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Singular units and isogenies between CM elliptic curves

Part of: Discontinuous groups and automorphic forms Abelian varieties and schemes Arithmetic algebraic geometry

Published online by Cambridge University Press:  29 April 2021

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

Keywords

singular modulihigher Green's function

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms 11G15: Complex multiplication and moduli of abelian varieties 14K02: Isogeny
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 5 , May 2021 , pp. 1022 - 1035
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Copyright
© The Author(s) 2021

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Footnotes

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

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