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Modularity of Elliptic Curves

Let $ E$ be an elliptic curve defined by a Weierstrass equation $ y^2 = x^3 + ax + b$ with $ a,b\in\mathbb{Q}$. For each prime $ p\nmid \Delta = -16(4a^3 + 27b^2)$, set

$\displaystyle a_p = p+1-\char93 E(\mathbb{Z}/p\mathbb{Z}).
$

Definition 4.1 (Modular)   $ E$ is modular if there exists a cuspidal modular form

$\displaystyle f(z) = \sum_{n=1}^{\infty} b_n q^n\in S_2(\Gamma_0(\Delta))
$

such that $ b_p = a_p$ for all $ p\nmid \Delta$.

At first glance, modularity appears to be a bizarre and unlikely property for an elliptic curve to have. When poor Taniyama (and Shimura) first suggested in 1955 that every elliptic curve is modular, people were dubious. But Taniyama was right. The proof is that conjecture is one of the crowning achievements of number theory.

Theorem 4.2 (Breuil, Conrad, Diamond, Taylor, Wiles)  

EVERY ELLIPTIC CURVE OVER #MATH98# IS MODULAR.

\includegraphics[width=1in]{wiles.eps}
Wiles



William A Stein 2001-11-30