David Zureick-Brown - Rational points and algebraic cycles
David Zureick-Brown - Rational points and algebraic cycles
David Zureick-Brown - Rational points and algebraic cycles
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Proofs<br />
Part (ii)<br />
P(a(M) = s) = p −(s+1 2 ) ·<br />
∞∏ (<br />
1 + p<br />
−i ) −1<br />
·<br />
i=1<br />
s∏ (<br />
1 − p<br />
−i ) −1<br />
.<br />
1 Duality implies that W 1 := ker(F ⊗ F p ) <strong>and</strong> W 2 := ker(V ⊗ F p ) are<br />
maximal isotropics.<br />
2 a(M) = dim (W 1 ∩ W 2 )<br />
i=1<br />
3 Argue that W 1 <strong>and</strong> W 2 are r<strong>and</strong>omly distributed.<br />
4 This expression is the probability that two r<strong>and</strong>om maximal isotropics<br />
intersect with dimension s.<br />
<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 21 / 29
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