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Counting rational points on cubic curves. (English) Zbl 1229.11052

Let \[ F(X_0,X_1,X_2)\in{\mathbb Z}[X_0,X_1,X_2] \] be a non-singular cubic form, so that \(F=0\) defines a smooth plane cubic curve \(C\), \(r\) the rank of the Jacobian \(\text{Jac}(C)\). The authors prove that for any \(B\geq 3\) and any positive integer \(m\) the number \(N(B)\) of points in \(C({\mathbb Q})\) with height \(\leq B\) satisfies \[ N(B)\ll m^{r+2}(\log^2 B+B^{\frac{2}{3m^2}}\log B) \] uniformly in \(C\), with an implied constant independent of \(m\). It holds that \(N(B)\ll (\log B)^{3+\frac r2}\) uniformly in \(C\). For \(\delta<\frac{1}{110}\) and for any smooth plane cubic curve \(C\) it holds that \(N(B)\ll B^{\frac 23 -\delta}\) uniformly in \(C\).

MSC:

11D25 Cubic and quartic Diophantine equations
11D45 Counting solutions of Diophantine equations
11G05 Elliptic curves over global fields
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References:

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