We use a global version of Heath-Brown's p−adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most B on non-singular cubic curves defined over Q. The bounds are uniform in the sense that they only depend on the rank of the corresponding Jacobian.
We give uniform upper bounds for the number of rational points of height at most B on non-singular complete intersections of two quadrics in P 3 defined over Q. To do this, we combine determinant methods with descent arguments.
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