Class Invariants for Quartic CM Fields
Annales de l'Institut Fourier, Volume 57 (2007) no. 2, p. 457-480
One can define class invariants for a quartic primitive CM field K as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to K. Such constructions were given by de Shalit-Goren and Lauter. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct S-units in certain abelian extensions of a reflex field of K, where S is effectively determined by K, and to bound the primes appearing in the denominators of the Igusa class polynomials arising in the construction of genus 2 curves with CM, as conjectured by Lauter.
On peut définir des invariants de classe pour un corps CM quartique primitif K comme valeurs spéciales de certaines fonctions modulaires de Siegel (ou Hilbert) aux points CM associés à K. De telles constructions ont été décrites par de Shalit-Goren et Lauter. Nous donnons des bornes explicites pour les idéaux premiers divisant les dénominateurs de ces nombres algébriques. Cela nous permet, en particulier, de construire des S-unités dans certaines extensions abéliennes d’un corps réflexe de K, où S est explicitement determiné par K, et de borner les nombres premiers apparaissant aux dénominateurs des polynômes de classe d’Igusa qui interviennent dans la construction des courbes CM de genre 2, comme dans la conjecture de Lauter.
DOI : https://doi.org/10.5802/aif.2264
Classification:  11G15,  11G16,  11G18,  11R27
Keywords: Class invariant, modular form, complex multiplication, polarization, superspecial abelian variety, units, Igusa invariants, quaternion algebra
@article{AIF_2007__57_2_457_0,
     author = {Goren, Eyal Z. and Lauter, Kristin E.},
     title = {Class Invariants for Quartic CM Fields},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {2},
     year = {2007},
     pages = {457-480},
     doi = {10.5802/aif.2264},
     zbl = {1172.11018},
     mrnumber = {2310947},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2007__57_2_457_0}
}
Class Invariants for Quartic CM Fields. Annales de l'Institut Fourier, Volume 57 (2007) no. 2, pp. 457-480. doi : 10.5802/aif.2264. https://aif.centre-mersenne.org/item/AIF_2007__57_2_457_0/

[1] Bruinier, Jan Hendrik; Yang, Tonghai CM-values of Hilbert modular functions, Invent. Math., Tome 163 (2006) no. 2, pp. 229-288 | Article | MR 2207018 | Zbl 05013725

[2] Deligne, Pierre; Pappas, Georgios Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, Compositio Math., Tome 90 (1994) no. 1, pp. 59-79 | Numdam | MR 1266495 | Zbl 0826.14027

[3] Dokchitser, T. Deformations of p -divisible groups and p -descent on elliptic curves, Utrecht (2000) (Masters thesis)

[4] Dorman, David R. Singular moduli, modular polynomials, and the index of the closure of Z[j(τ)] in Q(j(τ)), Math. Ann., Tome 283 (1989) no. 2, pp. 177-191 | Article | MR 980592 | Zbl 0642.12014

[5] Eisenträger, A. K.; Lauter, K. E. A CRT algorithm for constructing genus 2 curves over finite fields (to appear in Proceedings of Arithmetic, Geometry and Coding Theory (AGCT 2005))

[6] Faltings, Gerd; Chai, Ching-Li Degeneration of abelian varieties, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 22 (1990) (With an appendix by David Mumford) | MR 1083353 | Zbl 0744.14031

[7] Goren, E. Z.; Lauter, K. E. Evil primes and superspecial moduli, International Mathematics Research Notices, Tome 2006 (2006), p. 1-19, Article ID 53864 | MR 2250004 | Zbl 05122738

[8] Goren, Eyal Z. On certain reduction problems concerning abelian surfaces, Manuscripta Math., Tome 94 (1997) no. 1, pp. 33-43 | Article | MR 1468933 | Zbl 0924.14023

[9] Gross, Benedict H.; Zagier, Don B. On singular moduli, J. Reine Angew. Math., Tome 355 (1985), pp. 191-220 | MR 772491 | Zbl 0545.10015

[10] Ibukiyama, Tomoyoshi; Katsura, Toshiyuki; Oort, Frans Supersingular curves of genus two and class numbers, Compositio Math., Tome 57 (1986) no. 2, pp. 127-152 | Numdam | MR 827350 | Zbl 0589.14028

[11] Igusa, Jun-Ichi Arithmetic variety of moduli for genus two, Ann. of Math. (2), Tome 72 (1960), pp. 612-649 | Article | MR 114819 | Zbl 0122.39002

