Relation between different types of elliptic functions

Question

Disclamer: all formulas in this post are intended to be schematic, coefficients signs etc. are mostly wrong.

I do various physics-motivated computations involving elliptic functions and the more sources I look at the bigger grows my confusion. It seems there are many different parametrizations of elliptic functions, for example

1. Elliptic integrals such as $K(x)=\int_0^1\frac{dt}{\sqrt{(1-t^2)(1-xt^2)}}$.
2. Jacobi theta functions like $\vartheta_3=\sum_{n\in\mathbb{Z}}q^{n^2}$
3. Double-periodic functions such as Weierstrass function $\wp(z)\sim\sum_{n,m}\frac{1}{(z-n-m\tau)^2}$
4. Modular forms like $E_2(q)=1-24\sum_n\frac{nq^n}{1-q^n}$ or $\eta=q^{1/24}\prod_n(1-q^n)$

For example, some relations between them are

1. $K(x)=\frac{\pi}{2}\vartheta_3^2(q)$ for $q=\exp\Big(-2\pi\frac{K(1-x)}{K(x)}\Big)$
2. $\vartheta_1'=\eta^3$
3. $\wp(z)\sim \frac1{z^2}+z^2E_2+z^4E_4+\dots$

Is it possible to give a concise summary of how different types of functions are related to each other? Make these relationships less surprising? For example I've known for a long time that $\vartheta_3^2(q)=K(x)$ but still learning that $\vartheta_2^2(q)=xK(x)$ surprised me.

Of course a deeper theory behind all these objects exist but learning it seems quite an investment. Are there some exposition intermediate between full-blown modern algebraic geometry and simple statement of definitions and formulas relating them? Especially I'm interested in some resource that would connect many of these thigs. For example Mumford's lectures on theta-functions are great but they mostly don't touch other types of functions mentioned above.

Answer 1

You might want to start from the complex torus to elliptic curve (Riemann surface and group) isomorphism given by $z\to (\wp_L(z),\wp_L'(z)),\Bbb{C}/L\to y^2=4x^3-g_2(L)x-g_3(L)$. The incomplete elliptic integrals are the inverse map, the complete elliptic integral gives the lattice from the elliptic curve. The elliptic functions are the doubly periodic functions, meromorphic on $\Bbb{C}/L$, or the $\log$ and $\exp$ of them. The modular forms and modular functions are the functions of lattices such that $F(rL)=r^{-k}F(L)$, eg. the coefficients of $\wp$ at $z=0$. If $k=0$ then $\tau\to F(\Bbb{Z+\tau Z})$ is $SL_2(\Bbb{Z})$ invariant. $\theta(a/b,\tau)$ has $k=1/2$. To make $\theta(z,\tau)=\sum_n e^{2i\pi nz}e^{i\pi n^2\tau}$ appear you may start from the Jacobi product $P(z,\tau)=\prod_n (1+e^{2i\pi z} e^{i\pi (2n-1)\tau})(1+e^{-2i\pi z} e^{i\pi (2n-1)\tau})$ : as a function of $z$ its Fourier series is $C(\tau) \theta(z,\tau)$. Then every elliptic function is of the form $B\prod_j P(z+a_j,\tau)^{e_j}$ with $\sum_j e_j=0$ and $\sum_j a_j=0$. This is because any holomorphic elliptic function is constant. The space of integer weight $k$ modular forms $M_k(SL_2(\Bbb{Z}))$ is finite dimensional (due to the analyticity condition and the existence of $\Delta$) and belongs to $\Bbb{C}[E_4,E_6]$: it suffices to check finitely many Fourier coefficients to compare two modular forms, that's how we know that $E_4^3-E_6^2=1728 \Delta$ where $\Delta$ is given by a seemingly unrelated infinite product. $\Delta$ appears everywhere because if $f \in M_k(SL_2(\Bbb{Z})$ vanishes at $i\infty$ then $f/\Delta\in M_{k-12}(SL_2(\Bbb{Z})$. The field of $SL_2(\Bbb{Z})$-invariant meromorphic functions is finitely generated, by $j(\tau)$, this is because the modular curve is a compact Riemann surface. Most relations between elliptic and modular stuffs have an interpretation in those terms.

Answer 2

There is a "simple" analytic definition of an elliptic function: it is a doubly-periodic meromorphic function of complex variable. Number of poles (a.k.a. order) of such a function is important - every such function may be expressed as a rational function of an elliptic function of 2nd order, and its derivate.

So, algebraically, it looks a bit like trigonometry, where $\sin(x)$ and $\cos(x)=\sin(x)'$ play a similar role, and inverse trigonometric functions are akin to elliptic integrals - inverses of elliptic functions. But it's more complicated, as instead of just $\sin(x)$ you have infinitely many such "basic" functions.

Then you can start deriving relations between these functions and their inverses; this is an approach in e.g. (in Russian only, it seems) M. Lavrentiev and B. Shabat, Methods of the theory of functions of a complex variable.

Answer 3

One great book that I've found is Elliptic Curves Function Theory, Geometry, Arithmetic by Henry McKean and Victor Moll.