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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the two sheeted coverings of conics by elliptic curves


Author: R. E. MacRae
Journal: Trans. Amer. Math. Soc. 211 (1975), 277-287
MSC: Primary 14H05; Secondary 14H30
DOI: https://doi.org/10.1090/S0002-9947-1975-0379509-5
MathSciNet review: 0379509
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Abstract: Let K be the field of algebraic functions on an elliptic curve that can be described by an equation of the form $ {y^2} = f(x)$ where $ f(x)$ is a quartic polynomial over a field k. Moreover, assume that the Riemann surface for K contains no points rational over k. When k is the field of real numbers it is well known that K may also be expressed as a quadratic extension of a function field $ L = k(u,v)$ of algebraic functions on a conic whose Riemann surface also contains no points rational over k. We extend this result to p-adic ground fields k. Moreover, we describe the various subfields of index two and genus zero (conic subfields) in terms of the k-rational points on the Jacobian of K. This is done for arbitrary ground fields. In particular, the embedding of the projective class group of K (over k) is seen to describe exactly those conic subfields that possess k-rational points.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0379509-5
Article copyright: © Copyright 1975 American Mathematical Society

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