Pencils of quadrics and the arithmetic of hyperelliptic curves (original) (raw)
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From Separable Polynomials to Nonexistence of Rational Points on Certain Hyperelliptic Curves
Journal of the Australian Mathematical Society, 2014
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*}Usingthisseparabilitycriterion,weproveasufficientconditionusingtheBrauer–ManinobstructionunderwhichcurvesoftheformUsing this separability criterion, we prove a sufficient condition using the Brauer–Manin obstruction under which curves of the formUsingthisseparabilitycriterion,weproveasufficientconditionusingtheBrauer–Maninobstructionunderwhichcurvesoftheform\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.
The Hasse Principle for Certain Hyperelliptic Curves and Forms
The Quarterly Journal of Mathematics, 2012
We prove that, for each positive integer n ≥ 2, there is an infinite arithmetic family of hyperelliptic curves of genus n violating the Hasse principle explained by the Brauer-Manin obstruction. Using these families of curves, we show that, for any positive integer k ≥ 1, there are infinitely many algebraic and arithmetic families of forms in three variables of degree 4k + 2 such that they are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction.
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An irreducible smooth projective curve over mathbbFq\mathbb{F}_qmathbbFq is called \textit{pointless} if it has no mathbbFq\mathbb{F}_qmathbbFq-rational points. In this paper we study the lower existence bound on the genus of such a curve over a fixed finite field mathbbFq\mathbb{F}_qmathbbFq. Using some explicit constructions of hyperelliptic curves, we establish two new bounds that depend linearly on the number qqq. In the case of odd characteristic this improves upon a result of R. Becker and D. Glass. We also provide a similar new bound when qqq is even.
Rational Points on Certain Hyperelliptic Curves over Finite Fields
Bulletin of the Polish Academy of Sciences Mathematics, 2007
Let K be a field, a, b ∈ K and ab = 0. Let us consider the polynomials g1(x) = x n + ax + b, g2(x) = x n + ax 2 + bx, where n is a fixed positive integer. In this paper we show that for each k ≥ 2 the hypersurface given by the equation
The arithmetic of certain quartic curves
Monatshefte für Mathematik, 2012
We give a sufficient condition using the Brauer-Manin obstruction under which certain quartic curves have no rational points. Using this sufficient condition, we construct two families of genus one quartic curves violating the Hasse principle explained by the Brauer-Manin obstruction.
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Rocky Mountain Journal of Mathematics, 2014
Let C be a hyperelliptic curve of good reduction defined over a discrete valuation field K. Given d ∈ K * \ K * 2 , we find the minimal regular model of the quadratic twist of C by d. Then we prove that there exists an infinite family of hyperelliptic curves of genus 2 defined over Q violating the Hasse principle.
Pointless hyperelliptic curves
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In this paper we consider the question of whether there exists a hyperelliptic curve of genus g which is defined over F q but has no rational points over F q for various pairs (g, q).
Designs, Codes and Cryptography, 2015
Supersingular elliptic curves have played an important role in the development of elliptic curve crytography. Scott Vanstone introduced the first author to elliptic curve cryptography, a subject that continues to be a rich source of interesting problems, results and applications. One important fact about supersingular elliptic curves is that the number of rational points is tightly constrained to a small number of possible values. In this paper we present some similar results for curves of higher genus. We also present an application to the problem of determining abelian varieties that occur as jacobians.
The genus of curves over finite fields with many rational points
Manuscripta Mathematica, 1996
We prove the following result which was conjectured by Stichtenoth and Xing: letg be the genus of a projective, irreducible non-singular algebraic curve over the finite field mathbbFq2\mathbb{F}_{q^2 } mathbbFq2 and whose number of mathbbFq2\mathbb{F}_{q^2 } mathbbFq2 -rational points attains the Hasse-Weil bound; then either 4g≤(q−1)2 or 2g=(q−1)q.