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I am currently writing my master's thesis and I was wondering if the rank distribution conjecture was ever formally written down. Recall that it says that:

Half of all elliptic curves have rank $0$, half have rank $1$, and the rest have rank $\geq 2$.

I am looking for a reference for where the conjecture was made. Right now all I have for a reference is the linked talk and I'm not very comfortable with using that.

Thanks.

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  • $\begingroup$ Did you write to the person who wrote the slides at your link, or the person to whom he attributes the conjecture? $\endgroup$
    – KConrad
    Commented May 25, 2012 at 0:02
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    $\begingroup$ In the Bhargava–Shankar article (arxiv.org/abs/1006.1002), they say it "originates" in the work of Goldfeld and Katz–Sarnak and provide a reference for each. $\endgroup$
    – Rob Harron
    Commented May 25, 2012 at 1:44
  • $\begingroup$ Do you want to include your comment as an answer so I can close up this question? $\endgroup$
    – Eugene
    Commented May 25, 2012 at 3:42

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Goldfeld made the conjecture for the family of quadratic twists of a given curve; it would be interesting to check the paper to see if he wrote anything about the more general case (I don't have it at hand). For the latter, the possibility that the average rank might be 1/2 was certainly folklore quite a while before Katz-Sarnak, but the numerical evidence did not (still doesn't, I think) really support this, so it was not usually stated as a formal conjecture. But see, for instance, the introduction to "The average rank of elliptic curves, I" by Brumer (Invent. math. 109, 1992), or even the last lines of the first page of "The rank of elliptic curves", by Brumer-Kramer (Duke Math. J. 44, 1977):

Our bounds suggest that, in general, curves of prime conductor have the smallest rank compatible with the parity predictions of Birch and Swinnerton-Dyer.

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