Taniyama-Shimura theorem

For any natural number $N\ge 1$, define the modular group${\mathrm{\Gamma }}_0(N)$ to be the following subgroup of the group $\mathrm{SL}(2,\mathbb{Z})$ ofinteger coefficient matrices of determinant 1:

Let ${ℍ}^{*}$ be the subset of the Riemann sphere consisting of allpoints in the upper half plane (i.e., complex numbers with strictlypositive imaginary part), together with the rational numbers and thepoint at infinity. Then ${\mathrm{\Gamma }}_0(N)$ acts on ${ℍ}^{*}$, with groupaction given by the operation

Define $X_0(N)$ to be the quotient of ${ℍ}^{*}$ by the action of${\mathrm{\Gamma }}_0(N)$. The quotient space $X_0(N)$ inherits a quotienttopology and holomorphic structure from $ℂ$ making it into a compactRiemann surface. (Note: ${ℍ}^{*}$ itself is not a Riemann surface; onlythe quotient $X_0(N)$ is.) By a general theorem in complex algebraicgeometry, every compact Riemann surface admits a unique realization asa complex nonsingular projective curve; in particular, $X_0(N)$ hassuch 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 thisfact is beyond the scope of this entry.

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

This theorem was first conjectured (in a much more precise, but equivalent formulation) by Taniyama, Shimura, and Weil inthe 1970’s. It attracted considerable interest in the 1980’s whenFrey [2] proposed that the Taniyama-Shimura conjecture impliesFermat’s Last Theorem. In 1995, Andrew Wiles [3] proved aspecial case of the Taniyama-Shimura theorem which was strong enoughto yield a proof of Fermat’s Last Theorem. The full Taniyama-Shimuratheorem was finally proved in 1997 by a team of a half-dozenmathematicians who, building on Wiles’s work, incrementally chippedaway 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 http://abel.math.harvard.edu/ rtaylor/Richard Taylor’s preprints page.