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Abstract

We consider an elliptic surface π : Ᏹ → ‫ސ‬ 1 defined over a number field k and study the problem of comparing the rank of the special fibers over k with that of the generic fiber over k(‫ސ‬ 1 ). We prove, for a large class of rational elliptic surfaces, the existence of infinitely many fibers with rank at least equal to the generic rank plus two.