Kummer congruences and formal groups
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 \(\varepsilon_ j(I) \in R\). Now define an operator \(\Omega_ p\) on an invertible Hurwitz series \(f\) with the ``hypothesis \(K\)'' as follows: \[ \Omega_ p(f)= \left( {\partial^ p f_ i \over \partial t^ p_ j}(T)\right) - \left( {\partial f_ i \over \partial t_ j}(T) \right) \left( {\partial f_ i \over \partial t_ j} (0) \right)^{-1}\left( {\partial^ pf_ i \over \partial t^ p_ j} (0) \right) =\sum \eta_{ij} (I) \bigl( f(T) \bigr)^ I, \] where the hypothesis \(K\) guarantees that \(\eta_{ij}(I) \in R\). \(f(T)\) is said to have strong Kummer congruence at \({\mathfrak p}\) if \(\eta_{ij}(I) \equiv 0 \pmod {{\mathfrak p}}\) for all \(i\), \(j=1,\ldots,n\) and all \(I\). A characterization of an invertible Hurwitz series having strong Kummer congruences at \({\mathfrak p}\) is given. -- Main result: Theorem. Let \(F(X, Y)\) be a commutative formal group defined over \(R_{\mathfrak p}\). Let \(\omega_ 1, \ldots, \omega_ n\) be a basis for the formal invariant differentials of \(F\) defined over \(R_{\mathfrak p}\) and let \(\lambda\) be the corresponding logarithm. Then \(f(X)= \lambda^{-1} (X)\) has strong Kummer congruences at \({\mathfrak p}\). As an application, a higher dimensional analogue of the Atkin and Swinnerton-Dyer congruence for the coefficients of holomorphic differential 1-forms on elliptic curves is obtained. Let \(C\) be a complete non-singular algebraic curve of genus \(g\) defined over a number field \(K\), and let \(J\) be its Jacobian variety. Let \(R\) be the ring of integers of \(K\) and let \({\mathfrak p}\) be a prime ideal in \(R\) such that \(C\) has good reduction at \({\mathfrak p}\). Then there is a system \(\{ \omega_ 1, \ldots,\omega_ g\}\) of holomorphic differential 1-forms on \(C\) and a local parameter \(t\) such that \(\omega_ i= \sum^ \infty_{n=1}a_ i (n)t^{n-1} dt\), \(a_ i(n) \in R_{\mathfrak p}\). Assume that the Hasse- Witt = Cartier-Manin matrix of \(J\) \[ \left( \begin{matrix} a_ 1(1) & \cdots & a_ 1(g) \\ a_ 2(1) & \cdots & a_ 2(g) \\ \cdots & & \cdots \\ a_ g(1) & \cdots & a_ g(g) \end{matrix} \right) \] is invertible over \(R_{\mathfrak p}\). Let \(\ell_ i (t)\) be the integral of \(\omega_ i\) satisfying \(\ell_ i(0)=0\) for \(i=1,\ldots,g\), and let \(L_ i (T) = L_ i(t_ 1,\ldots, t_ g)= \ell_ 1(t_ 1) + \cdots+\ell_ g(t_ g)\). Then we may write \(L_ i(T) = \lambda_ i \bigl( s_ 1(T),\ldots,s_ g(T) \bigr)\) where \(s_ i\) is the \(i\)-th symmetric function on \(g\) letters. Then regarding \(\bigl( s_ 1 (T), \ldots, s_ g(T) \bigr)\) as a system of local parameters at the origin of \(J\), \(\{d \lambda_ 1, \ldots, d \lambda_ g\}\) gives a basis for the invariant differentials of the formal group of \(J\), so that \(\lambda(X) = \bigl( \lambda_ 1 (X), \ldots,\lambda_ g(X) \bigr)\) is the logarithm of the formal group of \(J\). Theorem. Suppose that the formal group of \(J\) is defined over \(R_{\mathfrak p}\). Then the coefficients of \(\omega_ 1,\ldots,\omega_ g\) satisfy \[ \begin{pmatrix} a_ 1(pn) \\ \vdots \\ a_ g(pn) \end{pmatrix} \;\equiv \bigl( a_ i(jp)\bigr) \bigl( a^ p_ i(j)\bigr) ^{-1} \begin{pmatrix} a^ p_ 1(n) \\ \vdots \\ a_ g^ p(n) \end{pmatrix} \;\pmod{{\mathfrak p}R_{\mathfrak p}} \] for all \(n\in\mathbb{Z}^ +\).
scientific article; zbMATH DE number 166055