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An algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.
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Endomorphism algebras of abelian surfaces with real multiplication
I had forgotten that I had posted this question, but in the time since I was pointed by John Voight to the following paper of Bruin, Flynn, Gonzalez, and Rotger: https://www-ma2.upc.edu/vrotger/docs/B …
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Endomorphism algebras of abelian surfaces with real multiplication
Given an abelian variety $A$ over a field $F$, one may consider the ring of endomorphisms $End(A)$, the ring of $F$-rational maps $A \to A$ respecting the group structure on $A$. We may also consider …