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A note on quadratic fields in which a fixed prime number splits completely

Published online by Cambridge University Press:  22 January 2016

Humio Ichimura
Humio Ichimura*
Affiliation:
Department of Mathematics, Faculty of Science, University of Tokyo, Hongo, Tokyo, 113, Japan
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Throughout this note, p denotes a fixed prime number and f denotes a fixed natural number prime to p.

It is easy to see and more or less known that for any natural number n, there exists an elliptic curve over p whose j-invariant is of degree n over Fp and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field. In this note, we consider a more precise problem: for any natural number n, decide whether or not there exists an elliptic curve over p whose j-invariant is of degree n over Fp and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field with conductor f.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 99 , September 1985 , pp. 63 - 71
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

References

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