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2019 Rabinowitsch times six
Pete L. Clark
Rocky Mountain J. Math. 49(2): 433-485 (2019). DOI: 10.1216/RMJ-2019-49-2-433

Abstract

We give an analogue of the Rabinowitsch criterion with $\mathbb{Z} $ replaced by the polynomial ring $k[t]$ over a field of characteristic different from $2$. In fact, we expose three different proofs of the Rabinowitsch criterion, using Dedekind-Hasse norms, binary quadratic forms and the Minkowski bound on ideal classes, and adapt each to prove our Polynomial Rabinowitsch criterion. Whereas there are precisely seven cases in which the classical Rabinowitsch criterion holds, working over an arbitrary ground field gives us much more latitude, e.g. recent results about genus $1$ curves yield infinitely many instances in which the Rabinowitsch criterion is satisfied over $k = \mathbb{Q} $. Finally, we take a geometric perspective and relate the Rabinowitsch criterion to the Mordell-Weil group of the Jacobian of the associated hyperelliptic curve.

Citation

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Pete L. Clark. "Rabinowitsch times six." Rocky Mountain J. Math. 49 (2) 433 - 485, 2019. https://doi.org/10.1216/RMJ-2019-49-2-433

Information

Received: 22 November 2016; Revised: 26 June 2018; Published: 2019
First available in Project Euclid: 23 June 2019

zbMATH: 07079978
MathSciNet: MR3973234
Digital Object Identifier: 10.1216/RMJ-2019-49-2-433

Subjects:
Primary: 11E16

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 2 • 2019
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