Timeline for Singular abelian surfaces that can be defined over $\mathbb Q$

5 events

when toggle format	what		by	license	comment
Jun 5, 2018 at 16:10	comment	added	Davide Cesare Veniani		Yes, exactly. I wouldn't know how to produce equations with coefficients $\mathbb Q$ for $E \times E'$ (or to prove that it is defined over $\mathbb Q$ in some other way).
Jun 5, 2018 at 15:57	comment	added	Joe Silverman		The abelian surface $E\times E'$ is isomorphic to itself under the action of Gal$(\overline{\mathbb Q}/\mathbb Q)$, since Galois just swaps the factors, so its field of moduli is $\mathbb Q$. I guess maybe one needs to do a bit more to prove that $\mathbb Q$ is a field of definition. Is that what you're worried about?
Jun 5, 2018 at 15:37	comment	added	Davide Cesare Veniani		Take a CM $\mathbb Q$-curve $E$ such that $\mathbb Q(j(E))$ is a quadratic extension of $\mathbb Q$. Let $E'$ be its Galois conjugate. Then $A = E\times E'$ is a singular abelian surface. You mean that $A$ is a good candidate for what I am looking for? How could one prove that $A$ is defined over $\mathbb Q$?
Jun 5, 2018 at 13:49	comment	added	Ariyan Javanpeykar		Yes, you are right. If $A$ is a singular abelian surface over $\mathbb{C}$ which can be defined over $\mathbb{Q}$, then $A$ is isomorphic to $E_1\times E_2$ with $E_1$ and $E_2$ isogenous CM elliptic $\mathbb{Q}$-curves.
Jun 5, 2018 at 13:06	history	answered	Joe Silverman	CC BY-SA 4.0