A univariate function $f$ is periodic with period $p_1,\ldots,p_k$ if
$f(z) = f(z + \sum_{i=1}^k n_i \cdot p_i)$
for all complex $z$ and integers $n_i$. Elliptic function is an example of doubly periodic function. It is claimed that a triply periodic univariate function cannot exist, but why? Can't we follow the construction of elliptic functions that uses Schwarz-Christoffel formula to map upper-plane onto a triangle, and use the reflection principle to create a lattice which corresponds to three periods?
Which step fails to hold when we mimic the construction of an elliptic function to create a triply period meromorphic univariate function?
Sorry if the question is stupid. I do not have an access to the original proof that a triply periodic function cannot exist, either. So references are welcome!