# Show that the Weierstrass $\wp$ function is absolutely and uniformly convergent

This question comes from Rational Points on Elliptic Curves (Silverman & Tate) exercise $$2.3$$ (a):

$$2.3$$ (a)

Show that the series $$\wp(u) = \frac{1}{u^2} + \sum_{\substack{\omega \in L \\ \omega \neq 0}}\left(\frac{1}{(u-\omega)^2}-\frac{1}{\omega^2}\right)$$ defining the Weierstrass $$\wp$$-function is absolutely and uniformly convergent on any compact subset of the complex $$u$$-plane that does not contain any of the points of $$L$$. Conclude that $$\wp$$ is a meromorphic function with a double pole at each point of $$L$$ and no other poles.

If the compact subspace we are summing contains no points in $$L$$, then how can we complete the sum for $$\omega \in L$$? Does this just mean that the function turns into $$\wp(u) = \dfrac 1 {u^2}$$?

How can we use this to show convergence?

The summation doesn't change: It runs over all $$\omega \in L$$. In the definition of absolute convergence over a compact subset $$K$$ that avoids $$L$$, it's the possible $$u$$ values that are being required to come from $$K$$.
It means that $$\wp(u)$$ has poles precisely on $$L$$. The poles are second order in this case and you can infer that $$\wp(u)=(z-\omega)^{-2}h(z)$$ for some holomorphic non-vanishing function $$h(z)$$ around each $$\omega \in L$$. So we only have $$\wp(u)=\frac{1}{u^2}h(z)$$ around $$0$$ as opposed to "$$\wp(u)=\frac{1}{u^2}$$".
In fact, it has poles nowhere else other than on the lattice points, since it converges uniformly on compact subsets and each partial sum is a holomorphic function on $$\mathbb{C} \setminus L$$.