Postprint version. Published in Journal of Number Theory, Volume 15, Issue 3, December 1, 1982, pages 318-330. Copyright © 1982 Elsevier. The definitive version is available at http://dx.doi.org/10.1016/0022-314X(82)90036-1.
Let O=limnZ/pnZ, , let A=O[g2,g3] Δ, where g2 and g3 are coefficients of the elliptic curve: Y2 = 4X3 − g2X − g3 over a finite field and Δ = g23 − 27g32 and let B=A[X,Y]/(Y2-4X3+g2X+g3). Then the p-adic cohomology theory will be applied to compute explicitly the zeta matrices of the elliptic curves, induced by the pth power map on the free AzQ -module H1(X, AzQ). Main results are; Theorem 1.1: X2 dY and Y dX are basis elements for H1(X, IA(X)zQ); Theorem 1.2: Y dX, X2 dY, Y−1 dX, Y−2 dX and XY−2 dX are basis elements for H1(X – (Y=0) IA(X)zQ), where X is a lifting of X, and all the necessary recursive formulas for this explicit computation are given.