Hilbert functions and Witt functions. An identity for the congruences of Atkin and of Swinnerton-Dyer type
Hilbert functions and Witt functions are studied. They describe the arithmetics of the (matrix) coefficients of logarithms of curvilinear formal group laws. Equivalently, they describe the arithmetics of covariant Dieudonné-Cartier modules for the formal groups, defined by them. Three theorems are given that describe the transition between both concepts. The first, theorem 1.12 interpretes the logarithm in terms of the exponential series, defined by the canonical curves. - The second, theorem 2.4, is an identity, relating logarithm and F-type. If one reduces this identity modulo \(p^ m\), congruences are obtaind that in case of elliptic curves, defined over \({\mathbb{Z}}\), are the well known congruences of Atkin and of Swinnerton-Dyer. - The third, theorem 4.3, expresses for potentially multiplicative groups the matrix \(\alpha\) of the Frobenius F in terms of the structure constants of the Hilbert function, obtained from the logarithm. It appears that \(\alpha\) can be described such that over p-Hilbert domains, the p-adic convergence of the entries is obvious.
logarithms of curvilinear formal group laws
scientific article; zbMATH DE number 4122128
scientific article