FROM SEPARABLE POLYNOMIALS TO NONEXISTENCE OF RATIONAL POINTS ON CERTAIN HYPERELLIPTIC CURVES
2014 ◽
Vol 96
(3)
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pp. 354-385
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Keyword(s):
AbstractWe 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.
1978 ◽
Vol 84
(2)
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pp. 219-223
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1990 ◽
Vol 32
(2)
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pp. 180-192
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2006 ◽
Vol 58
(1)
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pp. 115-153
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Keyword(s):
1990 ◽
Vol 42
(2)
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pp. 315-341
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1979 ◽
Vol 31
(2)
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pp. 255-263
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1989 ◽
Vol 41
(2)
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pp. 285-320
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2018 ◽
Vol 21
(3)
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pp. 923-956
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1978 ◽
Vol 26
(1)
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pp. 31-45
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