What even *are* elliptic functions?
The theory of elliptic functions started with elliptic integrals and the key players were Gauss, Legendre, Abel, Jacobi and finally Ramanujan.
A parallel approach using complex analysis was developed by Weierstrass.
I will present a brief outline of the approach based on elliptic integrals and at the end mention a thing or two about complex analytical approach.
Elliptic integrals arise while evaluating the arc length of an ellipse. If the equation of ellipse is $$x=a\cos t, y=b\sin t$$ then the arc length is given as $$L(t) =\int_{0}^{t}\sqrt{a^2\sin^2x +b^2\cos^2x}\,dx$$ The above is a typical (but slightly difficult) example of elliptic integral.
In standard notation we define elliptic integral of first kind via $$u=F(\phi, k) =\int_{0}^{\phi}\frac{dx} {\sqrt{1-k^2\sin^2x}}, \phi\in\mathbb {R}, k\in(0,1)$$ The parameter $k$ is a fixed constant called modulus. Sometimes one uses the parameter $m$ instead of $k^2$ and then the notation is $F(\phi\mid m) $.
Since the integrand is positive it follows that $u=F(\phi, k) $ is a strictly increasing function of $\phi$ and therefore is invertible. We write $\phi=\operatorname{am} (u, k) $ and say that $\phi$ is the amplitude of $u$. The elliptic functions are then defined by \begin{align} \operatorname {sn} (u, k) & =\sin\operatorname {am} (u, k) =\sin\phi\notag\\ \operatorname {cn} (u, k) & =\cos\operatorname {am} (u, k) =\cos\phi\notag\\ \operatorname {dn} (u, k) & =\sqrt{1-k^2\operatorname {sn} ^2(u,k)}\notag \end{align}
It appears from the above definition that the parameter $k$ is a silent spectator, but the most interesting aspects of the theory are hidden in $k$. But to deal with it we need to fix $\phi$ and we define two integrals $$K(k) =\int_{0}^{\pi/2}\frac{dx}{\sqrt{1-k^2\sin^2x}},E(k)=\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2x}\,dx$$ and introduce the complementary modulus $k'=\sqrt{1-k^2}$. The above integrals satisfy a key relation $$K(k) E(k') +K(k') E(k) - K(k) K(k') =\frac{\pi} {2}$$ which goes by the name of Legendre's identity. Usually if the value of $k$ is known from context one writes $K, K', E, E'$ instead of $K(k), K(k'), E(k), E(k') $.
If $k=0$ or $k=1$ the elliptic integrals reduce to elementary functions and the magical properties of elliptic integrals and functions (yet to be described) vanish.
Elliptic functions satisfy addition formulas like the circular functions. Thus functions of argument $u+v$ can be expressed using functions of $u$ and $v$. The key aspect is that the formula uses algebraic combination of functions of $u, v$. The converse also holds. Any sufficiently nice function (keyword is analytic) with an algebraic addition formula is necessarily an elliptic or a circular function. The key formula here is $$\operatorname {sn} (u+v) =\frac{\operatorname {sn} u\operatorname {cn} v\operatorname {dn} v+\operatorname {sn} v\operatorname {cn} u\operatorname {dn} u} {1-k^2\operatorname {sn} ^2u\operatorname {sn} ^2v} $$ The formula is usually proved via smart use of derivatives of elliptic functions (which themselves can be obtained using their definitions).
The truly magical property of elliptic functions (with no analogue for circular functions) is the transformation formulas between elliptic functions of two different but related moduli. For each positive integer $n>1$ there are two sets of transformation formulas: one which relates the elliptic functions of given modulus with those of a greater modulus (ascending transformation) and another which relates elliptic functions with those of a lesser modulus (descending transformation).
