×

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
PDFBibTeX XMLCite
Full Text: Numdam EuDML

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.