles

Missing Class Groups and Class Number Statistics for Imaginary Quadratic Fields

, , , &

References

  • [Bach 95] E. Bach. “Improved Approximations for Euler Products.” In Number Theory (Halifax, NS, 1994), vol. 15 of CMS Conf. Proc., pp. 13–28, Amer. Math. Soc., Providence, RI, 1995.
  • [Banks et al. 12] W. Banks, F. Pappalardi, and I. Shparlinski. “On Group Structures Realized by Elliptic Curves Over Arbitrary Finite Fields.” Experiment. Math. 21:1 (2012), 11–25.
  • [Belabas 97] K. Belabas. “A Fast Algorithm to Compute Cubic Fields.” Math. Comp. 66:219 (1997), 1213–1237.
  • [Bhargava et al. 13] M. Bhargava, A. Shankar, and J. Tsimerman. “On the Davenport–Heilbronn Theorems and Second Order Terms.” Invent. Math. 193:2 (2013), 439–499.
  • [Booker 06] A. R. Booker. “Quadratic Class Numbers and Character Sums.” Math. Comp. 75:255 (2006), 1481–1492 (electronic).
  • [Boyd and Kisilevsky 72] D. Boyd and H. Kisilevsky. “On the Exponent of the Ideal Class Groups of Complex Quadratic Fields.” Proc. Amer. Math. Soc. 31 (1972), 433–436.
  • [Buell 99] D. A. Buell. “The Last Exhaustive Computation of Class Groups of Complex Quadratic Number Fields.” In Number theory (Ottawa, ON, 1996), vol. 19 of CRM Proc. Lecture Notes, pp. 35–53, Amer. Math. Soc., Providence, RI, 1999.
  • [Chowla 34] S. Chowla. “An Extension of Heilbronn’s Class Number Theorem.” Quarterly J. Math. 5 (1934), 304–307.
  • [Claborn 66] L. Claborn. “Every Abelian Group is a Class Group.” Pacific J. Math. 18 (1966), 219–222.
  • [Cohen Preprint] H. Cohen. “High Precision Computation of Hardy-Littlewood Constants.” 1998 (http://www.math.u-bordeaux1.fr/cohen/hardylw.dvi).
  • [Cohen and Lenstra 84] H. Cohen and H. W. Lenstra. Heuristics on Class Groups of Number Fields, Lecture Notes in Mathematics, 1068. Springer, Berlin, 1984, pp. 33–62.
  • [Cornell 79] G. Cornell. “Abhyankar’s Lemma and the Class Group.” In Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), 82–88, Lecture Notes in Math., 751, Springer, Berlin, 1979.
  • [David and Smith 14] C. David and E. Smith. “A Cohen–Lenstra phenomenon for Elliptic Curves.” J. London Math. Soc. 89 (2014), 24–44.
  • [Ellenberg and Venkatesh 07] J. S. Ellenberg and A. Venkatesh. “Reflection Principles and Bounds for Class Group Torsion.” Int. Math. Res. Not. IMRN 18:1 (2007), Art. ID rnm002.
  • [Flajolet and Sedgewick 09] P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge: Cambridge University Press, 2009.
  • [Granville and Soundararajan 03] A. Granville and K. Soundararajan. “The Distribution of Values of L(1, χd).” Geom. Funct. Anal. 13 (2003), 992–1028.
  • [Hardy and Ramanujan 18] G. Hardy and S. Ramanujan. “Asymptotic Formulae in Combinatory Analysis.” Proc. London Math. Soc. 2 (1918), 75–115.
  • [Heath-Brown 08] D. R. Heath-Brown. “Imaginary Quadratic Fields with Class Group Exponent 5.” Forum Math. 20 (2008), 275–283.
  • [Helfgott and Venkatesh 06] H. A. Helfgott and A. Venkatesh. “Integral Points on Elliptic Curves and 3-Torsion in Class Groups.” J. Amer. Math. Soc., 19:3 (2006), 527–550 (electronic).
  • [Hillar and Rhea 07] C. Hillar and D. Rhea. “Automorphisms of Finite Abelian Groups.” Amer. Math. Monthly 114 (2007), 917–923.
  • [Hoeffding 63] W. Hoeffding. “Probability Inequalities for Sums of Bounded Random Variables.” J. Amer. Statist. Assoc. 58 (1963), 13–30.
  • [Holmin and Kurlberg XX] S. Holmin and P. Kurlberg. “List of F(h) for all Odd h < 106.” Available online (https://people.kth.se/kurlberg/class_group_data/class_group_orders.txt, 2015.
  • [Holmin and P. Kurlberg XX] S. Holmin and P. Kurlberg. “List of F(G) for all Noncyclic p-Groups G of Odd Order < 106.” Available online (https://people.kth.se/kurlberg/class_group_data/noncyclic_class_groups.txt), 2015.
  • [Holmin and P. Kurlberg XX] S. Holmin and P. Kurlberg. “List of (d, H(d)) for all Fundamental Discriminants d < 0 such that H(d) is a Noncyclic p-Group of Odd Order < 106.” Available online (https://people.kth.se/kurlberg/class_group_data/discriminants_of_noncyclic_groups.txt), 2015.
