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Ranks of elliptic curves with prescribed torsion over number fields

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Let d be a positive integer, and let Td be the set of isomorphism classes of groups that can occur as the torsion subgroup of E(K), where K is a number field of degree d and E is an elliptic curve over K. T1 is known by Mazur’s theorem, T2 is known as well, and for d equal to 3 or 4, it is known which groups occur infinitely often.

We shall study the following problem: given a d <= 4 and a group T in Td, what are the possibilities for the Mordell-Weil rank of E, where E is an elliptic curve over a number field K of degree d with the torsion subgroup of E(K) isomorphic to T. For d = 2 and T = Z/13Z or T = Z/18Z, and also for d = 4 and T = Z/22Z, it turns out that the rank is always even. This will be explained by a phenomenon we call “false complex multiplication”.

This is joint work with Peter Bruin, Andrej Dujella, and Filip Najman.

This talk is part of the Number Theory Seminar series.

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