Franklin's Notes

Modular Lambda Function

The modular lambda function $\lambda(\tau)$ is defined for arguments $\tau\in\mathbb H$ in the upper half-plane, and can be defined in terms of the critical values of the Weierstrass P function as follows: Since $\wp$ takes on the values $e_1,e_2,e_3$ each with multiplicity $2$, and it takes on all values in a fundamental parallelogram with the same number of times, we have that $e_1,e_2,e_3$ must all be distinct. This means that $\lambda(\tau)\ne 0,1$.




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