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SUMMARY:Modeling the distribution of Selmer groups\, Shafarevich-Tate grou
 ps\, and ranks of elliptic curves
DTSTART:20120920T111500
DTEND:20120920T123000
DTSTAMP:20260405T140052Z
UID:2c31ee119c65346d8dee82561e0eeed96f7dd50b3fc179f2a0afa8b0
CATEGORIES:Conferences - Seminars
DESCRIPTION:Bjorn Poonen\nUsing only linear algebra over Z_p\, we define a
  discrete probability distribution on the set of isomorphism classes of sh
 ort exact sequences of Z_p-modules\,\nand then conjecture that as E varies
  over elliptic curves over a fixed global field\, the distribution of 0 --
 > E(k) tensor Q_p/Z_p --> Sel_{p^infty} E --> Sha[p^infty] --\n> 0 is that
  one. This one conjecture would have the following consequences:\n1) Asymp
 totically\, 50% of elliptic curves have rank 0 and 50% have rank 1.\n2) Sh
 a[p^infty] is finite for 100% of elliptic curves.\n3) The Poonen-Rains con
 jecture on the distribution of Sel_p E holds.\n4) Delaunay's conjecture a 
 la Cohen-Lenstra on the distribution of Sha holds.\n(This is joint work wi
 th M. Bhargava\, D. Kane\, H. Lenstra\, and E. Rains.)
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 nem
STATUS:CONFIRMED
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