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Singular units and isogenies between CM elliptic curves

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

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.

References

Bilu, Y., Habegger, P. and Kühne, L., No singular modulus is a unit, Int. Math. Res. Not. IMRN 2020 (2020), 10005–10041; MR 4190395.CrossRefGoogle Scholar
Borcherds, R. E., Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), 491–562; 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), 1917–1967; 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), 497–562; MR 909238 (89i:11069).10.1007/BF01458081CrossRefGoogle Scholar
Gross, B. H. and Zagier, D. B., On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220; MR 772491 (86j:11041).Google Scholar
Gross, B. H. and Zagier, D. B., Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), 225–320.CrossRefGoogle Scholar
Habegger, P., Singular moduli that are algebraic units, Algebra Number Theory 9 (2015), 1515–1524; MR 3404647.CrossRefGoogle Scholar
Habegger, P. and Pazuki, F., Bad reduction of genus 2 curves with CM jacobian varieties, Compos. Math. 153 (2017), 2534–2576; MR 3705297.CrossRefGoogle Scholar
Kudla, S. S., Central derivatives of Eisenstein series and height pairings, Ann. of Math. (2) 146 (1997), 545–646; MR 1491448.CrossRefGoogle Scholar
Kudla, S. S. and Yang, T., Eisenstein series for SL(2), Sci. China Math. 53 (2010), 2275–2316; MR 2718827.CrossRefGoogle Scholar
Lauter, K. and Viray, B., On singular moduli for arbitrary discriminants, Int. Math. Res. Not. IMRN 2015 (2015), 9206–9250; 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), 1–36; 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), 305–340.CrossRefGoogle Scholar
Yang, T. and Yin, H., Difference of modular functions and their CM value factorization, Trans. Amer. Math. Soc. 371 (2019), 3451–3482; MR 3896118.CrossRefGoogle Scholar
Yang, T., Yin, H. and Yu, P., The lambda invariants at CM points, Int. Math. Res. Not. (IMRN), 2021 (2021), 5542–5603.CrossRefGoogle Scholar
Zhang, S., Heights of Heegner cycles and derivatives of $L$-series, Invent. Math. 130 (1997), 99–152; MR 1471887.CrossRefGoogle Scholar