This is question in mathematical exposition, not research, I hope this is ok.
I am writing a book about great theorems. My question is: what is the shortest formulation of the modularity theorem, which includes definitions of all concepts used, except of those included in standard undergraduate courses?
The theorem has many equivalent formulations. One example is given by wikipedia:
Let ${\mathbb H} = \{z \in {\mathbb C}, \text{Im}(z) > 0\}$, $j:{\mathbb H}\to{\mathbb C}$ be given by $j(z) = 1728\frac{20 G_4(z)^3}{20 G_4(z)^3-49G_6(z)^2}$, where $G_k(z)=\sum\limits_{(m,n)\neq (0,0)}(m+nz)^{-k}$. For every positive integer $n$, there exists a non-zero irreducible polynomial $P_n(x,y)$ with integer coefficients such that $P_n(j(nz),j(z))=0, \, z\in {\mathbb H}$. The set $X_0(n)$ of pairs of rational numbers $(x,y)$ such that $P_n(x,y)=0$ is called the classical modular curve over ${\mathbb Q}$.
Elliptic curve C over ${\mathbb Q}$ is the set of rational solution to equation $y^2=x^3+ax+b$, where $a,b \in {\mathbb Q}$ are such that $4a^3+27b^2 \neq 0$. It is called modular if it can be obtained via a rational map with integer coefficients from $X_0(n)$ for some positive integer $n$. The modularity theorem states that in fact every elliptic curve over ${\mathbb Q}$ is modular.
This is reasonably short but is it correct and rigorous? If not, how can it be corrected? Also, what exactly is meant by phrase "can be obtained via a rational map with integer coefficients"? If this formulation is not rigorous and cannot be easily fixed, can you suggest an alternative one?
Update: An even simpler formulation from Wolfram Mathworld http://mathworld.wolfram.com/Taniyama-ShimuraConjecture.html
According to [1, p.40], function $f:{\mathbb H}\to{\mathbb C}$ is a modular function if and only if $f(z)=\frac{P(j(z))}{Q(j(z))}$, where $j(z)$ is defined above, and $P$ and $Q$ are some polynomials with complex coefficients. Now, according to [2], moludar function $f$ has level $N$ if $f\left(\frac{az+b}{cz+d}\right)=f(z)$ for all integers $a,b,c,d$ with $ad-bc=1$ and $N|c$, and the modularity theorem states that, given elliptic curve $y^2=Ax^3+Bx^2+Cx+D$ over $\mathbb Q$, there are conconstant modular functions $f(z)$ and $g(z)$ of the same level N, such that $f(z)^2=Ag(z)^3+Bg(z)^2+Cg(z)+D$. Is this correct?
[1] Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, 1997.
[2] Cox, David A. "Introduction to Fermat's last theorem." The American Mathematical Monthly 101.1 (1994): 3-14
