I am wondering if any of you could point me to any books and/or lecture notes that explain the Weierstrass $\wp$ function for a self-studying student of elliptic curves and functions. I am interested in any resources that may give the history of the Weierstrass function and its derivation. I do understand the basics of lattices and doubly-periodic functions, but I am having trouble seeing the thought process that led to the creation of the function itself. I have tried searching through many books and while I have found several good ones, I haven't yet found one that shows the thought process that led to the Weierstrass $\wp$ function. (Sorry, but I am not sure how to format the letter that is traditionally associated with this function).

What resources are there to understand the derivation of $\wp$?

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    $\begingroup$ Put \wp(z) enclosed by the sign of dollar as usual and you will get $\wp(z)$ $\endgroup$ – Piquito Dec 17 '15 at 22:14

HINT.- here I give you a brief note, hoping that something you can help.

Knowing that all $L$-elliptic function (i.e. meromorphic and $L$-periodic) must have a finite order at least equal to $2$, the function $\wp$ gives a solution to the problem of construct a $L$-elliptic function taking exactly this minimum of twice all complex value $u$ in all fundamental parallelogram $P$ for the lattice $L$.

Assuming that such a function exists, it can have either just one pole of order 2 or two simple poles (two distinct kinds of these functions!); both exist in fact, the first one with a pole of order two are the $L$-functions $\wp$ of Weierstrass and the second one, the elliptic functions of Jacobi with two simple poles. The first function is defined by $$\wp(z)=\frac{1}{z^2}+\sum_{\omega\in L\setminus\{0\}}[\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}]$$ (a simpler form with $\sum \frac{1}{(z-\omega)^2}$ is useless because it is not convergent and the variation shown in $\wp$ has no other purpose than to ensure the necessary convergence).

If $P$ is the fundamental parallelogram of vertices $0,\omega_1,\omega_2,\omega_1+\omega_2$, the values $\wp(z_0)$ for $z_0=0,\frac{\omega_1}{2},\frac{\omega_2}{2},\frac{\omega_1+\omega_2}{2}$ are taken just once because $z_0$ is in each case a point of order two in $P$ (this because $0$ is a double pole with residue zero and the other three are double points since $\wp’=0$ and $\wp’’\ne 0$). All the other points $z\in P$ are simple and for all complex value $u\ne \wp(z_0)$ one has $\wp(z_1)=\wp(z_2)=u$ for certain unique $z_1$ and $z_2$ in $P$ which either are symmetric respect to $\frac {\omega_1+\omega_2}{2}$ (in whose case $z_1$ and $z_2$ are interior points of $P$) or are symmetric respect to $\frac{\omega_k}{2}$; $k=1,2$ (in whose case $z_1$ and $z_2$ are in the boundary of $P$, in the two sides assigned to $P$)

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►$\wp(z)$ has a derivative $\wp’(z)$ which is odd of order three and all $L$-elliptic function belongs to the field $\mathbb C(\wp, \wp’)$ of rational expressions of $\wp(z)$ and $\wp’(z)$ with complex coefficients; this field is a quadratic extension of the field $\mathbb C(\wp)$ formed by all the even $L$-elliptic functions.

►$\wp$ and $\wp’$ are transcendental functions but are not algebraically independent and it is verify that $$(\wp’)^2=4\wp^3-g_2 \wp-g_3$$ where the called invariants of $\wp$ are defined by $$g_2=60\sum_{\omega\in L\setminus\{0\}}\frac{1}{\omega^4}$$ and $$g_3=140\sum_{\omega\in L\setminus\{0\}} \frac{1}{\omega^6}$$ Besides one has an “addition theorem” given by $$\wp(u+v)=\frac 14[\frac{\wp’(u)-\wp’(v)}{\wp(u)-\wp(v)}]^2-\wp(u)-\wp(v)$$ These two equalities allow us go from the transcendent to algebraic and this is the link between the elliptic functions and the cubic curves of genus 1.


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