Modular Lambda Function as a Cross ratio What ie the meaning of the statement : Modular Lambda Function is such that,over any point $\tau$ in the upper half plane, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve $\mathbb{C}/\langle 1, \tau \rangle$
 A: The elliptic curve $\mathbb{C}/\langle 1, \tau \rangle$ is an abelian group with a "-1" automorphism, given by sending each point $z$ to $-z$.  If we take the quotient by this automorphism, we get a projective line.  Another way to say that is there is an action of a group of order two on the elliptic curve that yields a two-fold cover of a projective line that is ramified at four points (namely the four points of order dividing two that are fixed by the -1 automorphism).
If we choose an ordering on these points, and choose the first three  (say $a,b,c$), there is a unique automorphism of the projective line that sends $a \mapsto 0$, $b \mapsto 1$ and $c \mapsto \infty$.  The fourth point will go to some number $\lambda$, which is the cross ratio of the four initial points.  The value of $\lambda$ depends on the lattice generated by 1 and $\tau$ (i.e., on the isomorphism type of the elliptic curve), together with the ordering on the four branch points (which basically yields a full level two structure).
If we take a suitable symmetrization of the $\lambda$ function under changing the orders of the four points, we get the $j$-invariant.  The cross ratio is invariant under the normal even Klein 4-group in $S_4$, so we only need to symmetrize over the quotient $S_3$ that acts on the first three points $\{ a,b,c \}$.
