degree(xi) = di degree(F) = d s = d1 + d2 + d3 + d4 Omega = d1 x1 d(x2) /\ d(x3) /\ d(x4) - d2 x2 d(x3) /\ d(x4) /\ d(x1) + d3 x3 d(x4) /\ d(x1) /\ d(x2) - d4 x4 d(x1) /\ d(x2) /\ d(x3) = (x2 x3 x4)^(-d1+1) (x1)^(s-d1+1) d1^(-2) d(x2^d1/x1^d2) /\ d(x3^d1/x1^d3) /\ d(x4^d1/x1^d4). Eta = d1 x1 d(x2) /\ d(x3) - d2 x2 d(x1) /\ d(x2) + d3 x3 d(x1) /\ d(x2) = (x2 x3)^(-d1+1) (x1)^(d2+d3+1) d1^(-1) d(x2^d1/x1^d2) /\ d(x3^d1/x1^d3). We compute d( G Eta / F^j ) = d(G) /\ Eta / F^j + G d(Eta) / F^j - j G d(F) /\ Eta / F^(j+1) = G_1 d(x1) /\ d1 x1 d(x2) /\ d(x3) / F^j - G_2 d(x2) /\ d2 x2 d(x1) /\ d(x3) / F^j + G_3 d(x3) /\ d3 x3 d(x1) /\ d(x2) / F^j + G_4 d(x4) /\ Eta / F^j + (s-d4) G d(x1) /\ d(x2) /\ d(x3) / F^j - j G F_1 d(x1) /\ d1 x1 d(x2) /\ d(x3) / F^(j+1) - j G F_2 d(x2) /\ d2 x2 d(x3) /\ d(x1) / F^(j+1) - j G F_3 d(x3) /\ d3 x3 d(x1) /\ d(x2) / F^(j+1) - j G F_4 d(x4) /\ Eta / F^(j+1) = [(degree(G)+s-d4)G - d4 x4 G_4] d(x1) /\ d(x2) /\ d(x3) / F^j + G_4 d(x4) /\ Eta / F^j - j G[degree(F) F - d4 x4 F_4 ] d(x1) /\ d(x2) /\ d(x3) / F^(j+1) - j G F_4 d(x4) /\ Eta / F^(j+1) (*) = - d4 x4 G_4 d(x1) /\ d(x2) /\ d(x3) / F^j + G_4 d(x4) /\ Eta / F^j + j d4 G x4 F_4 d(x1) /\ d(x2) /\ d(x3) / F^(j+1) - j G F_4 d(x4) /\ Eta / F^(j+1) = G_4 Omega / F^j - j G F_4 Omega / F^(j+1) (*) Note that: degree(G) = j degree(F) - s + d4 Also note that the degree of GF_4 is equal to (j+1)d-s. The forms where cohomology lies are h Omega / F , h Omega / F^2 , h Omega / F^3 We define Delta by the formula Delta = (F^p - sigma(F))/p where sigma is the lift of frobenius mapping xi to xi^p. So the Frobenius maps sigma( h Omega / F^j ) = sigma(h) p^3 (x1x2x3x4)^(p-1) Omega ----------------------------------- (F^p - p Delta)^j = \sum_{i=0}^infty p^i Delta^i sigma(h) p^3 (x1x2x3x4)^(p-1) (i+j-1 choose j-1) Omega / F^(p(j+i))