## Rocky Mountain Journal of Mathematics

### Computing quadratic function fields with high 3-rank via cubic field tabulation

#### Abstract

In this paper, we present extensive numerical data on quadratic function fields with non-zero 3-rank. We use a function field adaptation of a method due to Belabas for finding quadratic number fields of high 3-rank. Our algorithm relies on previous work for tabulating cubic function fields of bounded discriminant \cite {Pieter3} but includes a significant novel improvement when the discriminants are imaginary. We provide numerical data for discriminant degree up to 11 over the finite fields $\mathbb{F}_5, \mathbb{F}_7, \mathbb{F}_11$ and $\mathbb{F}_13$ and $\mathbb{F}_13$. In addition to presenting new examples of fields of minimal discriminant degree with a given 3-rank, we compare our data with a variety of heuristics on the density of such fields with a given 3-rank, which in most cases supports their validity.

#### Article information

Source
Rocky Mountain J. Math., Volume 45, Number 6 (2015), 1985-2022.

Dates
First available in Project Euclid: 14 March 2016

https://projecteuclid.org/euclid.rmjm/1457960344

Digital Object Identifier
doi:10.1216/RMJ-2015-45-6-1985

Mathematical Reviews number (MathSciNet)
MR3473164

Zentralblatt MATH identifier
1350.11095

#### Citation

Rozenhart, P.; Jr, M.J. Jacobson,; Scheidler, R. Computing quadratic function fields with high 3-rank via cubic field tabulation. Rocky Mountain J. Math. 45 (2015), no. 6, 1985--2022. doi:10.1216/RMJ-2015-45-6-1985. https://projecteuclid.org/euclid.rmjm/1457960344

#### References

• Jeffrey D. Achter, The distribution of class groups of function fields, J. Pure Appl. Alg. 204 (2006), 316–333.
• ––––, Results of Cohen-Lenstra type for quadratic function fields, Contemp. Math. 463, American Mathermatical Society, Providence, RI, 2008.
• Emil Artin, Quadratische Körper im Gebiete der höheren Kongruenzen I, Math. Z. 19 (1924), 153–206.
• Mark Bauer, Michael J. Jacobson, Jr., Yoonjin Lee and Renate Scheidler, Construction of hyperelliptic function fields of high three-rank, Math. Comp. 77 (2008), 503–530.
• Karim Belabas, On quadratic fields with large $3$-rank, Math. Comp. 73 (2004), 2061–2074.
• Lisa Berger, Jing-Long Hoelscher, Yoonjin Lee, Jennifer Paulhus and Renate Scheidler, The $l$-rank structure of a global function field, Fields Inst. Comm. 60, American Mathematical Society, 2011.
• Bryden Cais, Jordan S. Ellenberg and David Zureick-Brown, Random Dieudonné modules, random $p$-divisible groups, and random curves over finite fields, J. Inst. Math. Jussieu 12 (2013), 651–676.
• Henri Cohen and Hendrik W. Lenstra, Jr., Heuristics on class groups, Lect. Notes Math. 1052, Springer, Berlin, 1984.
• ––––, Heuristics on class groups of number fields, Lect. Notes Math. 1068, Springer, Berlin, 1984.
• Francisco Diaz y Diaz, On some families of imaginary quadratic fields, Math. Comp. 32 (1978), 637–650.
• Iwan Duursma, Class numbers for some hyperelliptic curves, de Gruyter, Berlin, 1996.
• Jordan Ellenberg, Akshay Venkatesh and Craig Westerland, Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, available at http://arxiv.org/abs/0912.0325, 2009.
• Ke Qin Feng and Shu Ling Sun, On class number of quadratic function fields, World Scientific Publishers, Teaneck, NJ, 1990.
• Eduardo Friedman and Lawrence C. Washington, On the distribution of divisor class groups of curves over a finite field, de Gruyter, Berlin, 1989.
• Christian Friesen, Class group frequencies of real quadratic function fields: the degree $4$ case, Math. Comp. 69 (2000), 1213–1228.
• D. Garton, Random matrices, the Cohen-Lenstra heuristics, and roots of unity, Alg. Num. Theor. 9 (2015), 149–-171.
• Derek Garton, private communication, 2010. ––––, Random matrices and the Cohen-Lenstra statistics for global fields with roots of unity, 2012, available at http://gradworks.umi.com/35/21/3521978.html.
• Helmut Hasse, Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage, Math. Z. 31 (1930), 565–582.
• M.J. Jacobson, Jr., Y. Lee, R. Scheidler and H.C. Williams, Construction of all cubic function fields of a given square-free discriminant, Inter. J. Num. Theor. 11 (2015), 1839–1885.
• Michael J. Jacobson, Jr., Shantha Ramachandran and Hugh C. Williams, Numerical results on class groups of imaginary quadratic fields, Lect. Notes Comp. Sci. 4076, Springer, Berlin, 2006.
• Donald E. Knuth, The art of computer programming, Volume 2, third Addison-Wesley Publishing Co., Reading, Mass., 1998.
• Yoonjin Lee, Cohen-Lenstra heuristics and the Spiegelungssatz; function fields, J. Num. Theor. 106 (2004), 187–199.
• ––––, The Scholz theorem in function fields, J. Num. Theor. 122 (2007), 408–414.
• Gunter Malle, Cohen-Lenstra heuristic and roots of unity, J. Num. Theor. 128 (2008), 2823–2835.
• ––––, On the distribution of class groups of number fields, Exper. Math. 19 (2010), 465–474.
• Michael E. Pohst, On computing non-Galois cubic global function fields of prescribed discriminant in characteristic $>3$, Publ. Math. Debr. 79 (2011), 611–621.
• Pieter Rozenhart, Fast tabulation of cubic function fields, Ph.D. thesis, University of Calgary, 2009.
• Pieter Rozenhart, Michael J. Jacobson, Jr. and Renate Scheidler, Tabulation of cubic function fields via polynomial binary cubic forms, Math. Comp. 81 (2012), 2335–2359.
• Pieter Rozenhart and Renate Scheidler, Tabulation of cubic function fields with imaginary and unusual Hessian, Lect. Notes Comp. Sci. 5011, Springer, 2008.
• Victor Shoup, Number theory library $($NTL$)$, version 5.5.2, 2010, http://www.shoup.net/ntl.
• Herman te Riele and Hugh Williams, New computations concerning the Cohen-Lenstra heuristics, Exper. Math. 12 (2003), 99–113.
• Akshay Venkatesh and Jordan S. Ellenberg, Statistics of number fields and function fields, Volume II, Hindustan Book Agency, New Delhi, 2010.
• Colin Weir, Renate Scheidler and Everett W. Howe, Constructing and tabulating dihedral function fields, Mathematical Science Publishers, Berkeley, CA, 2013.