Class Invariants for Quartic CM Fields

Goren, Eyal Z.; Lauter, Kristin E.

Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 457-480.

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.

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.

Classification : 11G15, 11G16, 11G18, 11R27
Keywords: Class invariant, modular form, complex multiplication, polarization, superspecial abelian variety, units, Igusa invariants, quaternion algebra
Mot clés : invariant de classe, forme modulaire, multiplication complexe, polarisation, variété abélienne superspéciale, unités, invariants d’Igusa, algèbre de quaternions

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

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

[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 $\mathbf{Z}\left[j\right(\tau \left)\right]$ in $\mathbf{Q}\left(j\right(\tau \left)\right)$, Math. Ann., Volume 283 (1989) no. 2, pp. 177-191 | DOI | MR | Zbl

[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, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 22, Springer-Verlag, Berlin, 1990 (With an appendix by David Mumford) | MR | Zbl

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

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

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

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

[11] Igusa, Jun-ichi Arithmetic variety of moduli for genus two, Ann. of Math. (2), Volume 72 (1960), pp. 612-649 | DOI | MR | Zbl

[12] Igusa, Jun-ichi On Siegel modular forms of genus two, I, Amer. J. Math., Volume 84 (1962), pp. 175-200 | DOI | MR | Zbl

[13] Igusa, Jun-ichi On Siegel modular forms of genus two, II, Amer. J. Math., Volume 86 (1964), pp. 392-412 | DOI | MR | Zbl

[14] Igusa, Jun-ichi Modular forms and projective invariants, Amer. J. Math., Volume 89 (1967), pp. 817-855 | DOI | MR | Zbl

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

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

[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., Volume 295 (1993) no. 2, pp. 201-222 | DOI | MR | Zbl

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

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

[21] Pizer, Arnold An algorithm for computing modular forms on ${\Gamma }_0\left(N\right)$, J. Algebra, Volume 64 (1980) no. 2, pp. 340-390 | DOI | MR | Zbl

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

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

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

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

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

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

[28] Vallières, D. Class Invariants, McGill, October (2005) (Masters thesis Available from http://www.math.mcgill.ca/goren/Students/students.html)

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

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

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

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