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 $\mathbb{C}/\langle 1, \tau \rangle$



Modular Lambda Function as a Cross ratio
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 $\{ a,b,c \}$.


