Saturday, 3 July 2010

modular forms - Fourier coefficients for elliptic curves on average

As others have mentioned, if p is fixed then you're really looking at elliptic curves over a fixed finite field.



From some points of view an interesting variant would be to look at elliptic curves say Ea,b:y2=x3+ax+b where a and b vary over integers in a box, say |a|leqA and |b|leqB and relatively small compared to p. The one might try to find asymptotic results that hold as p, A, B get large together. If A and B aren't too big then this is giving more information about individual curves. For example, in bounding the average analytic rank of elliptic curves it is important to get a good bound on frac1ABsump<Psum|a|leqAsum|b|leqBaP(Ea,b)

with A and B as small as possible. For example, see A. Brumer, The average rank of elliptic curves. I, Invent. Math. 109(3), 445–472 (1992).



In a different but related direction, there is a paper of David and Pappalardi, Average Frobenius distributions of elliptic curves (it's the fourth from the bottom) on this subject. They get a kind of Lang-Trotter on average, so they are varying both p and the coefficients defining the elliptic curves. Stephan Baier later made some improvements on this problem here.

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