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We know that for each elliptic curve over rationals, we can define the Dirichlet series of the Hasse–Weil $L$-function, i.e., the function associated with an elliptic curve over rationals.

Then my question is: Consider a motivic $L$-functions $f$, can we find a set of elliptic curves over rationals associated with $f$?. Simply I ask about the inverse of the first statement in the motivation of this question.

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up vote 4 down vote accepted

This has already had some votes to close, but I'll see if I can answer it anyway...

The answer is "no". There are lots of motivic L-functions that are not elliptic curve L-functions, just because there are lots of motives that are not $H^1$ of an elliptic curve! For instance, the L-function attached to a modular form of weight $k > 2$ (which is motivic, by a theorem of Scholl) does not have anything to do with the L-function of any elliptic curve, because the form of the $\Gamma$-factors is different.

A nontrivial relevant statement that might interest you is perhaps this one: if $L(s) = \sum_{n \ge 1} a_n n^{-s}$ is a Dirichlet series with coefficients $a_n \in \mathbf{Q}$, and $L$ and all of its twists by Dirichlet characters have analytic continuation to all $s \in \mathbf{C}$ and satisfy a functional equation of the same kind as the $L$-function of an elliptic curve (in particular, with the same $\Gamma$-factors), then $L(s)$ is indeed the $L$-function of an elliptic curve. This follows from Weil's converse theorem (which is essentially the same statement with "modular form" in place of "elliptic curve") together with the fact that one can attach an elliptic curve to any weight 2 modular form with coefficients in $\mathbf{Q}$.

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@RH: I don't understand, what exactly are you asking? –  David Loeffler Apr 27 '13 at 20:32
    
David, if you don't know the backstory to this MO user, feel free to email me –  Yemon Choi Apr 29 '13 at 18:22
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