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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!

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    Ah, thanks @Mariano. Maybe I should assume the function to be meromorphic?2011-03-04

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The problem is with the lattice of periods you want to consider in the construction...

One can easily show that if a subgroup $\Gamma$ of the additive group $\mathbb C$ contains three linearly independent elemtents over $\mathbb Z$, then it is not closed; so there is no sensible way to do all the reflections you are planning to do... To actually prove triply periodic meromorphic functions do not exist, using that fact about subgroups of $\mathbb C$ and properties of meromorphic functions should get you to where you want.

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    Now I get it! Thanks for the quick and nice reply!2011-03-04