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Isogeny classes of rational squares of CM elliptic curves Fite, Francesc


Let $A$ be an abelian surface defined over $\Q$. It is well known that the $\Q$-algebra $End(A) \otimes \Q$ is either $\Q$, $\Q \times \Q$, $M_2(\Q)$, a real quadratic field $E$, the product of $\Q$ and a quadratic imaginary field of class number 1, a quartic CM field $L$, a definite division quaternion algebra $B$, or $M_2(K)$, where $K$ is a quadratic imaginary field. It is conjectured that there are only finitely many possibilities for $End(A) \otimes \Q$. While almost nothing is known about the sets of possibilities for $E$ and $B$, Murabashy and Umegaki have proved that there are precisely 19 possibilities for the quartic CM field $L$. In this talk I will report on a joint work with Xevi Guitart in which we show that there are at most 52 possibilities for the quadratic imaginary field $K$. An equivalent way to formulate the result is by saying that there are at most 52 $\bar\Q$-isogeny classes of abelian surfaces defined over $\Q$ that are isogenous to the square of an elliptic curve with CM.

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