## 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

### MSC:

 14L05 Formal groups, $$p$$-divisible groups 14E99 Birational geometry 14H45 Special algebraic curves and curves of low genus 14H52 Elliptic curves

### Citations:

Zbl 0156.041; Zbl 0321.14014
Full Text:

### References:

 [1] V.G. Drinfeld : Elliptic modules (Russian). Math. Sbornik 94 (136) (1974), 594-627, 656; English translation: Math. USSR-Sob. 23 (1976), 561-592. · Zbl 0321.14014 · doi:10.1070/SM1974v023n04ABEH001731 [2] Y. Fujiwara : On Galois actions on p-power torsion points of some one-dimensional formal groups over Fp[[t]] , J. Algebra 113 (1988) 491-510. · Zbl 0644.14017 · doi:10.1016/0021-8693(88)90175-5 [3] B. Gross : On canonical and quasi-canonical liftings , Invent. Math. 84 (1986) 321-326. · Zbl 0597.14044 · doi:10.1007/BF01388810 [4] B. Gross : Ramification in p-adic Lie extensions , Astérisque 65 (1979) 81-102. · Zbl 0423.14030 [5] N. Katz and B. Mazur : Arithmetic Moduli of Elliptic Curves , Princeton University Press (1985). · Zbl 0576.14026 · doi:10.1515/9781400881710 [6] K. Keating : Galois characters associated to formal A-modules , Comp. Math. 67 (1988) 241-269. · Zbl 0655.14016 [7] K. Keating : Lifting endomorphisms of formal groups , Harvard Ph. D. thesis, 1987. [8] J. Lubin , J.-P. Serre and J. Tate : Seminar at Woods Hole Institute on algebraic geometry (1964). [9] J. Lubin and J. Tate : Formal complex multiplication in local fields , Ann. of Math. (2) 81 (1965) 380-387. · Zbl 0128.26501 · doi:10.2307/1970622 [10] J. Lubin and J. Tate : Formal moduli for one-parameter formal Lie groups , Bull. Soc. Math. France 94 (1966) 49-59. · Zbl 0156.04105 · doi:10.24033/bsmf.1633 [11] Y. Fujiwara : On divisibilities of special values of real analytic Eisenstein series , to appear in J. Fac. Sci. Univ. Tokyo. · Zbl 0669.10049
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