[12] Igusa, Jun-Ichi On Siegel modular forms of genus two, I, Amer. J. Math., Tome 84 (1962), pp. 175-200 | Article | MR 141643 | Zbl 0133.33301

[13] Igusa, Jun-Ichi On Siegel modular forms of genus two, II, Amer. J. Math., Tome 86 (1964), pp. 392-412 | Article | MR 168805 | Zbl 0133.33301

[14] Igusa, Jun-Ichi Modular forms and projective invariants, Amer. J. Math., Tome 89 (1967), pp. 817-855 | Article | MR 229643 | Zbl 0159.50401

[15] Kottwitz, Robert E. Points on some Shimura varieties over finite fields, J. Amer. Math. Soc., Tome 5 (1992) no. 2, pp. 373-444 | Article | MR 1124982 | Zbl 0796.14014

[16] Lang, S. Complex multiplication, Springer-Verlag, New York, Grundlehren der Mathematischen Wissenschaften, Tome 255 (1983) | MR 713612 | Zbl 0536.14029

[17] Lauter, K. E. Primes in the denominators of Igusa class polynomials (2003) (Preprint, Available from http://www.arxiv.org/abs/math.NT/0301240)

[18] Liu, Qing Courbes stables de genre 2 et leur schéma de modules, Math. Ann., Tome 295 (1993) no. 2, pp. 201-222 | Article | MR 1202389 | Zbl 0819.14010

[19] Mumford, David Abelian varieties, Published for the Tata Institute of Fundamental Research, Bombay, Tata Institute of Fundamental Research Studies in Mathematics, No. 5 (1970) | MR 282985 | Zbl 0223.14022

[20] Oort, Frans Finite group schemes, local moduli for abelian varieties, and lifting problems, Compositio Math., Tome 23 (1971), pp. 265-296 | Numdam | MR 301026 | Zbl 0223.14024

[21] Pizer, Arnold An algorithm for computing modular forms on Γ 0 (N), J. Algebra, Tome 64 (1980) no. 2, pp. 340-390 | Article | MR 579066 | Zbl 0433.10012

[22] Rapoport, M. Compactifications de l’espace de modules de Hilbert-Blumenthal, Compositio Math., Tome 36 (1978) no. 3, pp. 255-335 | Numdam | MR 515050 | Zbl 0386.14006

[23] Rodriguez-Villegas, Fernando Explicit models of genus 2 curves with split CM, Algorithmic number theory (Leiden, 2000), Springer, Berlin (Lecture Notes in Comput. Sci.) Tome 1838 (2000), pp. 505-513 | MR 1850629 | Zbl 1032.11026

[24] De Shalit, E.; Goren, E. Z. On special values of theta functions of genus two, Ann. Inst. Fourier (Grenoble), Tome 47 (1997) no. 3, pp. 775-799 | Article | Numdam | MR 1465786 | Zbl 0974.11027

[25] Shimura, Goro; Taniyama, Yutaka Complex multiplication of abelian varieties and its applications to number theory, The Mathematical Society of Japan, Tokyo, Publications of the Mathematical Society of Japan, Tome 6 (1961) | MR 125113 | Zbl 0112.03502

[26] Spallek, A.-M. Kurven vom Geschlecht 2 und ihre Anwendung in Public-Key-Kryptosystemen, Universität Gesamthochschule Essen (1994) (Ph. D. Thesis) | Zbl 0974.11501

[27] Spearman, B. K.; Williams, K. S. Relative integral bases for quartic fields over quadratic subfields, Acta Math. Hungar., Tome 70 (1996) no. 3, pp. 185-192 | Article | MR 1374384 | Zbl 0853.11090

[28] Vallières, D. Class Invariants, McGill (2005) (Masters thesis)

[29] Vignéras, Marie-France Arithmétique des algèbres de quaternions, Springer, Berlin, Lecture Notes in Mathematics, Tome 800 (1980) | MR 580949 | Zbl 0422.12008

[30] Van Wamelen, Paul Examples of genus two CM curves defined over the rationals, Math. Comp., Tome 68 (1999) no. 225, pp. 307-320 | Article | MR 1609658 | Zbl 0906.14025

[31] Weil, André Zum Beweis des Torellischen Satzes, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa., Tome 1957 (1957), pp. 33-53 | MR 89483 | Zbl 0079.37002

[32] Weng, Annegret Constructing hyperelliptic curves of genus 2 suitable for cryptography, Math. Comp., Tome 72 (2003) no. 241, p. 435-458 (electronic) | Article | MR 1933830 | Zbl 1013.11023