Hilbert functions and Witt functions. An identity for the congruences of Atkin and of Swinnerton-Dyer type
type: Item
instance of: scholarly article
title: Hilbert functions and Witt functions. An identity for the congruences of Atkin and of Swinnerton-Dyer type
zbMATH Open document ID: 0685.14026
DOI: 10.1007/BF02571239
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174172
review text: 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.
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 4122128
zbMATH Keywords: Atkin congruence
zbMATH Keywords: Swinnerton-Dyer congruence
zbMATH Keywords: Hilbert functions
zbMATH Keywords: Witt functions
zbMATH Keywords: logarithms of curvilinear formal group laws
zbMATH Keywords: Dieudonné- Cartier modules
author: Bert E. J. Ditters
MaRDI profile type: Publication
cites work: Q5644042
cites work: On the connected part of the covariant Tate p-divisible group and the \(\zeta\)-function of the family of hyperelliptic curves \(y^ 2=1+\mu x^ N\) modulo various primes
cites work: On the covariant Dieudonne-module of an abelian variety of dimension two over W(k)
cites work: On the theory of commutative formal groups
cites work: Q4047662
cites work: Arithmetic variety of moduli for genus two
cites work: Q5661948
cites work: Q4748285
cites work: Q4075566
cites work: Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences
cites work: Formal Groups and Congruences for L-Functions
OpenAlex ID: W178425785
description: scientific article; zbMATH DE number 4122128