Timeline for Historical Articles about zeta functions of curves
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Mar 13, 2011 at 14:13 | comment | added | Victor Miller | Keith: thanks for the pointer to Lang's article. It was instructive (and hilarious -- cf. Weil's postscript to his letter to Lang at the end). | |
| Mar 13, 2011 at 14:11 | vote | accept | Victor Miller | ||
| Mar 13, 2011 at 9:19 | history | edited | KConrad | CC BY-SA 2.5 |
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| Mar 13, 2011 at 7:49 | history | edited | KConrad | CC BY-SA 2.5 |
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| Mar 13, 2011 at 7:24 | history | edited | KConrad | CC BY-SA 2.5 |
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| Mar 13, 2011 at 7:13 | comment | added | KConrad | Victor: The original formulation of the Taniyama--Shimura conjecture was in terms of the L-series of modular form. See the Lang reference I added in my edited comments to my answer. | |
| Mar 13, 2011 at 7:05 | comment | added | KConrad | David: Deuring's work on this spanned several years and started nearly simultaneously with Weil: Die Zetafunktion einer algebraischen Kurve vom Geschlecht Eins I,II,III,IV (1953--1957). I don't know how much Deuring accomplished in his 1953 paper alone. Does anyone who comfortably reads German want to comment further? | |
| Mar 13, 2011 at 4:25 | comment | added | David Hansen | @ KConrad. Wait, I thought Deuring proved something of the form "L-functions of CM elliptic curves are grossencharakter L-functions", before Weil. | |
| Mar 13, 2011 at 0:27 | comment | added | Victor Miller | @KConrad: I just looked at Weil's paper. He has no references! It looks like (for elliptic curves), he only treats the case of complex multiplication. What I'm after is who (possibly Artin?) had the idea of making a global $L$-function by taking the product of the local factors? On a related note, when the modularity conjecture was first stated (say by Taniyama and/or Shimura) was it in terms of uniformization of elliptic curves by modular functions or the fact that the global $L$-series was the Mellin transform of a particular eigen-form on $\Gamma_0(N)$ of weight 2? | |
| Mar 12, 2011 at 21:29 | comment | added | Felipe Voloch | The story goes (I think I read this in the commentary of Weil's collected works) that Hasse suggested as a thesis problem to a student to prove that the zeta function of an elliptic curve satisfied a functional equation. He asked Weil his opinion of the suitability of problem and Weil thought that it might a bit too hard for a student. | |
| Mar 12, 2011 at 20:53 | comment | added | KConrad | On p. 303 of Ireland and Rosen (section 2 of Chapter 18) they say that the analytic continuation of zeta-functions of elliptic curves over Q to the whole complex plane was conjectured by Hasse, although they don't cite a paper. They also say that the first proof of special cases of the conjecture was in Weil's paper "Jacobi sums as Grossencharakete", which is in Transactions AMS 73 (1952), 487--495. So maybe in that paper you'll find a citation by Weil to the appropriate earlier paper of Hasse (the same one in which Hasse proved RH for ell. curves over finite fields?). | |
| Mar 12, 2011 at 20:43 | comment | added | Victor Miller | So, since it's called the Hasse-Weil zeta function did Hasse previously define it for elliptic curves? | |
| Mar 12, 2011 at 20:37 | history | answered | KConrad | CC BY-SA 2.5 |