Let $E/\mathbb C$ be an elliptic curve. It is known that if $E$ has CM, then $j(E)$ is an algebraic integer. My first question is: what about the converse? Is there a way to identify the subset of algebraic integers whose elements correspond to [isomorphism classes of] elliptic curves with CM?
The second question is: fixed a number field $K$, does there always exist an elliptic curve over $K$ with CM? And without CM?


Thank you very much!



elliptic curves with and without CM
1)If an elliptic curve has integral $j$-invariant it absolutely DOES NOT NEED to have CM. The class of curves with integral $j$-invariant (let's call that the class of IM Elliptic curves for Integral Modulus) is MUCH MUCH larger than the class of CM Elliptic curves. In fact, one can use Heilbronn's Theorem that class numbers of imaginary quadratic fields tend to infinity to show that over any given number field, there are only finitely many elliptic curves with CM. In particular over $\mathbf{Q}$, there are only 13 CM $j$-invariants. Even for all number fields of a certain degree, there are only a finite number $N(d)$ of elliptic curves with CM over any number field of degree $d$. This gives a way to enumerate all the CM $j$-invariants or "singular moduli," which is about as good as you can hope for in terms of describing the complex numbers which are $j$-invariants of CM elliptic curves. To do this explicitly (say for number fields of degree up to 100) see Mark Watkins' enumeration of imaginary quadratic fields of class number up to 100.


Meanwhile, for any regular integer $n$ (1,2,3, etc) there is at least one elliptic curve over $\mathbf{Q}$ whose $j$ invariant is $n$. Therefore over $\mathbf{Q}$ and therefore over any number field, there are infinitely many non-isomorphic IM elliptic curves.


If you want to say something general about elliptic curves with IM, consider the theorem of Deuring that an elliptic curve has IM if and only if it has *potential good reduction*. For this and much much more see Serre-Tate's "Good reduction of abelian varieties"


2) Easy proof that the answer is yes, at least as long as you mean "has a CM $j$-invariant" when you say CM: $y^2 = x^3 + 1$ is a CM elliptic curve with $j$-invariant zero defined over any number field. On the other hand, if we take an elliptic curve with $j$-invariant equal to 1/2, say this one: y^2 + x\*y = x^3 + 72\*x + 13822 , well it doesn't have integral $j$-invariant and therefore can't have CM!


