Student Algebra Seminar: Elliptic Curves Over Quadratic Fields with An Odd-torsion Subgroup

Thursday, March 4 at 12:00pm to 1:00pm

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Irmak Balcik, USC

Abstract: Let $E$ be an elliptic curve over a number field $K$. The set of points on $E$ that are defined over $K$ has a group structure. More interestingly $E(K)$ is a  finitely generated abelian group. We study how the torsion subgroup of $E(K)$ grows in a quadratic extension of $K$. When $|E(K)_{\text{tor}}|$ is odd, we answer this question for the special set of quadratic  fields.
In order to achieve this we study the torsion structures which occur on the quadratic twist of $E$ and examine $N$-isogenies defined over $K$ for some values of $N.$

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