The simplest case is for $n=2$ which is famous by the name Landen transformation. The transformation is neither obvious nor easy to prove. The ascending transformation (by John Landen) starts with $$u=\int_{0}^{\phi}\frac{dx}{\sqrt{1-k^2\sin^2x}}$$ and uses substitution $$\sin(2t-x)=k\sin x$$ and after reasonable algebra one gets $$\frac{dx} {\sqrt{1-k^2\sin^2x}}=\frac{2}{1+k}\cdot\frac{dt}{\sqrt{1-l^2\sin^2t}}$$ where $l=2\sqrt{k}/(1+k)$. The corresponding formula for elliptic functions is $$\operatorname {sn} (u, k) =\frac{2}{1+k}\cdot\dfrac{\operatorname {sn} \left(\dfrac{(1+k)u} {2},\dfrac{2\sqrt{k}}{1+k} \right)\operatorname {cn} \left(\dfrac{(1+k)u} {2},\dfrac{2\sqrt{k}}{1+k} \right) }{\operatorname {dn} \left(\dfrac{(1+k)u} {2},\dfrac{2\sqrt{k}}{1+k} \right)}$$ The descending transformation uses the substitution (given by Gauss) $$\sin t=\frac{(1+k)\sin x} {1+k\sin^2x}$$ to get $$\frac{dt} {\sqrt{1-l^2\sin^2t} }= (1+k)\frac{dx}{\sqrt{1-k^2\sin^2x}}$$ The corresponding formula for elliptic functions is $$\operatorname {sn} \left((1+k)u, \frac{2\sqrt{k}}{1+k}\right)=\frac{(1+k)\operatorname {sn} (u, k) } {1+k\operatorname {sn} ^2(u,k)} $$ Even more important is the relationship between $K(k), K(k'),K(l), K(l') $ (these are typically denoted by $K, K', L, L'$) $$L=(1+k) K, K=\frac{1+l'}{2}\cdot L$$ It can be seen that the relationship between $k, l$ is same as that between $l', k'$ and hence we get $$K'=(1+l') L', L'=\frac{1+k}{2}\cdot K'$$ From the above two relations we get $$\frac{K'} {K} =2\cdot\frac{L'}{L}$$ Jacobi further gave transformation formulas when $n$ is prime and showed that the relationship between $k, l$ is algebraic and $K'/K=nL'/L$. The theory can be easily extended to all values of $n$ and the above result holds. Given a positive integer finding an algebraic relationship between moduli $k, l$ such that $K'/K=nL'/L$ is a computational challenge. Such a relationship is called modular equation of degree $n$.
Using transformation theory Jacobi derived infinite product and series representations for elliptic functions. A key parameter in such representations is $q=e^{-\pi K'/K} $ which is called nome corresponding to modulus $k$. Jacobi introduced his theta functions which use the nome $q$ and expressed elliptic functions as ratios of theta functions. Theta functions themselves are very interesting with wide applications in other fields (number theory for example) and their beauty lies in large number of algebraic relationships between them. Jacobi gave a complete description of theta functions and a host of formulas related to elliptic and theta functions.
Ramanujan somehow fell in love with these topics and developed his theory of theta functions and elliptic functions using different notation and technique and went far ahead of Jacobi. He had almost magical powers in this field and till date no one knows how he derived a large number of modular equations and related formulas. Most of his results have only been verified using symbolic software. Notable here is the fact that both Jacobi and Ramanujan avoided complex analytic techniques (Ramanujan had no serious idea of complex analysis but his achievements remain still unparalleled in the field of elliptic function theory).
Liouville and Weierstrass on the other hand championed the methods of complex analysis to deal with elliptic functions. The starting point in this approach is the study of doubly periodic functions and one learns that elliptic functions are doubly periodic and conversely doubly periodic functions can be expressed in terms of elliptic functions. In this approach elliptic integrals take backstage and transformation theory of Jacobi and modular equations of Ramanujan are presented in a very different framework called modular forms.
