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The elliptic curve $-4 x^3 + 4 x^2 y + 16 x - y^3 + 9 y$ goes through $21$ integer points in the range $-9$ to $9$. Is that the maximum?

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Is that the maximum in the range $[-9,9]$? Why make that cutoff? –  Aaron May 31 '11 at 17:35
    
[-9,9] is arbitrary. Any non-tangent line going through two rational points on an elliptic curve will go through a third. Multiplying by the GCD gives an elliptic curve going through more integer lattice points, for a sufficiently large lattice. I was wondering about maxima on a small lattice, so I picked [-9,9] arbitrarily. –  Silas Pike May 31 '11 at 20:43

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This is a bit of a longshot, but have a look at Matthew Baker and Clayton Petsche, Global discrepancy and small points on elliptic curves, http://arxiv.org/pdf/math/0507228v1 and at some of the papers in the references that have titles that suggest they may be relevant to this problem.

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There is a demonstration, Elliptic Curves on a Small Lattice that gives equations for 5 elliptic curves with 21 lattice points on a [-9,9] lattice, and hundreds more with 20-15 points. –  Silas Pike Jun 2 '11 at 14:12

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