18 . 783 Elliptic Curves Spring 2013 Lecture # 18 04 / 18 / 2013

converge absolutely for any fixed τ ∈ H, by Lemma 16.11, and uniformly over τ in any compact subset of H. The proof of this last fact is straight-forward but slightly technical; see [1, Thm. 1.15] for the details. It follows that g2(τ) and g3(τ) are both holomorphic on H, and therefore ∆(τ) = g2(τ) 3 − 27g3(τ) is also holomorphic on H. Since ∆(τ) is nonzero for all τ ∈ H, by Lemma 16.21, the j-function j(τ) is holomorphic on H as well. The lattices [1, τ ] and [1,−1/τ ] = −1/τ [1, τ ] are homothetic, and the lattices [1, τ + 1] and [1, τ ] are equal; thus j(−1/τ) = j(τ) and j(τ + 1) = j(τ), by Theorem 17.6.