Examples of genus two CM curves defined over the rationals

van Wamelen, Paul

doi:10.1090/S0025-5718-99-01020-0

Examples of genus two CM curves defined over the rationals
HTML articles powered by AMS MathViewer

by Paul van Wamelen PDF

Math. Comp. 68 (1999), 307-320 Request permission

Abstract:

We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the well-known example $y^2 = x^5 - 1$ we find 19 non-isomorphic such curves. We believe that these are the only such curves.

References

Erhard Gottschling, Explizite Bestimmung der Randflächen des Fundamentalbereiches der Modulgruppe zweiten Grades, Math. Ann. 138 (1959), 103–124 (German). MR 107020, DOI 10.1007/BF01342938
Kenneth Hardy, Richard H. Hudson, David Richman, and Kenneth S. Williams, Determination of all imaginary cyclic quartic fields with class number $2$, Trans. Amer. Math. Soc. 311 (1989), no. 1, 1–55. MR 929663, DOI 10.1090/S0002-9947-1989-0929663-9
Jun-ichi Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. (2) 72 (1960), 612–649. MR 114819, DOI 10.2307/1970233
Alfred Rosenblatt, Sur les points singuliers des équations différentielles, C. R. Acad. Sci. Paris 209 (1939), 10–11 (French). MR 85
Herbert Lange and Christina Birkenhake, Complex abelian varieties, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992. MR 1217487, DOI 10.1007/978-3-662-02788-2
Stéphane Louboutin and Ryotaro Okazaki, Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one, Acta Arith. 67 (1994), no. 1, 47–62. MR 1292520, DOI 10.4064/aa-67-1-47-62
Jean-François Mestre, Construction de courbes de genre $2$ à partir de leurs modules, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 313–334 (French). MR 1106431
David Mumford, Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. MR 742776, DOI 10.1007/978-0-8176-4578-6
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216, DOI 10.1007/978-1-4684-9884-4
Bennett Setzer, The determination of all imaginary, quartic, abelian number fields with class number $1$, Math. Comp. 35 (1980), no. 152, 1383–1386. MR 583516, DOI 10.1090/S0025-5718-1980-0583516-2
Goro Shimura and Yutaka Taniyama, Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, Mathematical Society of Japan, Tokyo, 1961. MR 0125113
A.-M. Spallek. Kurven vom geschlecht 2 und ihre anwendung in public-key-kryptosystemen. Preprint 18, Universität GH Essen, Ellernstraße 29, 45326 Essen, Germany, 1994.
P. B. van Wamelen. Equations for the algebraic Jacobian of a hyperelliptic curve. Submitted to Trans. Amer. Math. Soc.
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2