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On the Schertz Conjecture

Part of: Algebraic number theory: global fields Arithmetic algebraic geometry

Published online by Cambridge University Press:  08 February 2019

Ja Kyung Koo  and
Dong Sung Yoon
Ja Kyung Koo
Affiliation:
Department of Mathematical Sciences, KAIST, Daejeon 34141, Republic of Korea ([email protected])
Dong Sung Yoon
Affiliation:
Department of Mathematics Education, Pusan National University, Busan 46241, Republic of Korea ([email protected])
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Abstract

Schertz conjectured that every finite abelian extension of imaginary quadratic fields can be generated by the norm of the Siegel–Ramachandra invariants. We present a conditional proof of his conjecture by means of the characters on class groups and the second Kronecker limit formula.

Keywords

class field theorymodular units

MSC classification

Primary: 11R37: Class field theory
Secondary: 11G16: Elliptic and modular units
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 837 - 845
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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References

1Janusz, G. J., Algebraic number fields, Graduate Studies in Mathematics, Volume 7, 2nd edn (American Mathematical Society, Providence, RI, 1996).Google Scholar
2Jung, H. Y., Koo, J. K. and Shin, D. H., Ray class invariants over imaginary quadratic fields, Tohoku Math. J. 63 (2011), 413–426.Google Scholar
3Koo, J. K. and Yoon, D. S., Construction of ray class fields by smaller generators and applications, Proc. Roy. Soc. Edinburgh Sect. A 147(4) (2017), 781–812.Google Scholar
4Kubert, D. and Lang, S., Modular units, Grundlehren der Mathematischen Wissenschaften, Volume 244 (Springer-Verlag, New York-Berlin, 1981).Google Scholar
5Lang, S., Elliptic functions, 2nd edn (Springer-Verlag, New York, 1987).Google Scholar
6Ramachandra, K., Some applications of Kronecker's limit formulas, Ann. Math. (2) 80 (1964), 104–148.Google Scholar
7Schertz, R., Complex multiplication, New Mathematical Monographs, Volume 15 (Cambridge University Press, Cambridge, 2010).Google Scholar
8Serre, J.-P., A course in arithmetic, Graduate Texts in Mathematics, Volume 7 (Springer-Verlag, New York-Heidelberg, 1973).Google Scholar
9Siegel, C. L., Lectures on advanced analytic number theory, Tata Institute of Fundamental Research Lectures on Mathematics, Volume 23 (Tata Institute of Fundamental Research, Bombay, 1965).Google Scholar