  • [Jacobson et al. 06] M. J. Jacobson, Jr, S. Ramachandran, and H. C. Williams. “Numerical Results on Class Groups of Imaginary Quadratic Fields.” In Algorithmic number theory, vol. 4076, pp. 87–101, Lecture Notes in Comput. Sci. Berlin: Springer, 2006.
  • [Lagarias and Odlyzko 77] J. C. Lagarias and A. M. Odlyzko. “Effective Versions of the Chebotarev Density Theorem.” In Algebraic Number Fields, edited by A. Frohlich, pp. 409–464. Academic Press, London, 1977.
  • [Lamzouri XX] Y. Lamzouri. “On the Average of the Number of Imaginary Quadratic Fields with a Given Class Number.” arXiv:1512.07134., XXXX.
  • [Lamzouri XX] Y. Lamzouri. “The Number of Imaginary Quadratic Fields with Prime Discriminant and Class Number Up to H.”arXiv:1701.05267., Ramanujan J. 44 (2017), 411–416.
  • [Lamzouri et al. 15] Y. Lamzouri, X. Li, and K. Soundararajan. “Conditional Bounds for the Least Quadratic Non-Residue and Related Problems.” Math. Comp. 84:295 (2015), 2391–2412.
  • [Leedham-Green 72] C. R. Leedham-Green. “The Class Group of Dedekind Domains.” Trans. Amer. Math. Soc. 163 (1972), 493–500.
  • [Lengler 10] J. Lengler. “The Cohen-Lenstra Heuristic: Methodology and Results.” J. Algebra 323 (2010), 2960–2976.
  • [Lengler 12] J. Lengler. “The Global Cohen-Lenstra Heuristic.” J. Algebra 357 (2012), 247–269.
  • [Mosunov and Jacobson XX] A. S. Mosunov and M. J. Jacobson, Jr. “Unconditional Class Group Tabulation of Imaginary Quadratic Fields to Δ < 240.” arXiv:1502.07953., 85 (2016), 1983–2009.
  • [Oesterlé 85] J. Oesterlé. “Nombres de Classes Des Corps Quadratiques Imaginaires.” Astérisque. 1983/84:121–122 (1985), 309–323.
  • [Ozaki 11] M. Ozaki. “Construction of Maximal Unramified p-Extensions with Prescribed Galois Groups.” Invent. Math., 183:3 (2011), 649–680.
  • [The PARI Group 15] The PARI Group. “Bordeaux.” PARI/GP version 2.7.3. Available online (http://pari.math.u-bordeaux.fr/pub/pari/unix/pari-2.7.3.tar.gz) 2015.
  • [Perret 99] M. Perret. “On the Ideal Class Group Problem for Global Fields.” J. Number Theory 77:1 (1999), 27–35.
  • [Pierce 06] L. B. Pierce. “A Bound for the 3-Part of Class Numbers of Quadratic Fields by Means of the Square Sieve.” Forum Math. 18:4 (2006), 677–698.
  • [Ranum 07] A. Ranum. “The Group of Classes of Congruent Matrices with Application to the Group of Isomorphisms of Any Abelian Group.” Trans. Amer. Math. Soc. 8 (1907), 71–91.
  • [Roberts 01] D. P. Roberts. “Density of Cubic Field Discriminants.” Math. Comp. 70:236 (2001), 1699–1705 (electronic).
  • [Lowell Schoenfeld 76] L. Schoenfeld. “Sharper Bounds for the Chebyshev Functions θ(x) and ψ(x). II.” Math. Comp. 30:134 (1976), 337–360.
  • [Sellers 03] J. A. Sellers. “Extending a Recent Result of Santos on Partitions Into Odd Parts.” Integers 3:A4 (2003), 5 pp. (electronic).
  • [Sellers 04] J. A. Sellers. “Corrigendum to: “Extending a Recent Result of Santos on Partitions into Odd Parts.” Integers 4:A8 (2004), 1 pp. (electronic).
  • [Serre 81] J.-P. Serre. “Quelques Applications du théorème de densité de Chebotarev.” Publ. Math. I. H. E. S. 54 (1981), 123–201.
  • [Soundararajan 07] K. Soundararajan. “The Number of Imaginary Quadratic Fields with a Given Class Number.” Hardy-Ramanujan J. 30 (2007), 13–18.
  • [Sutherland 07] A. V. Sutherland. “Order Computations in Generic Groups.” Ph.D. Thesis, Massachusetts Institute of Technology, Ann Arbor, MI: ProQuest LLC, 2007.
  • [Taniguchi and Thorne 13] T. Taniguchi and F. Thorne. “Secondary Terms in Counting Functions for Cubic Fields.” Duke Math. J. 162 (2013), 2451–2508.
  • [Walisch XX] K. Walisch. “primesieve. Fast C/C++ Prime Number Generator.” Available online (http://primesieve.org/), 2015.
  • [Watkins 03] M. Watkins. “Class Numbers of Imaginary Quadratic Fields.” Math. Comp. 73 (2003), 907–938.
  • [Weinberger 73] P. Weinberger. “Exponents of the Class Groups of Complex Quadratic Fields.” Acta Arith. 22 (1973), 117–124.
  • [Yahagi 78] O. Yahagi. “Construction of Number Fields with Prescribed l-Class Groups.” Tokyo J. Math. 1:2 (1978), 275–283.
  • [Yamamoto 70] Y. Yamamoto. “On Unramified Galois Extensions of Quadratic Number Fields.” Osaka J. Math. 7 (1970), 57–76.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.