The Dedekind eta function is defined by , for in the complex upper half-plane. If is regarded as the ratio of half-periods of an elliptic curve and are the corresponding Weierstrass coefficients, then .
Under fractional linear transformations of the complex plane with integer coefficients, the Dedekind eta function transforms in simple ways. The most obvious example of such a transformation (apart from the identity) is ; in this case .
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