In the literature, an elliptic curve $E/\mathbb{Q}$ is defined to be modular in two different ways

1) if there exists a nonconstant morphism $X_0(N) \to E$,

2) if there exists a modular form $f$ with $L(E,s) = L(f,s)$.

I get that from the first definition, one can deduce the second by pullback of a holomorphic differential on $E$ to a newform which has the same $L$-function, so my question is how does one get from the second to the first? Are both definitions equivalent? They seem to be used interchangeably by some authors but I have not been able to find any reference on this.

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    $\begingroup$ Did you look at Diamond–Shuramn book, in particular p. 362? It uses Faltings' isogeny theorem. $\endgroup$ – Watson Dec 4 '18 at 13:56
  • $\begingroup$ @Watson Thanks, I guess that will do. $\endgroup$ – hrt Dec 4 '18 at 13:59

The book Diamond–Shurman "A first course in modular forms", in particular p. 362, tells you how $2)$ implies $1)$. This is actually a deep result: it requires to use Faltings' isogeny theorem.

If $L(f,s) = L(E,s)$ for some normalized newform $f \in S_2(\Gamma_0(N_E))$, then $a_p(f) = a_p(E)$ for every prime $p$ and then all the Fourier coefficients of $f$ at $i\infty$ are integers. In particular, the abelian variety $A_f$ constructed by Eichler and Shimura has dimension $[K_f : \Bbb Q] = 1$, where $K_f$ is the coefficient field of $f$.

Thus, we have an elliptic curve $A_f$ such that $L_p(A_f, s) = L_p(E,s)$ for almost every prime $p$ (equation 8.42 in Diamond–Shurman). Faltings' isogeny theorem asserts that you have an isogeny $\phi : A_f \to E$. On page 247 of Diamond–Shurman, using the decomposition of $J_0(N)$, one gets an isogeny $J_0(N) \to E$, and on page 216, we get a non-constant holomorphic map $X_0(N) \to E(\Bbb C)$. Finally, it is mentioned on page 292 that we get a non-constant morphism $X_0(N) \to E$ defined over $\Bbb Q$ (and work of Carayol ensures that we may take $N = N_E$).

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    $\begingroup$ The OP's condition (2) did not specify that $f$ has level equal to the conductor $N_E$ of $E$. A sequence of steps to get from this weaker form of (2) to condition (1) is set out on the 3rd page of the BCDT paper that finished the proof of modularity of elliptic curves over $\mathbf Q$ here: in the notation of the 6 conditions there, follow $(1) \Rightarrow (4) \Rightarrow (3) \Rightarrow (2) \Rightarrow (6)$. $\endgroup$ – KCd Dec 4 '18 at 14:16
  • $\begingroup$ @KCd : thank you. Anyway, OP's condition (1) did not specify the integer $N$, so I think it is not necessary to use Carayol's work for answer OP's question. $\endgroup$ – Watson Dec 4 '18 at 14:22

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