1School of Mathematics and Statistical Sciences Arizona State University Tempe, AZ 85287-1804, U.S.A. 2Institute of Mathematics Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland
Acta Arithmetica, Tome 179 (2017) no. 1, pp. 55-78
Given an elliptic quartic of type $Y^2=f(X)$ representing an elliptic curve of positive rank over $\mathbb Q$, we investigate the question of when the $Y$-coordinate can be represented by a quadratic form of type $ap^2+bq^2$. In particular, we give examples of equations of surfaces of type
$c_0+c_1x+c_2x^2+c_3x^3+c_4x^4=(ap^2+bq^2)^2$, $a,b,c \in \mathbb Q$, where we can deduce the existence of infinitely many rational points. We also investigate surfaces of type $Y^2=f(a p^2+b q^2)$ where the polynomial $f$ is of degree $3$.
Keywords:
given elliptic quartic type representing elliptic curve positive rank mathbb investigate question y coordinate represented quadratic form type particular examples equations surfaces type mathbb where deduce existence infinitely many rational points investigate surfaces type where polynomial degree nbsp
Affiliations des auteurs :
Andrew Bremner
1
;
Maciej Ulas
2
1
School of Mathematics and Statistical Sciences Arizona State University Tempe, AZ 85287-1804, U.S.A.
2
Institute of Mathematics Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland
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title = {On representing coordinates of points on elliptic curves by quadratic forms},
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Andrew Bremner; Maciej Ulas. On representing coordinates of points on elliptic curves by quadratic forms. Acta Arithmetica, Tome 179 (2017) no. 1, pp. 55-78. doi: 10.4064/aa8597-1-2017
