Integer points on curves of genus two and their Jacobians

Abstract

Let <italic>C</italic> be a curve of genus 2 defined over a number field, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Theta"> <mml:semantics> <mml:mi mathvariant="normal">Θ<!-- Θ --></mml:mi> <mml:annotation encoding="application/x-tex">\Theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the image of <italic>C</italic> embedded into its Jacobian <italic>J</italic>. We show that the heights of points of <italic>J</italic> which are integral with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 2 right-bracket Subscript asterisk Baseline normal upper Theta"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">[</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mo stretchy="false">]</mml:mo> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> </mml:mrow> <mml:mi mathvariant="normal">Θ<!-- Θ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{[2]_\ast }\Theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be effectively bounded. As a result, if <italic>P</italic> is a point on <italic>C</italic>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P overbar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>P</mml:mi> <mml:mo stretchy="false">¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> its image under the hyperelliptic involution, then the heights of points on <italic>C</italic> which are integral with respect to <italic>P</italic> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P overbar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>P</mml:mi> <mml:mo stretchy="false">¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be effectively bounded, in such a way that we can isolate the dependence on <italic>P</italic>, and show that if the height of <italic>P</italic> is bigger than some bound, then there are no points which are <italic>S</italic>-integral with respect to <italic>P</italic> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P overbar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>P</mml:mi> <mml:mo stretchy="false">¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We relate points on <italic>C</italic> which are integral with respect to <italic>P</italic> to points on <italic>J</italic> which are integral with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Theta"> <mml:semantics> <mml:mi mathvariant="normal">Θ<!-- Θ --></mml:mi> <mml:annotation encoding="application/x-tex">\Theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and discuss approaches toward bounding the heights of the latter.

Locations

• Transactions of the American Mathematical Society - View - PDF