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The average elliptic curve has few integral points

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  • UserLevent Alpoge (Cambridge)
  • ClockTuesday 26 May 2015, 16:15-17:15
  • HouseMR13.

If you have a question about this talk, please contact Jack Thorne.

It is a theorem of Siegel that the Weierstrass model y^2 = x^3 + A x + B of an elliptic curve has finitely many integral points. A “random” such curve should have no points at all. I will show that the average number of integral points on such curves (ordered by height) is bounded—in fact, by 66. The methods combine a Mumford-type gap principle, LP bounds in sphere packing, and results in Diophantine approximation. The same result also holds (though I have not computed an explicit constant) for the families y^2 = x^3 + A x, y^2 = x^3 + B, and y^2 = x^3 - n^2 x.

This talk is part of the Number Theory Seminar series.

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