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I'm looking for the answer of who first proved modularity of CM curves? That is if $E$ is an elliptic curve over $\mathbb{Q}$ which has complex multiplication then there's a non-constant morphism from $X_0(N)$ to $E$ for some $N$ (not necessarily the conductor). The possible names that I've thought of are Hecke, Deuring, Weil or Shimura. Does anybody know something more definite?

edited Jun 1 2011 at 22:37

asked Jun 1 2011 at 19:57

Victor Miller
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Emerton · Accepted Answer · 2011-06-01 20:09:18Z UTC

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Dear Victor,

I believe it was Shimura in the paper On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J. 43 (1971), 199–208.

Regards,

Matthew

answered Jun 1 2011 at 20:09

Emerton
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@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. – Victor Miller Jun 1 2011 at 21:55

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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 ... – Emerton Jun 1 2011 at 23:57

... 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 ... – Emerton Jun 2 2011 at 0:00

... giving rise to factors of modular Jacobians dates from Shimura's work in the early 70s. Regards, Matthew – Emerton Jun 2 2011 at 0:01

@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). – Victor Miller Jun 2 2011 at 0:29

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Historical question about modularity of CM curves

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