Another important feature of elliptic functions is complex multiplication. Using addition formula for elliptic functions it is easy to see that if $n$ is a positive integer then we can express elliptic functions of argument $nu$ in terms of elliptic functions of argument $u$. However it turns out that for some values of $k$ there exist complex number $\alpha\in\mathbb{C} \setminus \mathbb{R} $ such that elliptic functions of argument $\alpha u$ can be expressed in terms of functions of argument $u$. This happens only when the value of $k$ is such that $K'/K$ is the square root of a rational number. Under these circumstances $k$ turns out to be an algebraic number.
Let $n$ be a positive integer and $F$ be the smallest subfield of $\mathbb{C} $ which contains the imaginary number $i\sqrt{n} $ and let $\mathbb{Z} _{F} $ be the set of algebraic integers in $F$. Let $k\in(0,1)$ such that $K'/K=\sqrt{n} $. Then for any $\alpha\in\mathbb {Z} _F$ we can express $\operatorname {sn} (\alpha u, k) $ in terms of $\operatorname {sn} (u, k) $.
The link between elliptic function theory and imaginary quadratic extensions of $\mathbb{Q} $ is the most fascinating and difficult one. Abel worked in this direction and Kronecker understood its importance well enough. Kronecker was working on his theorem regarding abelian extensions of $\mathbb{Q} $ and realized that a similar result would hold for abelian extensions of imaginary quadratic extensions of $\mathbb{Q} $ and elliptic functions would play a central role there. All of this later developed into class field theory.
For a lattice $L = u\Bbb{Z}+v\Bbb{Z}$ in $\Bbb{C}$ let $ \wp_L(z)$ be the only $L$-periodic function with only one double pole at $z=0$ where $\wp_L(z)=z^{-2}+O(z^2)$. Since analytic $L$-periodic functions are constant then $$\wp_L'(z)^2=4\wp_L(z)^3+g_2(L)\wp_L(z)+g_3(L)$$ with $g_2,g_3$ found from the $z^4,z^6$ coefficient of $\wp_L(z)-z^{-2}$.
Thus $z\mapsto(\wp_L(z),\wp_L'(z))$ is an isomorphism from the complex torus $\Bbb{C}/L$ to the elliptic curve $E:y^2=4x^3+g_2(L)x+g_3(L)$.
$$K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2t}}= \int_0^1 \frac{dw}{\sqrt{(1-k^2 w^2) (1-w^2)}}$$ $$=\int_0^1 \frac{ds}{2\sqrt{(1-k^2 s) (1-s)s}}=\frac1{ik}\int_{-(k^{-2}+1)/3}^{1-(k^{-2}+1)/3} \frac{dx}{\sqrt{4x^3 + a(k)x+b(k)x}}$$ where $a(k)=4\frac{-k^4 + k^2 - 1}{3k^4},b(k)=4\frac{-2k^6 + 3k^4 + 3k^2 - 2}{27k^6}$.
Take the lattice $L$ such that $a(k)=g_2(L),b(k)=g_3(L)$. From the $z\mapsto(\wp_L(z),\wp_L'(z))$ isomorphism we get $$\frac{dx}{ \sqrt{4x^3 + a(k)x+b(k)x}} = \frac{d\wp_L(z)}{\wp_L'(z)} = dz$$
In the group law of $E$: $-(x,y)=(x,-y)$ thus the points $(k^{-2}+1)/3,0), (1-(k^{-2}+1)/3,0)$ are in the 2-torsion of $E$. On the complex torus side it is obvious the 2-torsion is : $L,u/2+L,v/2+L,(u+v)/2+L$, altogether it means $$K(k) = \frac1{ik}\int_{u/2}^{(u+v)/2+\ell} dz = \frac{v/2+\ell}{ik}\qquad\text{ for some } \ell \in L$$
In other words the complete elliptic integrals are the way to recover the lattice from the elliptic curve $y^2=(1-k^2 s) (1-s)s$.