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

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic 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. 

References

Cohen, H., Number Theory, Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics, 239 (Springer, New York, 2007).Google Scholar
Colliot-Thélène, J. -L., Coray, D. F. and Sansuc, J. -J., ‘Descente et principe de Hasse pour certaines variétés rationnalles’, J. reine angew. Math 320 (1980), 150–191.Google Scholar
Coray, D. and Manoil, C., ‘On large Picard groups and the Hasse principle for curves and K3 surfaces’, Acta. Arith. 76 (1996), 165–189.Google Scholar
Dong Quan, N. N., ‘The Hasse principle for certain hyperelliptic curves and forms’, Q. J. Math. 64 (2013), 253–268.Google Scholar
Dong Quan, N. N., ‘Algebraic families of hyperelliptic curves violating the Hasse principle’, 2013. Available at http://www.math.ubc.ca/∼dongquan/JTNB-algebraic-families.pdf.Google Scholar
Dong Quan, N. N., ‘Nonexistence of rational points on certain varieties’, PhD Thesis, University of Arizona, 2012.Google Scholar
Iwaniec, H., ‘Primes represented by quadratic polynomials in two variables’, Acta Arith. 24 (1974), 435–459.Google Scholar
Jahnel, J., ‘Brauer groups, Tamagawa measures, and rational points on algebraic varieties’, Habilitationsschrift, Georg-August-Universität Göttingen, 2008.Google Scholar
Lind, C. E., ‘Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins’, Thesis, University of Uppasala, 1940.Google Scholar
Manin, Yu. I., ‘Le groupe de Brauer-Grothendieck en géométrie Diophantienne’, Actes, Congres. Intern. Math. 1 (1970), 401–411.Google Scholar
Reichardt, H., ‘Einige im Kleinen überall lösbre, im Grossen unlösbare diophantische Gleichungen’, J. reine angew. Math. 184 (1942), 12–18.Google Scholar
Skorobogatov, A. N., Torsors and Rational Points, Cambridge Tracts in Mathematics, 144 (Cambridge University Press, Cambridge, 2001).Google Scholar
Viray, B., ‘Failure of the Hasse principle for Châtelet surfaces in characteristic 2’, J. Théor. Nombres Bordeaux 24 (2012), 231–236.Google Scholar