Mathematics of Computation

Author(s): M. Bauer; M. J. Jacobson Jr.; Y. Lee; R. Scheidler.
Journal: Math. Comp. 77 (2008), 503-530.
MSC (2000): Primary 11R11; Secondary 11R65, 11Y16, 11Y40, 14H05, 14H40
Posted: July 26, 2007
Retrieve article in: PDF DVI PostScript

Abstract: We present several explicit constructions of hyperelliptic function fields whose Jacobian or ideal class group has large

-rank. Our focus is on finding examples for which the genus and the base field are as small as possible. Most of our methods are adapted from analogous techniques used for generating quadratic number fields whose ideal class groups have high

-rank, but one method, applicable to finding large

-ranks for odd primes $l \geq 3,$ is new and unique to function fields. Algorithms, examples, and numerical data are included.

1.: J. Achter, The distribution of class groups of function fields. J. Pure Appl. Algebra 204 (2006), 316-333. MR 2184814 (2006h:11132)
2.: E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen. Math. Zeitschrift 19 (1924), 153-206. MR 1544651
3.: M. L. Bauer, M. J. Jacobson, Jr., Y. Lee and R. Scheidler, Construction of Hyperelliptic Function Fields of High Three-Rank. University of Calgary Yellow Series 849. Available at www.math.ucalgary.ca/files/publications/3443849.pdf.
4.: K. Belabas, On quadratic fields with large -rank. Math. Comp. 73 (2004), 2061-2074. MR 2059751 (2005c:11132)
5.: J. Buchmann, M. J. Jacobson, Jr., and E. Teske, On some computational problems in finite abelian groups. Math. Comp. 66 (1997), 1663-1687. MR 1432126 (98a:11185)
6.: H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups of number fields. In Number Theory (Noordwijkerhout, 1983), Lect. Notes Math. 1068, 33-62, Springer, Berlin, 1984. MR 756082 (85j:11144)
7.: M. Craig, A type of class group for imaginary quadratic fields. Acta Arith. XXII (1973), 449-459. MR 0318098 (47:6647)
8.: M. Craig, A Construction for irregular discriminants. Osaka J. Math. 14 (1977), 365-402. MR 0450226 (56:8522)
9.: H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields (ii), Proc. Roy. Soc. London A 322 (1971), 405-420. MR 0491593 (58:10816)
10.: F. Diaz y Diaz, On some families of imaginary quadratic fields. Math. Comp. 32 (1978), 637-650. MR 0485775 (58:5582)
11.: F. Diaz y Diaz, D. Shanks and H. C. Williams, Quadratic fields with -rank equal to 4. Math. Comp. 33 (1979), 836-840. MR 521299 (80i:12004)
12.: G. W.-W. Fung, Computational Problems in Complex Cubic Fields. Doctoral Dissertation, University of Manitoba, 1990.
13.: E. Friedman and L. C. Washington, On the distribution of divisor class groups of curves over a finite field. In Théorie des Nombres (Québec, PQ, 1987), 227-239, de Gruyter, Berlin, 1989. MR 1024565 (91e:11138)
14.: H. Hasse, Arithmetische Theorie der kubischen Einheiten. Math. Zeitschrift 31 (1930), 565-582. MR 1545136
15.: E. W. Howe, F. Leprévost, and B. Poonen, Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Math. 12 (2000), 315-364. MR 1748483 (2001e:11071)
16.: T. W. Hungerford, Algebra. Springer-Verlag, New York 1974. MR 600654 (82a:00006)
17.: Y. Lee, Cohen-Lenstra heuristics and the Spiegelungssatz: function fields. J. Number Theory 106 (2004), 187-199. MR 2059070 (2005b:11177)
18.: Y. Lee, The Scholz theorem in function fields. J. Number Theory 122 (2007), 408-414. MR 2059070 (2005b:11177)
19.: Y. Lee and A. M. Pacelli, Class groups of imaginary function fields: the inert case. Proc. Amer. Math. Soc. 133 (2005), 2883-2889. MR 2159765 (2006e:11177)
20.: P. Llorente, Cubic fields and class fields of real quadratic fields (in Spanish). Publ. Sec. Mat. Univ. Autònoma Barcelona 26 (1982), 93-109. MR 768364 (85j:11138)
