Skip to main content Accessibility help
×
Home
Hostname: page-component-747cfc64b6-dwt4q Total loading time: 0.517 Render date: 2021-06-15T10:21:30.355Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Singular units and isogenies between CM elliptic curves

Published online by Cambridge University Press:  29 April 2021

Yingkun Li
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany li@mathematik.tu-darmstadt.de
Corresponding

Abstract

In this note, we will apply the results of Gross–Zagier, Gross–Kohnen–Zagier and their generalizations to give a short proof that the differences of singular moduli are not units. As a consequence, we obtain a result on isogenies between reductions of CM elliptic curves.

Type
Research Article
Copyright
© The Author(s) 2021

Access options

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

Footnotes

The author is partially supported by the LOEWE research unit USAG.

References

Bilu, Y., Habegger, P. and Kühne, L., No singular modulus is a unit, Int. Math. Res. Not. IMRN 2020 (2020), 1000510041; MR 4190395.CrossRefGoogle Scholar
Borcherds, R. E., Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), 491562; MR 1625724 (99c:11049).CrossRefGoogle Scholar
Bruinier, J. H., Borcherds products on O(2, l) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, vol. 1780 (Springer, Berlin, 2002); MR 1903920 (2003h:11052).CrossRefGoogle Scholar
Bruinier, J. H., Ehlen, S. and Yang, T., CM values of higher automorphic Green functions for orthogonal groups, Invent. Math., to appear. Preprint (2019), arXiv:1912.12084.Google Scholar
Bruinier, J. H., Kudla, S. S. and Yang, T., Special values of Green functions at big CM points, Int. Math. Res. Not. IMRN 2012 (2012), 19171967; MR 2920820.Google Scholar
Cohn, H., Introduction to the construction of class fields, Cambridge Studies in Advanced Mathematics, vol. 6 (Cambridge University Press, Cambridge, 1985); MR 812270.Google Scholar
Gross, B., Kohnen, W. and Zagier, D., Heegner points and derivatives of $L$-series. II, Math. Ann. 278 (1987), 497562; MR 909238 (89i:11069).10.1007/BF01458081CrossRefGoogle Scholar
Gross, B. H. and Zagier, D. B., On singular moduli, J. Reine Angew. Math. 355 (1985), 191220; MR 772491 (86j:11041).Google Scholar
Gross, B. H. and Zagier, D. B., Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), 225320.CrossRefGoogle Scholar
Habegger, P., Singular moduli that are algebraic units, Algebra Number Theory 9 (2015), 15151524; MR 3404647.CrossRefGoogle Scholar
Habegger, P. and Pazuki, F., Bad reduction of genus 2 curves with CM jacobian varieties, Compos. Math. 153 (2017), 25342576; MR 3705297.CrossRefGoogle Scholar
Kudla, S. S., Central derivatives of Eisenstein series and height pairings, Ann. of Math. (2) 146 (1997), 545646; MR 1491448.CrossRefGoogle Scholar
Kudla, S. S. and Yang, T., Eisenstein series for SL(2), Sci. China Math. 53 (2010), 22752316; MR 2718827.CrossRefGoogle Scholar
Lauter, K. and Viray, B., On singular moduli for arbitrary discriminants, Int. Math. Res. Not. IMRN 2015 (2015), 92069250; MR 3431591.CrossRefGoogle Scholar
Li, Y., Average CM-values of higher Green's function and factorization, Preprint (2018), arXiv:1812.08523.Google Scholar
Schofer, J., Borcherds forms and generalizations of singular moduli, J. Reine Angew. Math. 629 (2009), 136; MR 2527412.CrossRefGoogle Scholar
Viazovska, M., CM values of higher Green's functions, Preprint (2011), arXiv:1110.4654.Google Scholar
Yang, T., CM number fields and modular forms, Pure Appl. Math. Q. 1 (2005), 305340.CrossRefGoogle Scholar
Yang, T. and Yin, H., Difference of modular functions and their CM value factorization, Trans. Amer. Math. Soc. 371 (2019), 34513482; MR 3896118.CrossRefGoogle Scholar
Yang, T., Yin, H. and Yu, P., The lambda invariants at CM points, Int. Math. Res. Not. (IMRN), 2021 (2021), 55425603.CrossRefGoogle Scholar
Zhang, S., Heights of Heegner cycles and derivatives of $L$-series, Invent. Math. 130 (1997), 99152; MR 1471887.CrossRefGoogle Scholar

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.

Singular units and isogenies between CM elliptic curves
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.

Singular units and isogenies between CM elliptic curves
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.

Singular units and isogenies between CM elliptic curves
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *