I am reading a proof of the fact that every even elliptic function $f$ with periods $1$ and $\tau$ is a rational function of the Weierstrass $\wp$ function.
The proof seems to use this fact often, but I don't understand where it comes from:
If $f$ is an even elliptic function with a zero (or a pole) at $0$, $1/2$, $\tau/2$, or $(1+\tau)/2$, the order of the zero (or pole) is even.
What I do know is that for elliptic functions [not necessarily even], (a) the number of poles (up to congruency) is $\ge 2$, and (b) the number of zeros (up to congruency) is the same as the number of poles. I don't understand why evenness of $f$ forces the order of the lattice points and half periods to be even. Why can't for example, one have a simple pole at $0$ and a simple pole at $1/2$? Apologies if this is very obvious...
So, if I have stated the above fact correctly, does this imply the following?
If $f$ is an even elliptic function, it has even order.
Thanks in advance!