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An elliptic curve over a number field $K$ is a smooth projective curve $E$ with a point defined over $K$. By the Mordell–Weil theorem, the $K$-rational points on elliptic curves form a finitely generated abelian group. Isogenies of elliptic curves are maps between elliptic curves which are also group homomorphisms. We can organize isogeny classes into graphs and label the vertices with the torsion structure of corresponding elliptic curves. These graphs are called isogeny-torsion graphs, and in 2021, Chiloyan and Lozano-Robledo classified all isogeny torsion graphs over $\mathbb{Q}$. In this talk, we explore progress on this classification question over other number fields, using work done by Banwait, Najman, and Padurariu extending Mazur’s theorem. Of particular interest is the quadratic field $K = \mathbb{Q}(\sqrt{213})$. This talk is based on joint work with Clayton Boothe, Michael Logal, and Lance E Miller.

Event contact: Richard Ng and Gene Kopp

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