Timeline for Historical question about modularity of CM curves
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 2, 2011 at 12:19 | comment | added | Victor Miller | The paper math.berkeley.edu/~ribet/Articles/korea.pdf by Ken Ribet gives a good summary as of the date just before Wiles' proof of modularity for semi-stable curves. | |
| Jun 2, 2011 at 0:29 | comment | added | Victor Miller | @Matthew, Thanks. I took down my well-thumbed copy of Shimura's book (which I bought right after it came out) from my bookshelf, and noted that it doesn't appear to contain the result in the paper (which looks like it was written after the book) but is close (not surprisingly). | |
| Jun 2, 2011 at 0:01 | comment | added | Emerton | ... giving rise to factors of modular Jacobians dates from Shimura's work in the early 70s. Regards, Matthew | |
| Jun 2, 2011 at 0:00 | comment | added | Emerton | ... operators. (I think these operators also appear in Atkin--Lehner, when they discuss newforms fixed under twisting by a character, i.e. newforms with CM, as we would now say.) Also, I think that there is another paper of Shimura with a similar title from a year or two later (1973?) in which he discusses more generally the factors of modular Jacobians attached to general weight 2 eigenforms. He also discusses these in his book, I think, which came out in the early 70s as well. So it seems that (at least insofar as appearing in print is concerned) the general idea of wt. 2 forms ... | |
| Jun 1, 2011 at 23:57 | comment | added | Emerton | Dear Victor, You're welcome! In fact, one can view Shimura's paper as proving a case of the Tate conjecture: he knows (by Hecke, as you wrote) that the $L$-series of the CM curve corresponds to a weight 2 modular form, and now he wants to physically realize the CM form in the Jacobian. For this, he defines correspondences on the Jacobian that concretely realize the CM, and uses them to split off the CM curve. When Ribet verified the Tate conjecture for all abelian variety factors of Jacobians of modular curves (in a paper in the 80s, if I remember correctly), he uses these same twist ... | |
| Jun 1, 2011 at 21:55 | vote | accept | Victor Miller | ||
| Jun 1, 2011 at 21:55 | comment | added | Victor Miller | @Matthew, Thanks! I was a bit surprised that it was so recent. I was in grad school in 1971, and remembered hearing it mentioned then. I didn't realize that it was "hot off the press". So, according to Shimura's paper, Deuring proved that the $L$-series of a CM curve was a Hecke $L$-series, and so was the Mellin transform of a modular form of weight 2, but it wasn't until Shimura's paper that he showed that this implied that the curve was a quotient of the modular curve. | |
| Jun 1, 2011 at 20:09 | history | answered | Emerton | CC BY-SA 3.0 |