For a complex number τ with positive imaginary part, we consider the quotient E_{τ} := C/(Z + Zτ). This quotient can be embedded in the complex projective plane via the Weierstrass P -function and its derivative. The image of E_{τ} in the projective plane is then a complex elliptic curve given by an equation of the form y^{2} = x^{3} + a_{τ} x + b_{τ}. To this equation one associates the complex number

j(τ) := 1728\frac{4a_{τ}^{3}}{4a_{τ}^{3} + 27b_{τ}^{2}} ,

the so-called j-invariant of E_{τ}. The j-invariant classifies complex elliptic curves up to isomorphism: E_{τ} is isomorphic to E_{τ′} if and only if j(τ) = j(τ′). The complex analytic maps from E_{τ} to itself that respect the group structure are those of the form z ↦ cz, with c a complex number such that c · (Z + Zτ) is contained in Z + Zτ. All such c are easily seen to be integers, except when the field extension of Q generated by τ has degree two, in which case E_{τ} is said to have complex multiplications. Let now J be the set of all complex numbers of the form j(τ), with τ quadratic over Q. The aim of the talk is to classify all irreducible polynomials f in C[x, y] for which there exist infinitely many (a, b) in J × J with f(a, b) = 0. Examples of such f are x − a and y − b for a and b in J. For each poitive integer n one has Kronecker’s polynomial F_{n} in Z[x, y], characterized (up to sign) by the properties that it is irreducible and satisfies F_{n}(j(τ), j(nτ)) = 0 for all τ in the upper half plane, giving some more examples. It will be shown that the generalized Riemann hypothesis implies that these examples are in fact the only ones, up to multiplication by scalars. It will also be explained how this result gives evidence for a very general conjecture made by F. Oort. The proof uses some algebraic number theory, some analytic number theory and some real analytic group theory.