Timeline for Does $L$-functions of elliptic curves over $\mathbb{Q}$ being meromorphic obviously imply modularity?
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| Oct 29, 2021 at 6:48 | answer | added | David Loeffler | timeline score: 3 | |
| Oct 28, 2021 at 10:28 | comment | added | Wojowu | @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). | |
| Oct 28, 2021 at 10:21 | history | edited | novler | CC BY-SA 4.0 |
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| Oct 28, 2021 at 10:19 | comment | added | GH from MO | @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). | |
| Oct 28, 2021 at 9:46 | comment | added | Wojowu | 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. | |
| S Oct 28, 2021 at 9:00 | review | First questions | |||
| Oct 28, 2021 at 11:13 | |||||
| S Oct 28, 2021 at 9:00 | history | asked | novler | CC BY-SA 4.0 |