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If I somehow know that for each elliptic curve over $\mathbb{Q}$ the $L$-function has a meromorphic continuation to the whole plane can I easily deduce modularity from that?

If not is there a way to establish meromorphic continuation without going through modularity?

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    $\begingroup$ I don't think any such implication is known. However, Weil's converse theorem implies that meromorphic continuation and functional equation for the L-function and all of its twists implies modularity. I believe this is considered the first "solid" evidence towards the modularity conjecture. $\endgroup$
    – Wojowu
    Commented Oct 28, 2021 at 9:46
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    $\begingroup$ @Wojowu: In Weil's converse theorem one needs more than meromorphic continuation. One needs analytic continuation of the completed $L$-function (i.e. with the gamma factors present) with controlled poles. These assumptions have been weakened by Booker-Krishnamurthy (2014) and Booker (2019). $\endgroup$
    – GH from MO
    Commented Oct 28, 2021 at 10:19
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    $\begingroup$ @GHfromMO Thank you for the clarification, yes, we need the meromorphic continuation with some specific conditions (I think there was also some moderate growth assumption, but perhaps these are not necessary here). $\endgroup$
    – Wojowu
    Commented Oct 28, 2021 at 10:28

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No, this does not work; we need analytic, not just meromorphic, continuation. If meromorphic continuation were enough, then we would know modularity of elliptic curves in a great deal more generality than we do now.

As a consequence of Taylor's "potential modularity" theorem, we know that for any totally real field $F$, and any elliptic curve $E/F$, the $L$-series $L(E/F, s)$ has meromorphic continuation and the expected functional equation (and I think this works for character twists as well). If that were enough to deduce modularity of $E$, then we'd know that elliptic curves over $F$ were modular, which we don't.

(We do now know modularity of all $E / F$ if $[F : \mathbf{Q}] \le 3$, by results of Siksek et al; but that came well after Taylor's potential-modularity theorem and required a great deal of new ideas.)

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  • $\begingroup$ Suppose we know that for each elliptic curve over $\mathbb{Q}$ the $L$-function multiplied by the $\Gamma$-factor extends to a holomorphic function on the whole plane. Does that obviously imply modularity? $\endgroup$
    – novler
    Commented Oct 29, 2021 at 7:23
  • $\begingroup$ No, except in the vacuous sense that anything "implies" a statement which is already known to be true. You need the functional equation as well (and you need non-trivial character twists). May I suggest reading the very thorough account of Weil's converse theorem in the book by Miyake? $\endgroup$
    – David Loeffler
    Commented Oct 29, 2021 at 7:41
  • $\begingroup$ OK thanks I just got that impression from your first sentence. $\endgroup$
    – novler
    Commented Oct 29, 2021 at 7:44
  • $\begingroup$ @novler Do you consider Taylor's potential modularity theorem to be "obvious"? $\endgroup$
    – David Loeffler
    Commented Oct 29, 2021 at 7:45
  • $\begingroup$ No it's not obvious to me (maybe to a Fields medalist it would be). $\endgroup$
    – novler
    Commented Oct 29, 2021 at 8:06

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