Skip to main content Accessibility help
×
Home
Hostname: page-component-78dcdb465f-bcmtx Total loading time: 0.305 Render date: 2021-04-18T18:51:35.633Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Algebraic integers as special values of modular units

Published online by Cambridge University Press:  01 November 2011

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)
Corresponding

Abstract

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2011

Access options

Get access to the full version of this content by using one of the access options below.

References

1.Berndt, B. C., Ramanujan's notebooks, III (Springer, 1991).CrossRefGoogle Scholar
2.Berndt, B. C., Chan, H. H. and Zhang, L. C., Ramanujan's remarkable product of theta-functions, Proc. Edinb. Math. Soc. 40 (1997), 583612.CrossRefGoogle Scholar
3.Cox, D. A., Primes of the form x2 + ny2: Fermat, class field, and complex multiplication (Wiley–Interscience, New York, 1989).Google Scholar
4.Deuring, M., Die Klassenkörper der Komplexen Multiplikation, Enzyklopädie der mathematischen Wissenschaften, Volume 12, Issue 10, Part II (Teubner, Stuttgart, 1958).Google Scholar
5.Jung, H. Y., Koo, J. K. and Shin, D. H., Ray class invariants over imaginary quadratic fields, Tohoku Math. J. 63 (2011), 413426.CrossRefGoogle Scholar
6.Koo, J. K. and Shin, D. H., On some arithmetic properties of Siegel functions, Math. Z. 264 (2010), 137177.CrossRefGoogle 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 (Spinger, 1987).CrossRefGoogle Scholar
9.Ramachandra, K., Some applications of Kronecker's limit formula, Annals Math. (2) 80 (1964), 104148.CrossRefGoogle Scholar
10.Ramanujan, S., Notebooks, two volumes (Tata Institute of Fundamental Research, Bombay, 1957).Google Scholar
11.Shimura, G., Introduction to the arithmetic theory of automorphic functions (Iwanami Shoten and Princeton University Press, 1971).Google Scholar
12.Stevenhagen, P., Hilbert's 12th problem, complex multiplication and Shimura reciprocity, in Class field theory: its centenary and prospect, Advanced Studies in Pure Mathematics, Volume 30, pp. 161176 (Mathematical Society of Japan, Tokyo, 2001).Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 22 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 18th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Algebraic integers as special values of modular units
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Algebraic integers as special values of modular units
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Algebraic integers as special values of modular units
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *