Hostname: page-component-7dd5485656-wlg5v Total loading time: 0 Render date: 2025-10-31T22:27:21.967Z Has data issue: false hasContentIssue false
  • English
  • Français

Some applications of modular units

Part of: Discontinuous groups and automorphic forms Arithmetic algebraic geometry

Published online by Cambridge University Press:  03 June 2015

Ick Sun Eum ,
Ja Kyung Koo  and
Dong Hwa Shin
Ick Sun Eum
Affiliation:
Department of Mathematical Sciences, KAIST, Daejeon 305–701, Republic of Korea (zandc@kaist.ac.kr; jkkoo@math.kaist.ac.kr)
Ja Kyung Koo
Affiliation:
Department of Mathematical Sciences, KAIST, Daejeon 305–701, Republic of Korea (zandc@kaist.ac.kr; jkkoo@math.kaist.ac.kr)
Dong Hwa Shin
Affiliation:
Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si, Gyeonggi-do 449–791, Republic of Korea (dhshin@hufs.ac.kr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. Furthermore, we find a necessary and sufficient condition for a meromorphic Siegel modular function of degree g to have neither a zero nor a pole on a certain subset of the Siegel upper half-space .

Keywords

modular unitsmodular functionsSiegel modular forms

MSC classification

Secondary: 11G16: Elliptic and modular units 11F03: Modular and automorphic functions 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 1 , February 2016 , pp. 91 - 106
Copyright
Copyright © Edinburgh Mathematical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1. Bruinier, J. H., van der Geer, G., Harder, G. and Zagier, D., The 1-2-3 of modular forms, in Lectures from the Summer School on Modular Forms and Their Applications, Nordfjordeid, Norway, Universitext (Springer, 2008).Google Scholar
2. Diamond, F. and Shurman, J., A first course in modular forms, Graduate Texts in Mathematics, Volume 228 (Springer, 2005).Google Scholar
3. Fine, N. J., Basic hypergeometric series and applications, Mathematical Surveys and Monographs, Volume 27 (American Mathematical Society, Providence, RI, 1988).Google Scholar
4. Igusa, J.-I., On the graded ring of theta-constants, II, Am. J. Math. 88(1) (1966), 221–236.Google Scholar
5. Kim, C. H. and Koo, J. K., Arithmetic of the modular function j 1,4 , Acta Arith. 84(2) (1998), 129–143.Google Scholar
6. Koo, J. K. and Shin, D. H., On some arithmetic properties of Siegel functions, Math. Z. 264(1) (2010), 137–177.Google Scholar
7. Kubert, D. and Lang, S., Modular units, Grundlehren der mathematischen Wissenschaften, Volume 244 (Spinger, 1981).Google Scholar
8. Lang, S., Elliptic functions, 2nd edn, Graduate Texts in Mathematics, Volume 112 (Spinger, 1987).CrossRefGoogle Scholar
9. Shimura, G., Introduction to the arithmetic theory of automorphic functions (Iwanami Shoten/Princeton University Press, 1971).Google Scholar
10. Shimura, G., Theta functions with complex multiplication, Duke Math. J. 43(4) (1976), 673–696.CrossRefGoogle Scholar
11. Silverman, J. H., The arithmetic of elliptic curves, Graduate Texts in Mathematics, Volume 106 (Springer, 1992).Google Scholar