From: Richard Pinch <rgep@cam.ac.uk>
Newsgroups: sci.math
Subject: Re: Rational points on a^3+b^3=k
Date: Sat, 22 Nov 1997 12:57:34 +0000

James Buddenhagen wrote:
> 
> Dave Rusin wrote:
> [..interesting stuff deleted..]
> > The question, "Can you predict the rank of the elliptic curves in the
> > following 1-parameter family..." usually has answer "no". I'm sure
> > there has been work on this particular family, since it's so well
> > known, but I don't think there's a conclusive answer. In some sense
> > the "average" rank is supposed to be 1/2, I believe.
> 
> This brings to mind again a question I have been wondering about:
> given a family of elliptic curves, how can one develop heuristics
> for the distribution of ranks, i.e. for the percentages p_i that
> have rank i, i = 0,1,2,... ?

Armand Brumer has written a number of papers on this subject:
for example, Duke Math. J. 44 (1977) 715--743; Bull. Amer. Math. 
Soc. 23 (1990) 375--382; Inventiones Math. 109 (1992) 445--472;
Asterisque 222 (1995) 41--68.

There are results by Roger Heath-Brown on the order of the
Selmer group, which is not quite so hard to handle.  See
Iventiones Math. 111 (1993) 171--195 and 118 (1994) 331-370.

-- 
Richard Pinch		Queens' College, Cambridge
rgep@cam.ac.uk		http://www.dpmms.cam.ac.uk/~rgep
Looking for a job from Oct'98: 
		http://www.dpmms.cam.ac.uk/~rgep/cv.html
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Generic v. specific ranks of elliptic curves (was: Re: Rational points on a^3+b^3=k)
Date: 23 Nov 1997 05:30:00 GMT

In article <19971122195301.OAA25092@ladder02.news.aol.com>,
KRamsay <kramsay@aol.com> wrote:
>
>>given a family of elliptic curves, how can one develop heuristics
>>for the distribution of ranks, i.e. for the percentages p_i that
>>have rank i, i = 0,1,2,... ?
>
>This is not easy.
>
>There was this heuristic that suggested half of the ranks would be 
>even, and half odd, but that otherwise the rank would remain
>as low as reasonable.

Right, more or less, but not without exception. I thought I had an
interesting example along these lines and got some helpful details
from  Joe Silverman, who wrote:

"There are curves over Q(T) such that (i) the rank over Q(T) is 0.
(ii) the sign of the functional equation is odd for every rational
value of T, hence conjecturally (via Birch-Swinnerton-Dyer) the
rank over Q is odd for every rational T. The first example of this
sort is due to Cassels and Schinzel, and Rohrlich produced lots more
examples."

The Cassels-Schinzel example (Bull London Math Soc 14(1982)345-348) is
	y^2 = x ( x^2-(1+t^4)^2 ).

From  Silverman's site (www.math.brown.edu/~jhs) you can retrieve
preprints with current work along these lines. Here's one abstract:
                
"The Average Rank of an Algebraic Family of Elliptic Curves

Let $E/Q(T)$ be a one-parameter family of elliptic curves.
Assuming various standard conjectures, we give an upper bound for the
average rank of the fibers $E_t(Q)$ with $t\in Z$, improving
earlier estimates of Fouvry-Pomykala and Michel. We also show how
reasonable assumptions about the distribution of zeros of $L$-series
can be used to explain the experimentally observed fact that the
average rank of the fibers appears to be strictly larger than the naive
expected value of $\rank E(Q(T))+1/2$."

I think it's fair to say the generic rank more or less bounds the ranks
of the specializations on average, but the bounds are not quite as
tight as one might like, and of course these are only averages: specific
ranks can be lower sometimes and frequently much higher than the generic rank.

dave

(Something is terribly out of the ordinary if worthwhile current
research is being discussed on sci.math!)