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One version of Modularity Theorem says that

The elliptic curves with rational $j$-values arise from modular forms.



I know every terminology in the statement, but I don't know what "arise from" mean. Can anyone tell me?

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up vote 6 down vote accepted

The construction of Eichler-Shimura associates to a weight $2$ newform $f\in S_2(\Gamma_0(N))$ (trivial character) an abelian variety quotient $A_f$ of the modular Jacobian $J_0(N)$ (Jacobian of $X_0(N)$). Explicitly one takes the kernel $I_f$ of the homomorphism from the Hecke algebra to $\mathbf{C}$ determined by $f$ and then $A_f$ is $J_0(N)/I_fJ_0(N)$ (defined appropriately). The dimension of this abelian variety is equal to the degree over $\mathbf{Q}$ of the number field generated by the Hecke eigenvalues of $f$. In particular, if the Fourier coefficients of $f$ are rational, $A_f$ is an elliptic curve. The meaning of the boxed statement is that every elliptic curve over $\mathbf{Q}$ is isogenous to $A_f$ for some $f\in S_2(\Gamma_0(N))^{\mathrm{new}}$ with rational Fourier coefficients and some integer $N$ (which can actually be taken to be the conductor of $E$).

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well. I will learn more to understand what you are saying... – hxhxhx88 Jan 8 '13 at 1:33

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