Most binary forms come from a pencil of quadrics
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- by Brendan Creutz HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 3 (2016), 18-27
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.References
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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
- © Copyright 2016 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- 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
- MathSciNet review: 3579585