We prove Manin's conjecture for two del Pezzo surfaces of degree four which are split over Q and whose singularity types are respectively 3A 1 and A 1 + A 2 . For this, we study a certain restricted divisor function and use a result about the equidistribution of its values in arithmetic progressions. In this task, Weil's bound for Kloosterman sums plays a key role.
We establish estimates for the number of solutions of certain affine
congruences. These estimates are then used to prove Manin's conjecture for a
cubic surface split over Q and whose singularity type is D_4. This improves on
a result of Browning and answers a problem posed by Tschinkel
We prove that a positive proportion of squarefree integers are congruent numbers such that the canonical height of the lowest non-torsion rational point on the corresponding elliptic curve satisfies a strong lower bound.
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