Walter Misar
modularity of elliptic curves
Equivalent conditions for an elliptic curve of conductor N to be modular:
- The L-function L(E,s) equals the L-function L(f,s) for a newform of weight 2 and level N.
- The elliptic curve is isogenous to a strong Weil curve which is isomorphic to a one-dimensional factor of the Jacobian of $X_0(N)$.
- The galois representation $\rho_E,l$ is modular for one/all primes $l$.
CM case:
Shimura, Goro ''On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields'' Nagoya Math. J. Volume 43 (1971), 199-208.
semistable case:
Wiles, Andrew ''Modular elliptic curves and Fermat's last theorem.'' Ann. of Math. (2) 141 (1995), no. 3, 443--551.
Taylor, Richard; Wiles, Andrew ''Ring-theoretic properties of certain Hecke algebras.'' Ann. of Math. (2) 141 (1995), no. 3, 553--572
27|N:
Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor: On the modularity of elliptic curves over '''Q''': Wild 3-adic exercises, Journal of the American Mathematical Society 14 (2001), pp. 843–939.