\(p\)-typical dynamical systems and formal groups (Q2781943)

The author gives a new criterion for deciding whether a stable serie \(f\) is an endomorphism of a formal group over a \(\mathbb Z_p\)-algebra \(A\).NEWLINENEWLINEMore precisely let \(p\) be a prime number and \(A\) be a characteristic zero integral domain which is a \(\mathbb Z_p\)-algebra. A one-dimensional formal group law over \(A\) is a formal power series \(F(x,y)\in A[[x,y]]\) which satisfies the three conditions: NEWLINE\[NEWLINE\text{ (i)}\;F(x,0)=x,\quad \text{ (ii)}\;F(x,y)=y,\quad\text{ (iii)} \;F(x,F(y,z))=F(F(x,y),z).NEWLINE\]NEWLINE And an endomorphism of the formal group \(F(x,y)\) is merely a power series \(f(x) \in xA[[x]]\), which verifies the identity: \(f(F(x,y))=F(f(x),f(y))\). Now given a stable series \(f(x)\) (i.e. such that \(f'(0)\) is neither zero nor a root of unity), there exists an unique series \(L_f(x) \in x+x^2A[[x]]\), the so-called logarithm of \(f(x)\), which verifies: \(L(f(x))=f'(0)L(x)\). Moreover, according to \textit{J. Lubin} [Compos. Math. 94, 321--346 (1994; Zbl 0843.58111)], the set of stable series \(g(x)\) which commute with \(f(x)\), i.e. the dynamical system over \(A\) arising from \(f(x)\), is characterized by the logarithmic identity \(L_g(x) = L_f(x)\).NEWLINENEWLINEIn the paper under review the author observes that if \(L_f(x) = x + \sum_{k\geq 2}a_k x^k\) is the logarithm of a formal group over \(A\), the \(p\)-typical series \(l_f(x) = x + \sum_{n \in \mathbb N} a_{p^n} x^{p^n}\) is also the logarithm of a strictly isomorphic formal group. So, by using the \(p\)-typical description of formal groups given by \textit{M. Hazewinkel} [{Formal groups and applications}, Academic Press, New-York (1978; Zbl 0454.14020)], he proves the following main result: a dynamical system \(\mathcal S\) over \(A\) containing a stable series \(g(x) \in xA[[x]]\) with \(g'(x)\) in \(\mathbb Z_p\) and \(g'(0)^p-g'(0) \in p\mathbb Z_p \setminus p^2\mathbb Z_p\) arises from a formal group over \(A\) if and only \(\mathcal S\) is isomorphic to a \(p\)-typical dynamical system. In other words, a stable series \(f(x)\) is an endomorphism of a formal group over \(A\) if and only one has: {(i)} \(L_f^{-1}(l_f(x)) \in A[[x]]\) and {(ii)} \(f(x)\) commutes with some element \(g(x) \in A[[x]]\) with \(g'0)=p\).NEWLINENEWLINEIn the final section, the author gives an interesting example of two stable series \(f(x)\) and \(g(x)\) over \(\mathbb Z_p\), with \(g(x)\) invertible , \(f(x)\) non invertible and \(f(g(x)) = g(f(x))\), such that the iterates of \(f(x)\) have in total only finite many roots. This last example contradicts the precise version of the so-called Lubin conjecture that the author stated in a previous article [J. Number Theory 62, 284--297 (1997; Zbl 0871.11085)].