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Proceedings of the American Mathematical Society Series B

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Most binary forms come from a pencil of quadrics


Author: Brendan Creutz
Journal: Proc. Amer. Math. Soc. Ser. B 3 (2016), 18-27
MSC (2010): Primary 11D09, 11G30; Secondary 14H25, 14L24
DOI: https://doi.org/10.1090/bproc/24
Published electronically: December 6, 2016
MathSciNet review: 3579585
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Abstract: A pair of symmetric bilinear forms $A$ and $B$ determine a binary form $f(x,y) := \operatorname {disc}(Ax-By)$. We prove that the question of whether a given binary form can be written in this way as a discriminant form generically satisfies a local-global principle and deduce from this that most binary forms over $\mathbb {Q}$ are discriminant forms. This is related to the arithmetic of the hyperelliptic curve $z^2 = f(x,y)$. Analogous results for nonhyperelliptic curves are also given.


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Additional Information

Brendan Creutz
Affiliation: School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
Email: brendan.creutz@canterbury.ac.nz

Received by editor(s): January 24, 2016
Received by editor(s) in revised form: March 14, 2016, July 26, 2016, and August 16, 2016
Published electronically: December 6, 2016
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2016 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)