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Okay so we are trying to understand the mapping properties of the the modular function lambda which is you know invariant under the action of the congruence mode to subgroup of basil to Z so let me let me quickly recall what we are trying to prove so we have this function lambda of tau which is which is given by III tau is z2 of tau by even of tau minus e2 of tau lambda is defined on the upper half plane and taking values in the complex plane and in fact it does not assume the value zero and one okay and this is analytic never is equal to 0 or 1 okay and the claim was the following that if we take the following region in the the tau plane this is the tau plane we take this we take this line segment which is given by real part of tau is equal to 1 this is the point 1 this is origin and then we draw we take this point qn by half and consider the semi circle centered at the point half and radius half we take this region Omega this this region the boundary of Omega consists of this positive part of the imaginary axis and part of this sensor this this same circular arc and then this this positive part of this line parallel to the imaginary axis and then the claim was that so the theorem we are trying to prove is that is lambda Maps Omega a holomorphic holomorphic alee isomorphic follow more freakily onto onto the upper half plane k in in a you know one in a 1 a 1 to 1 manner okay which means which is same as saying that lambda from Omega to the upper half plane is a holomorphic a sum of some okay and further and further extends to the boundary of Omega okay continuously so that you see infinitely the point at infinity the point the origin and one are mapped mapped on to 0 1 infinity respectively so this is the theorem we are trying to prove that land lambda maps c lambda is defined of the upper half plane and it takes complex values what we want to say is that if you consider only this region then lambda maps the interior of this region by omega i mean the interior of this region it perhaps the interior of this region on to the upper half plane in a one-to-one manner and you know when a holomorphic map is 12 and you know it is a holomorphic isomorphism onto the image which is an open set and and further you can extend lambda to the boundary of Omega so that this extension is continuous and the point at infinity 0 and 1 are mapped on to 0 1 and infinity in the target plane okay in the lambda plane all right so this is the theorem we are trying to prove and as it happened the proof of the CM is extending to a few lectures because one has to do X extensive computations okay so let me recall what we did what we have done so far okay so what you see what you have done so far is the following what we have proved so far is that the first thing is lambda is is real on the boundary of Omega except the point 0 & 1 okay so you see what I want you to understand is lambda is already defined and analytic on the per half plane so there are really difficult...