Title

Zeta Matrices of Elliptic Curves

Author Info

Goro Kato, California Polytechnic State University, San Luis ObispoFollow
Saul Lubkin, University of Rochester

Recommended Citation

Postprint version. Published in Journal of Number Theory, Volume 15, Issue 3, December 1, 1982, pages 318-330.

The definitive version is available at https://doi.org/10.1016/0022-314X(82)90036-1.

Abstract

Let O=lim_nZ/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.

Disciplines

Mathematics

Copyright

1982 Elsevier.