Kummer congruences and formal groups
type: Item
instance of: scholarly article
title: Kummer congruences and formal groups
zbMATH Open document ID: 0781.14034
author: Margaret N. Freije
published in: Journal of Number Theory
publication date: 1993-05-16T00:00:00Z
review text: Let \(R\) be an integral domain of characteristic 0 and let \(K\) be its field of quotients. Let \(f(X)= \bigl( f_ 1(X),\ldots,f_ n(X) \bigr)\) be an \(n\)-tuple of power series in \(n\) variables \(X=(x_ 1,\ldots,x_ n)\) over \(K\). Suppose that \(df_ j \in \bigoplus RXdx_ i\), \(f(X) \equiv 0\) (mod deg 1) and that the Jacobian \(J(f)=( \partial f_ i/ \partial x_ j) (0)\) is invertible over \(R\). Put \(F_ f(X,Y) = f^{-1} \bigl( f(X)+f(Y) \bigr)\). Then \(F_ f\) is a formal group a priori defined over \(K\) having \(f\) as its logarithm. A sufficient condition for the integrality of \(F_ f\) was given by \textit{T. Honda} [J. Math. Soc. Japan 22, 213-246 (1970; Zbl 0202.031)]. In this paper the converse question is addressed. That is, given a formal group \(F(X,Y)\) over \(R\), find necessary conditions for the coefficients of the logarithm of \(F\) should satisfy. These conditions are described in terms of Hurwitz series and Kummer congruences. To state them, we need to set up some definitions and notations. If \(I=(i_ 1, \ldots,i_ n)\) is an index set, \(I\)! stands for \(i_ 1! \cdots i_ n!\), \(X!\) for \(x_ 1^{i_ 1} \cdots x_ n ^{i_ n}\) and \(g(I)\) for the g.c.d. of \(i_ 1,\ldots,i_ n\). Let \(p\) be a rational prime, and let \({\mathfrak p}\) be a prime ideal in \(R\) above \(p\). An \(n\)-dimensional Hurwitz series over \(R\) is an \(n\)-tuple of power series \(f(T)= \bigl( f_ 1(T),\ldots,f_ n(T) \bigr)\) where each \(f_ i(T)\) is of the form \(f_ i(T)= \sum_{I \geq 0}{a_ i(I) \over I!}T^ I\), \(a_ i(I) \in R\). \(f(T)\) is said to be invertible if \(f(0)=0\) and \(J(f)\) is invertible in \(R\). Let \(\lambda(T)= \bigl( \lambda_ 1(T), \ldots, \lambda_ n(T) \bigr)\) denote the inverse of \(f\), and we write \(\lambda_ j(T) = \sum_{I \geq 0}{c_ j(I) \over I!}T^ I\), \(c_ j(I) \in R\). Further, one confines oneself to invertible Hurwitz series satisfying the ``hypothesis \(K\)'' that \({c_ j(I) \over I!}={1 \over g(I)} \varepsilon_ j(I)\) with \(\varepsil…
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 166055
zbMATH Keywords: formal group
zbMATH Keywords: logarithm
zbMATH Keywords: Hurwitz series
zbMATH Keywords: hypothesis \(K\)
zbMATH Keywords: formal invariant differentials
zbMATH Keywords: strong Kummer congruences
zbMATH Keywords: Atkin and Swinnerton-Dyer congruence
MaRDI profile type: Publication
full work available at URL: jnth.1993.1004
OpenAlex ID: W1981158358
DOI: 10.1006/JNTH.1993.1004
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description: scientific article; zbMATH DE number 166055