de Shalit, Ehud; Goren, Eyal Z. On special values of theta functions of genus two. (English) Zbl 0974.11027 Ann. Inst. Fourier 47, No. 3, 775-799 (1997). Summary: 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 special values are related to the arithmetic of the Siegel moduli space. Cited in 3 Documents MSC: 11G15 Complex multiplication and moduli of abelian varieties 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11R16 Cubic and quartic extensions Keywords:theta functions; complex multiplication; units PDF BibTeX XML Cite \textit{E. de Shalit} and \textit{E. Z. Goren}, Ann. Inst. Fourier 47, No. 3, 775--799 (1997; Zbl 0974.11027) Full Text: DOI Numdam EuDML References: [1] T. EKEDHAL, On supersingular curves and abelian varieties, Math. Scand. 60 (1987), 151-178. · Zbl 0641.14007 [2] G. FALTINGS, C.-L. CHAI, Degeneration of abelian varieties, Springer-Verlag, Berlin-Heidelberg, 1990. · Zbl 0744.14031 [3] Eyal Z. GOREN, Ph. D. thesis, Hebrew University of Jerusalem (1996). [4] J.I. IGUSA, On Siegel modular forms of genus two (II), Am. J. Math., 86 (1964), 392-412. · Zbl 0133.33301 [5] D. KUBERT, S. LANG, Modular units, Springer-Verlag, Berlin-Heidelberg-New York, 1981. · Zbl 0492.12002 [6] S. LANG, Elliptic functions, Addison-Wesley, Reading, 1973. · Zbl 0316.14001 [7] F. OORT, Which abelian surfaces are products of elliptic curves? Math. Ann., 214, 1975, 35-47. · Zbl 0291.14014 [8] K. RAMACHANDRA, Some applications of Kronecker’s limit formulas, Ann. Math., 80 (1964), 104-148. · Zbl 0142.29804 [9] G. ROBERT, Unités elliptiques, Bull. Soc. Math. France, Mémoire, 36 (1973). · Zbl 0314.12006 [10] G. SHIMURA, Y. TANIYAMA, Complex multiplication of abelian varieties and its applications to number theory, Math. Soc. Japan (1991). · Zbl 0112.03502 [11] G. SHIMURA, Theta functions with complex multiplication, Duke Math. J., 43 (1976), 673-696. · Zbl 0371.14022 [12] G. SHIMURA, On certain reciprocity laws for theta functions and modular forms, Acta Math., 141 (1978), 35-71. · Zbl 0402.10030 [13] G. SHIMURA, Arithmetic of alternating forms and quaternion Hermitian forms, J. Math. Soc. Japan, 15 (1963). · Zbl 0121.28102 [14] C. L. SIEGEL, Lectures on advanced analytic number theory, Tata Institute for Fundamental Research (1961). [15] J. TATE, LES conjectures de Stark sur LES fonctions L d’Artin en s=0, Progress in Math. vol. 47, Birkhauser (1984). · Zbl 0545.12009 [16] G. VAN DER GEER, Hilbert modular surfaces, Springer-Verlag, Berlin-Heidelberg-New York, 1988. · Zbl 0634.14022 [17] L. WASHINGTON, Introduction to cyclotomic fields, Springer-Verlag, 1982. · Zbl 0484.12001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.