no. 3
On special values of theta functions of genus two
Annales de l'Institut Fourier, Tome 47 (1997) no. 3, pp. 775-799.

Nous étudions un certain sous-groupe multiplicatif de type fini du corps de classes d’un corps quartique de type CM. Il est constitué de valeurs spéciales de certaines fonctions thêta de genre deux, et il est l’analogue du groupe des unités de Siegel. Les questions d’intégralité de ces valeurs spéciales sont reliées à l’arithmétique de l’espace des modules de Siegel de genre deux.

We study a certain finitely generated multiplicative subgroup of the Hilbert class field of a quartic CM field. It consists of special values of certain theta functions of genus 2 and is analogous to the group of Siegel units. Questions of integrality of these specials values are related to the arithmetic of the Siegel moduli space.

@article{AIF_1997__47_3_775_0,
     author = {Shalit, Ehud De and Goren, Eyal Z.},
     title = {On special values of theta functions of genus two},
     journal = {Annales de l'Institut Fourier},
     pages = {775--799},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {3},
     year = {1997},
     doi = {10.5802/aif.1580},
     mrnumber = {98g:11071},
     zbl = {0974.11027},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1580/}
}
TY  - JOUR
AU  - Shalit, Ehud De
AU  - Goren, Eyal Z.
TI  - On special values of theta functions of genus two
JO  - Annales de l'Institut Fourier
PY  - 1997
SP  - 775
EP  - 799
VL  - 47
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1580/
DO  - 10.5802/aif.1580
LA  - en
ID  - AIF_1997__47_3_775_0
ER  - 
%0 Journal Article
%A Shalit, Ehud De
%A Goren, Eyal Z.
%T On special values of theta functions of genus two
%J Annales de l'Institut Fourier
%D 1997
%P 775-799
%V 47
%N 3
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1580/
%R 10.5802/aif.1580
%G en
%F AIF_1997__47_3_775_0
Shalit, Ehud De; Goren, Eyal Z. On special values of theta functions of genus two. Annales de l'Institut Fourier, Tome 47 (1997) no. 3, pp. 775-799. doi : 10.5802/aif.1580. http://www.numdam.org/articles/10.5802/aif.1580/

[Ek] T. Ekedhal, On Supersingular Curves and Abelian Varieties, Math. Scand. 60 (1987), 151-178. | MR | Zbl

[FC] G. Faltings, C.-L. Chai, Degeneration of Abelian Varieties, Springer-Verlag, Berlin-Heidelberg, 1990. | MR | Zbl

[G] Eyal Z. Goren, Ph. D. Thesis, Hebrew University of Jerusalem (1996).

[Ig] J.I. Igusa, On Siegel Modular Forms of Genus Two (II), Am. J. Math., 86 (1964), 392-412. | MR | Zbl

[KL] D. Kubert, S. Lang, Modular Units, Springer-Verlag, Berlin-Heidelberg-New York, 1981. | MR | Zbl

[L] S. Lang, Elliptic Functions, Addison-Wesley, Reading, 1973. | MR | Zbl

[Oo] F. Oort, Which Abelian Surfaces are Products of Elliptic Curves? Math. Ann., 214, 1975, 35-47. | MR | Zbl

[Ra] K. Ramachandra, Some Applications of Kronecker's Limit Formulas, Ann. Math., 80 (1964), 104-148. | MR | Zbl

[Ro] G. Robert, Unités Elliptiques, Bull. Soc. Math. France, Mémoire, 36 (1973). | Numdam | Zbl

[ShTa] G. Shimura, Y. Taniyama, Complex Multiplication of Abelian Varieties and its Applications to Number Theory, Math. Soc. Japan (1991). | Zbl

[Sh1] G. Shimura, Theta Functions with Complex Multiplication, Duke Math. J., 43 (1976), 673-696. | MR | Zbl

[Sh2] G. Shimura, On Certain Reciprocity Laws for Theta Functions and Modular Forms, Acta Math., 141 (1978), 35-71. | MR | Zbl

[Sh3] G. Shimura, Arithmetic of Alternating Forms and Quaternion Hermitian Forms, J. Math. Soc. Japan, 15 (1963). | MR | Zbl

[Sie] C. L. Siegel, Lectures on Advanced Analytic Number Theory, Tata Institute for Fundamental Research (1961).

[Ta] J. Tate, Les Conjectures de Stark sur les Fonctions L d'Artin en s=0, Progress in Math. vol. 47, Birkhauser (1984). | Zbl

[vdG] G. Van Der Geer, Hilbert Modular Surfaces, Springer-Verlag, Berlin-Heidelberg-New York, 1988. | MR | Zbl

[Wa] L. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, 1982. | MR | Zbl

Cité par Sources :