130 research outputs found
Extending Elliptic Curve Chabauty to higher genus curves
We give a generalization of the method of "Elliptic Curve Chabauty" to higher
genus curves and their Jacobians. This method can sometimes be used in
conjunction with covering techniques and a modified version of the Mordell-Weil
sieve to provide a complete solution to the problem of determining the set of
rational points of an algebraic curve .Comment: 24 page
Twists of X(7) and primitive solutions to x^2+y^3=z^7
We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent
argument involving the simple group of order 168 reduces the problem to the
determination of the set of rational points on a finite set of twists of the
Klein quartic curve X. To restrict the set of relevant twists, we exploit the
isomorphism between X and the modular curve X(7), and use modularity of
elliptic curves and level lowering. This leaves 10 genus-3 curves, whose
rational points are found by a combination of methods.Comment: 47 page
Rational points on
We compute the rational points on the Atkin-Lehner quotient
using the quadratic Chabauty method. Our work completes the study of
exceptional rational points on the curves of genus between 2 and 6.
Together with the work of several authors, this completes the proof of a
conjecture of Galbraith.Comment: 8 pages; minor changes following referee repor
Rational points on
We compute the rational points on the Atkin-Lehner quotient
using the quadratic Chabauty method. Our work completes the study of
exceptional rational points on the curves of genus between 2 and 6.
Together with the work of several authors, this completes the proof of a
conjecture of Galbraith.Comment: 8 pages; minor changes following referee repor
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