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Unfaking the fake Selmer group

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Let C be a smooth projective curve over a global field k with Jacobian J. Then the Mordell-Weil group J(k) of k-rational points on J is finitely generated. Knowing the torsion subgroup, which is usually relatively easy to find, the rank of J(k) can be read off from the size of the finite group J(k)/2J(k). This quotient injects into the so called Selmer group, which is abstractly defined as a certain subgroup of the cohomology group H1(k,J [ 2 ]). The Selmer group is finite, so the image of J(k)/2J(k) in it can be determined by deciding for each element of the Selmer group separately whether or not it is in the image of J(k)/2J(k). Unfortunately, the abstract definition of the Selmer group is not very amenable to explicit computations, which are in practice done with the fake Selmer group instead. In general the fake Selmer group is isomorphic to a quotient of the Selmer group by a subgroup of order 1 or 2. In this talk we will define all the groups just mentioned and we will introduce a new group, equally amenable to explicit computations as the fake Selmer group, that is always isomorphic to the Selmer group. This is joint work with Michael Stoll.

This talk is part of the Number Theory Seminar series.

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