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Algebraic integers as special values of modular units

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 November 2011

Ja Kyung Koo ,
Dong Hwa Shin  and
Dong Sung Yoon
Ja Kyung Koo
Affiliation:
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 373-1, Korea (jkkoo@math.kaist.ac.kr; shakur01@kaist.ac.kr; yds1850@kaist.ac.kr)
Dong Hwa Shin
Affiliation:
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 373-1, Korea (jkkoo@math.kaist.ac.kr; shakur01@kaist.ac.kr; yds1850@kaist.ac.kr)
Dong Sung Yoon
Affiliation:
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 373-1, Korea (jkkoo@math.kaist.ac.kr; shakur01@kaist.ac.kr; yds1850@kaist.ac.kr)
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Abstract

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Let , where η(τ) is the Dedekind eta function. We show that if τ0 is an imaginary quadratic argument and m is an odd integer, then is an algebraic integer dividing This is a generalization of a result of Berndt, Chan and Zhang. On the other hand, when K is an imaginary quadratic field and θK is an element of K with Im(θK) > 0 which generates the ring of integers of K over ℤ, we find a sufficient condition on m which ensures that is a unit.

Keywords

Dedekind eta functionmodular functionsautomorphic functions

MSC classification

Secondary: 11F20: Dedekind eta function, Dedekind sums 11F03: Modular and automorphic functions

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 1 , February 2012 , pp. 167 - 179
Copyright
Copyright © Edinburgh Mathematical Society 2011

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