Lifting endomorphisms of formal A-modules. (English) Zbl 0655.14017

Let \(F_ 0\) be a formal A-module (or formal group law) of height \(h<\infty\) over a field k of characteristic p, and let F be a formal A- module of height \(h-1\) over \(R=k[[ t]]\) whose special fiber is \(F_ 0\) (i.e., F is a deformation of \(F_ 0\) over R). The paper is concerned with lifting endomorphisms of \(F_ 0/k\) to endomorphisms of F over \(R_ n=:R/(t^{n+1})\), and with the structure of \(End_{R_ n}(F)\). The results are summarized as follows. Recall that if k is separably closed, \(End_ k(F_ 0)\) is isomorphic to the maximal order, B, of the division algebra \(D_{1/h}\) of degree h 2 over K with invariant 1/h. If k is arbitrary of characteristic p, then \(End_ k(F_ 0)\subset B.\)
Theorem A. Let F/R be a deformation of height \(g=:h-1\) of the formal A- module \(F_ 0/k\) of height h. Write \([\pi]_ F(x)=a_ 0x^{q^ g}+...\), and set \(e=\nu_ t(a_ 0)>0\). Choose \(f_ 0\in End_ k(F_ 0)\subset B\) which satisfies \(f_ 0\in (A+\pi^{\ell}_ BB)\setminus (A+\pi_ B^{\ell +1}B)\) for some \(\ell >0\). Write \(\ell =hm+b\) with \(0\leq b<h\). Then \(f_ 0\) lifts to \(End_{R_{n-1}}(F)\) but not to \(End_{R_ n}(F)\), where \(n=e[a(gm)+q^{gm}(q\) \(b-1)/(q-1)+1].\)
Theorem B. Let F be a deformation of \(F_ 0\) as in theorem A. Then \(End_{R_ n}(F)=End_ k(F_ 0)\cap (A+\pi_ B^{j(n)}B)\) where \(j(n)=hm+b\) whenever \(a(gm)-q^{gm}+1\leq n/e\leq a(gm)+1\) \((b=0)\); \(a(gm)+q^{gm}(q^{b-1}-1)/(q-1)+1\leq n/e<a(gm)+q^{gm}(q\) \(b-1)/(q- 1)+1\) \((0<b<h).\)
These results are obtained by using the formal cohomology theory of J. Lubin and J. Tate [Bull. Soc. Math. Fr. 94, 49-59 (1966; Zbl 0156.041)] and V. G. Drinfel’d [Math. USSR, Sb. 23(1974), 561-592 (1976); translation from Mat. Sb., Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014)].
Applications to elliptic curves are discussed: Theorem C. Let E be an elliptic curve over \(R=k[[ t]]\) whose reduction \(E_ 0=E(mod(t))\) is supersingular. Assume that \(p>2\). Let \(\phi \in End_ k(E_ 0)\setminus {\mathbb{Z}}\). Then \(\sup_{a\in {\mathbb{Z}}}\nu_ p(\deg (\phi -a))=\nu_ p((Tr(\phi))^ 2-4\cdot \deg(\phi)).\)
Therefore \(\phi\) lifts to \(End_{R_{n-1}}(E)\) but not to \(End_{R_ n}(E)\), where \(n=(a(m)+1)e\) if \(\nu_ p((Tr(\phi))^ 2-4\cdot \deg(\phi))=2m\), and \(n=(a(m)+p\) \(m+1)e\) if \(\nu_ p((Tr(\phi))^ 2- 4\cdot \deg(\phi))=2m+1\). (Here \(a(m)=[(p+1)(p\) m-1)]/(p-1).)
There is a corresponding theorem for \(p=2\) also.
Reviewer: N.Yui


14L05 Formal groups, \(p\)-divisible groups
14E99 Birational geometry
14H45 Special algebraic curves and curves of low genus
14H52 Elliptic curves
Full Text: Numdam EuDML


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