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 $\mathbb{C}/\langle 1, \tau \rangle$
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$\begingroup$ Which parts don't you understand? where have you read this? Have you read mathoverflow.net/faq? $\endgroup$– Yemon ChoiCommented Dec 15, 2010 at 5:13
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$\begingroup$ See also mathoverflow.net/howtoask $\endgroup$– Yemon ChoiCommented Dec 15, 2010 at 5:20
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$\begingroup$ I have read this in: en.wikipedia.org/wiki/Modular_lambda_function I can see it as cross ratio from the Legendre form of elliptic curve, but I do not see how it is that of the branch points of a ramified double cover of the projective line. $\endgroup$– kksCommented Dec 15, 2010 at 6:50
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$\begingroup$ An elliptic curve has (affine) equation $y^2=f(x)$ where $f$ is a cubic or a quartic. In either cases there is a map from $E$ to $P^1$ given by $(x,y)\mapsto x$. This is a double cover, ramified at the zeros of $f$, and also at $\infty$ when $f$ is cubic. $\endgroup$– Robin ChapmanCommented Dec 15, 2010 at 8:07
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1$\begingroup$ There is an excellent answer to you question in Yoshida's book "Hypergeometric Function, My Love". $\endgroup$– Wadim ZudilinCommented Dec 15, 2010 at 10:37
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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 $\{ a,b,c \}$.
answered Dec 15, 2010 at 8:25
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$\begingroup$ In his page math.ucr.edu/home/baez/week229.html John Baez explains the way 4 points occur only once in the double cover of the 2-sphere by a 2-torus, every elliptic curve can be imagined to be. The fact that these 4 points can be chosen along the equator or as vertices touching the sphere of an ideal tetrahedron implies a relationship of the cross-ratio to the hyperbolic space with the sphere as boundary at infinity, something I still find mysterious. $\endgroup$– kksCommented Dec 16, 2010 at 8:17