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Jean-Louis Colliot-Thélène, CNRS and Paris-Saclay University (Orsay)

December 11th, 2020 2:30-3:30 pm CET

Title : Jumps in the rank of the Mordell-Weil group

Abstract : Let k be a number field and U a smooth integral k-variety. Let XU be an abelian scheme. We consider the set U(k)+U(k) of k-rational points of U such that the Mordell-Weil rank of the fibre Xm is strictly bigger than the Mordell-Weil rank of the generic fibre over the function field k(U).
We prove : if the k-variety X is k-unirational, then U(k)+ is dense for the Zariski topology on U. Variants are given and compared with old and new results in the literature.