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FROM SEPARABLE POLYNOMIALS TO NONEXISTENCE OF RATIONAL POINTS ON CERTAIN HYPERELLIPTIC CURVES

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  16 June 2014

NGUYEN NGOC DONG QUAN
NGUYEN NGOC DONG QUAN*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada email [email protected]
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Abstract

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We give a separability criterion for the polynomials of the form

$$\begin{equation*} ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e). \end{equation*}$$
Using this separability criterion, we prove a sufficient condition using the Brauer–Manin obstruction under which curves of the form
$$\begin{equation*} z^2 = ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e) \end{equation*}$$
 have no rational points. As an illustration, using the sufficient condition, we study the arithmetic of hyperelliptic curves of the above form and show that there are infinitely many curves of the above form that are counterexamples to the Hasse principle explained by the Brauer–Manin obstruction.

Keywords

Azumaya algebrasBrauer groupsBrauer–Manin obstructionHasse principlehyperelliptic curves

MSC classification

Secondary: 14G05: Rational points 11G35: Varieties over global fields 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 96 , Issue 3 , June 2014 , pp. 354 - 385
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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