Number Theory Seminar - Asimina Hamakiotes

Asimina Hamakiotes
UConn

Title
Abelian extensions arising from elliptic curves with complex multiplication

Abstract
Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f \geq 1$. Let $E$ be an elliptic curve with complex multiplication by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$, where $j(E)$ is the $j$-invariant of $E$. Let $N\geq 2$ be an integer. The extension $\mathbb{Q}(j(E), E[N])/\mathbb{Q}(j(E))$ is usually not abelian; it is only abelian for $N=2,3$, and $4$. Let $p$ be a prime and let $n\geq 1$ be an integer. In this talk, we will classify the maximal abelian extension contained in $\mathbb{Q}(E[p^n])/\mathbb{Q}$.

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