21.: P. Llorente and J. Quer, On the 3-Sylow subgroup of the class group of quadratic fields. Math. Comp. 50 (1988), 321-333. MR 917838 (89b:11083)
22.: J.-F. Mestre, Courbes elliptiques et groupes de classes d'ideaux de certain corps quadratiques. J. Reine Angew. Math. 343 (1983), 23-25. MR 705875 (84m:12004)
23.: J. S. Milne, Abelian Varieties. In Arithmetic Geometry (G. Cornell and J. Silverman, eds.), Springer-Verlag, New York, 1986. MR 861974
24.: L. J. Mordell, Diophantine Equations. Pure Appl. Math. Ser. 30, Academic Press 1969. MR 0249355 (40:2600)
25.: A. M. Pacelli, Abelian subgroups of any order in class groups of global function fields. J. Number Theory 106 (2004), 26-49. MR 2029780 (2004m:11193)
26.: D. Glass and R. Pries, Hyperelliptic curves with prescribed -torsion. Manuscripta Math. 117 (2005), 299-317. MR 2154252 (2006e:14039)
27.: J. Quer, Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 215-218. MR 907945 (88j:11074)
28.: M. Rosen, Number Theory in Function Fields. Springer 2002. MR 1876657 (2003d:11171)
29.: A. Scholz, Über die Beziehung der Klassenzahlen quadratischer Körper zueinander. J. Reine Angew. Math. 166 (1932), 201-203.
30.: D. Shanks, New types of quadratic fields having three invariants divisible by 3. J. Number Theory 4 (1972), 537-556. MR 0313220 (47:1775)
31.: D. Shanks, Class groups of the quadratic fields found by F. Diaz y Diaz. Math. Comp. 30 (1976), 173-178. MR 0399039 (53:2890)
32.: D. Shanks, Determining all cubic fields having a given fundamental discriminant. Unpublished manuscript.
33.: D. Shanks and R. Serafin, Quadratic fields with four invariants divisible by 3. Math. Comp. 27 (1973), 183-187. MR 0330097 (48:8436a)
34.: D. Shanks and P. Weinberger, A quadratic field of prime discriminant requiring three generators for its class group, and related theory. Acta Arith. XXI (1972), 71-87. MR 0309899 (46:9003)
35.: V. Shoup, NTL: A library for doing number theory. Software, 2001. Available from http://www.shoup.net/ntl.
36.: A. Stein and E. Teske, Explicit bounds and heuristics on class numbers in hyperelliptic function fields. Math. Comp. 71 (2002), 837-861. MR 1885633 (2002k:11210)
37.: A. Weil, Variétés Abéliennes de Courbes Algébriques. Hermann, Paris 1948. MR 0029522 (10:621d)
38.: Y. Yamamoto, On unramified Galois extensions of quadratic number fields. Osaka J. Math. 7 (1970), pp. 57-76. MR 0266898 (42:1800)
39.: N. Yui, On the Jacobian varieties of hyperelliptic curves over fields of characteristic J. Algebra 52 (1978), 378-410. MR 0491717 (58:10920)
40.: X.-K. Zhang, Algebraic function fields of type $(2,2,\cdots,2).$ Sci. Sinica Ser. A 31 (1988), 521-530. MR 962269 (90a:11143)

M. Bauer
Affiliation: Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
Email: mbauer@math.ucalgary.ca

M. J. Jacobson Jr.
Affiliation: Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
Email: jacobs@cpsc.ucalgary.ca

Y. Lee
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada, V5A 1S6
Email: yoonjinl@sfu.ca

R. Scheidler
Affiliation: Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
Email: rscheidl@math.ucalgary.ca

DOI: 10.1090/S0025-5718-07-02001-7
PII: S 0025-5718(07)02001-7
Keywords: Hyperelliptic function field, ideal class group, Jacobian, 3-rank
Received by editor(s): July 26, 2005
Received by editor(s) in revised form: November 8, 2006
Posted: July 26, 2007
Additional Notes: The first, second, and fourth authors were supported by NSERC of Canada
The third author was supported by an AWM-NSF Mentoring Grant
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS