For any natural number $N \geq 1$, define the modular group $\Gamma_0(N)$ to be the following subgroup of the group $\operatorname{SL}(2,{\mathbb{Z}})$ of integer coefficient matrices of determinant 1:

$$\Gamma_0(N) := \left\{ \left. \begin{pmatrix}a & b \\ c & d \end{... ... \operatorname{SL}(2,{\mathbb{Z}})\ \right\vert\ c \equiv 0 \pmod{N} \right\}.$$

Let $\H ^*$ be the subset of the Riemann sphere consisting of all points in the upper half plane (i.e., complex numbers with strictly positive imaginary part), together with the rational numbers and the point at infinity. Then $\Gamma_0(N)$ acts on $\H ^*$, with group action given by the operation

$$\begin{pmatrix}a & b \\ c & d \end{pmatrix} \cdot z := \frac{az+b}{cz+d}.$$

Define $X_0(N)$ to be the quotient of $\H ^*$ by the action of $\Gamma_0(N)$. The quotient space $X_0(N)$ inherits a quotient topology and holomorphic structure from ${\mathbb{C}}$ making it into a compact Riemann surface. (Note: $\H ^*$ itself is not a Riemann surface; only the quotient $X_0(N)$ is.) By a general theorem in complex algebraic geometry, every compact Riemann surface admits a unique realization as a complex nonsingular projective curve; in particular, $X_0(N)$ has such a realization, which by abuse of notation we will also denote $X_0(N)$. This curve is defined over ${\mathbb{Q}}$, although the proof of this fact is beyond the scope of this entry ¹.

Taniyama-Shimura Theorem (weak form): For any elliptic curve $E$ defined over $\mathbb{Q}$, there exists a positive integer $N$ and a surjective algebraic morphism $\phi: X_0(N) \to E$ defined over $\mathbb{Q}$.

This theorem was first conjectured (in a much more precise, but equivalent formulation) by Taniyama, Shimura, and Weil in the 1970's. It attracted considerable interest in the 1980's when Frey [2] proposed that the Taniyama-Shimura conjecture implies Fermat's Last Theorem. In 1995, Andrew Wiles [3] proved a special case of the Taniyama-Shimura theorem which was strong enough to yield a proof of Fermat's Last Theorem. The full Taniyama-Shimura theorem was finally proved in 1997 by a team of a half-dozen mathematicians who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved. As of this writing, the proof of the full theorem can still be found on Richard Taylor's preprints page.

Bibliography

1

Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard; On the modularity of elliptic curves over $\mathbf{Q}$: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843-939

2

Frey, G. Links between stable elliptic curves and certain Diophantine equations. Ann. Univ. Sarav. 1 (1986), 1-40.

3

Wiles, A. Modular elliptic curves and Fermat's Last Theorem. Annals of Math. 141 (1995), 443-551.

Footnotes

... entry¹

Explicitly, the curve $X_0(N)$ is the unique nonsingular projective curve which has function field equal to ${\mathbb{C}}(j(z), j(Nz))$, where $j$ denotes the elliptic modular $j$-function. The curve $X_0(N)$ is essentially the algebraic curve defined by the polynomial equation $\Phi_N(X,Y) = 0$ where $\Phi_N$ is the modular polynomial, with the caveat that this procedure yields singularities which must be resolved manually. The fact that $\Phi_N$ has integer coefficients provides one proof that $X_0(N)$ is defined over ${\mathbb{Q}}$.