## Computing quadratic function fields with high 3-rank via cubic field tabulation.(English)Zbl 1350.11095

This paper is continuation of previous work of the authors [Math. Comput. 81, No. 280, 2335–2359 (2012; Zbl 1290.11154)] and of the PhD Thesis of the first author [Fast tabulation of cubic function fields. Calgary: University of Calgary (PhD Thesis) (2009)]. Let $$D$$ be a square-free polynomial in $${\mathbb F}_q [t]$$ of positive degree. A polynomial $$F\in {\mathbb F}_q[t]$$ is called imaginary if $$\deg (F)$$ is odd, unusual if $$\deg(F)$$ is even and $$\text{sgn}(F)$$ (the leading coefficient of $$F$$) is a non-square in $${\mathbb F}^{\ast}_q$$ and real if $$\deg(F)$$ is even and $$\text{sgn}(F)$$ is a square in $${\mathbb F}^{\ast}_q$$. The main purpose of this paper is to obtain extensive numerical data on quadratic function fields of discriminant $$D$$ with bounded degree, where $$-3D$$ is imaginary or unusual, and non-zero $$3$$-rank. The algorithm is given in Section 3 (Algorithm 3.1). The rapid tabulation of all cubic function fields with bounded discriminant degree is key to this approach. In Section 4, the improved algorithm for quadratic function fields where $$-3D$$ is imaginary is presented (Algorithm 4.4). Its complexity is studied in Section 5.
The Friedman-Washington heuristics [E. Friedman and L. C. Washington, in: Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 227–239 (1989; Zbl 0693.12013)] attempt to explain statistical observations about divisor class groups of quadratic function fields. The data produced in this paper yields evidence for the Friedman-Washington heuristics for $$q=5,11$$ but for $$q=7,13$$ the data does not agree closely with the Friedman-Washington heuristics. This is studied in the last three sections.

### MSC:

 11R11 Quadratic extensions 11R58 Arithmetic theory of algebraic function fields 11R65 Class groups and Picard groups of orders 11Y40 Algebraic number theory computations

### Keywords:

quadratic function field; ideal class group; three rank

### Citations:

Zbl 1290.11154; Zbl 0693.12013

NTL
Full Text:

### References:

 [1] Jeffrey D. Achter, The distribution of class groups of function fields , J. Pure Appl. Alg. 204 (2006), 316-333. · Zbl 1134.11042 [2] —-, Results of Cohen-Lenstra type for quadratic function fields , Contemp. Math. 463 , American Mathermatical Society, Providence, RI, 2008. · Zbl 1166.11018 [3] Emil Artin, Quadratische Körper im Gebiete der höheren Kongruenzen I, Math. Z. 19 (1924), 153-206. · JFM 50.0107.01 [4] 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. · Zbl 1131.11073 [5] Karim Belabas, On quadratic fields with large $$3$$-rank , Math. Comp. 73 (2004), 2061-2074. · Zbl 1051.11055 [6] 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. · Zbl 1247.11143 [7] 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. · Zbl 1284.14055 [8] Henri Cohen and Hendrik W. Lenstra, Jr., Heuristics on class groups , Lect. Notes Math. 1052 , Springer, Berlin, 1984. [9] —-, Heuristics on class groups of number fields , Lect. Notes Math. 1068 , Springer, Berlin, 1984. · Zbl 0558.12002 [10] Francisco Diaz y Diaz, On some families of imaginary quadratic fields , Math. Comp. 32 (1978), 637-650. · Zbl 0383.12005 [11] Iwan Duursma, Class numbers for some hyperelliptic curves , de Gruyter, Berlin, 1996. · Zbl 0901.11020 [12] 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. · Zbl 1342.14055 [13] Ke Qin Feng and Shu Ling Sun, On class number of quadratic function fields , World Scientific Publishers, Teaneck, NJ, 1990. · Zbl 0924.11089 [14] Eduardo Friedman and Lawrence C. Washington, On the distribution of divisor class groups of curves over a finite field , de Gruyter, Berlin, 1989. · Zbl 0693.12013 [15] Christian Friesen, Class group frequencies of real quadratic function fields: the degree $$4$$ case , Math. Comp. 69 (2000), 1213-1228. · Zbl 1042.11071 [16] D. Garton, Random matrices, the Cohen-Lenstra heuristics, and roots of unity , Alg. Num. Theor. 9 (2015), 149–171. · Zbl 1326.11068 [17] 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. · Zbl 1326.11068 [18] Helmut Hasse, Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage , Math. Z. 31 (1930), 565-582. · JFM 56.0167.02 [19] 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. · Zbl 1390.11134 [20] 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. · Zbl 1143.11370 [21] Donald E. Knuth, The art of computer programming , Volume 2, third Addison-Wesley Publishing Co., Reading, Mass., 1998. · Zbl 0895.68054 [22] Yoonjin Lee, Cohen-Lenstra heuristics and the Spiegelungssatz; function fields , J. Num. Theor. 106 (2004), 187-199. · Zbl 1049.11121 [23] —-, The Scholz theorem in function fields , J. Num. Theor. 122 (2007), 408-414. · Zbl 1155.11052 [24] Gunter Malle, Cohen-Lenstra heuristic and roots of unity , J. Num. Theor. 128 (2008), 2823-2835. · Zbl 1225.11143 [25] —-, On the distribution of class groups of number fields , Exper. Math. 19 (2010), 465-474. · Zbl 1297.11139 [26] Michael E. Pohst, On computing non-Galois cubic global function fields of prescribed discriminant in characteristic $$>3$$ , Publ. Math. Debr. 79 (2011), 611-621. · Zbl 1249.11112 [27] Pieter Rozenhart, Fast tabulation of cubic function fields , Ph.D. thesis, University of Calgary, 2009. · Zbl 1232.11129 [28] 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. · Zbl 1290.11154 [29] Pieter Rozenhart and Renate Scheidler, Tabulation of cubic function fields with imaginary and unusual Hessian , Lect. Notes Comp. Sci. 5011 , Springer, 2008. · Zbl 1232.11129 [30] Victor Shoup, Number t heory library $$($$NTL$$)$$, version 5.5.2, 2010, http://www.shoup.net/ntl. [31] Herman te Riele and Hugh Williams, New computations concerning the Cohen-Lenstra heuristics , Exper. Math. 12 (2003), 99-113. · Zbl 1050.11096 [32] Akshay Venkatesh and Jordan S. Ellenberg, Statistics of number fields and function fields , Volume II, Hindustan Book Agency, New Delhi, 2010. · Zbl 1259.11106 [33] Colin Weir, Renate Scheidler and Everett W. Howe, Constructing and tabulating dihedral function fields , Mathematical Science Publishers, Berkeley, CA, 2013. · Zbl 1345.11079
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.