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SUMMARY:The average elliptic curve has few integral points - Levent Alpoge
  (Cambridge)
DTSTART:20150526T151500Z
DTEND:20150526T161500Z
UID:TALK58797@talks.cam.ac.uk
CONTACT:Jack Thorne
DESCRIPTION:It is a theorem of Siegel that the Weierstrass model <code>y^2
  = x^3 + A x + B </code> of an elliptic curve has finitely many integral p
 oints. A "random" such curve should have no points at all. I will show tha
 t the\naverage number of integral points on such curves (ordered by height
 )\nis bounded -- in fact\, by 66. The methods combine a Mumford-type gap\n
 principle\, LP bounds in sphere packing\, and results in Diophantine appro
 ximation. The same result also holds (though I have not computed\nan expli
 cit constant) for the families <code>y^2 = x^3 + A x</code>\, <code>y^2 = 
 x^3 + B</code>\,\nand <code>y^2 = x^3 - n^2 x</code>.
LOCATION:MR13
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