= Solution
The inverse is nonconstant; locally at zero,
$$
\mathcal H(0)=0,
\qquad
\mathcal H'(0)=\frac1{H'(0)}=1.
$$
If the assumed <meromorphic function> $\mathcal H$ had no <poles>, it would be <entire function>[entire]. It is bounded on the compact closure of a <fundamental parallelogram>, and its two <periods> then make it bounded throughout the <complex plane>. The <Liouville theorem> would force it to be constant, a contradiction. Therefore $\mathcal H$ has at least one pole, as stated by the general result that a <nonconstant elliptic function has a pole>.
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