18

数理解析研究所講究録
第 713 巻 1990 年 18-23

On unramified coverings of the affine line
in positive characteristics*
Tatsuji Kambayashi (上林達治)
Faculty of Science and Engineering
Tokyo Denki University
Hiki-gun, Saitama-ken, 350-03 /JAPAN

October 10, 1989
In 1957 a fundamental paper by Abhyankar [A] dealing with covering ofalgebraic curves in the modular’ ( $i.e.$ , positive-characteristic) case appeared.
In that paper, among other things, he made a conjecture that “any finitegroup generated by its p-Sylow subgroups can occur as Galois group of an\’etale Galois covering of the affine line in characteristic. The conjecture,if proven true, would imply that any finite simple group whose order isdivisible by should occur as Galois group of such a covering.
In the early $1980’ s$ , after a long period of inaction, certain problems
related to Abhyankar’s Conjecture began to receive some attention (see,for instance, Harbater [H], Kambayashi-Srinivas [KS]). More recently, JeanPierre Serre appears to have taken considerable interest in the conjecture,and Ram Abhyankar has rejoined the ranks to resolve his old problem. They
have already made significant progress, though as far as known to this writernone of their results have been published yet.
The purpose of this note is to explain the situation underlying Abhyankar’s Conjecture and to review what has been done in this area sofar, including very recent results due to Abhyankar and Serre as gatheredfrom their letters to the writer.
$p’$

$p$

’A summary of the writer) talk given at the “Symposium on Frobenius Maps in Commutative Algebra“ held at Research Institute for Mathematical Sciences, Kyoto Universityin September 1989$s$

19\’Etale coverings of the affine space.

1We shall work over a fixed algebraically closed ground field whose characteristic may be for the time being, and shall consider only normal( $i.e,$ , normal, integral schemes of finite type over ). A
varieties overmorphism $Yarrow X$ of such varieties will be called an \’etale covering of $X$
by if it is \’etale and finite. Such an \’etale covering is by definition Galoisian,or is a Galois covering, if the fields of rational functions $k(Y)\supseteq k(X)$ onand $X$ form a Galois extension. In case $p=char(k)>0$ , a Galois covering$Yarrow X$ is said to be tame if its degree $[k(Y) : k(X)]$ is not divisible by .
The starting point of all is the following well-known fact:
$k$

$0$

$p$

$k$

$k$

$Y$

$Y$

$p$

has no nontrivial \’etale coverings.
Theorem 1 $(a)$ The projective space$(b)$ In characteristic $p=0$ , thehas no nontrivial \’etale
affine spacecoverin $gs$ .
$(c)$ In case $p>0_{f}$ thehas no nontrivial \’etale tame Galois
affine spacecoverings.
$P^{n}$$A^{n}$$A^{n}$The proof, via Bertini’s First Theorem, comes down to the $n=1$ level,
has no nontrivial covand one then shows using Hurwitz’ Theorem thaterings at worst tamely ramified over one point on the line (cf. [KS] forinstance).
From now on we confine ourselves to the characteristic $p>0$ case.
It is important to note in part (c) above that both ‘tame’ and ‘Galoisian’conditions are required to assure the non-existence of said coverings.
Example 1. (a) Drop the condition of tameness (or not dividing thedegree); then one gets the classical Artin-Schreier coverings:$P^{1}$$p$$Y=(y^{p}-y-f(x)=0)arrow X=A^{1}=$ ( $the$ x-axis)

with $f(x)\in k[x],$ $f(x)\neq g(x)^{p}-g(x)$ for any $g(x)\in k[x]$ , which is an \’etale
Galois covering of degree of the affine line.
(b) Drop the condition of being a Galois covering; then one gets Abhyankar’s example (and many others like it):$p$

$Y=(y^{p+1}-xy^{p}+1=0)arrow X=A^{1}=$ ( $the$ x-axis),which is an \’etale, non-Galois covering of the affine line of degree $p+1$ . See[ ; Th. 1, p. 830].
and,For further investigation of \’etale coverings of the affine n-space, one should obviously start with coveringsin particular, of the affine line$A$$A^{n}$$A^{1}$2

20of degree . There is a basic fact about this, apparently known to experts
alre ady:$p$

Theorem 2 Let $X=SpecR$ be an affine variety, and let $Yarrow X$ be an\’etale Galois covering of degree . Then, $Y$ is an Artin-Schreier covering:$p$$Y=Spec(R[T]/(T^{p}-T-f))$ ,$f\in R$.

as a $Z/p$-torsor
The proof of this theorem is achieved by viewing
over $X$ with respect to the \’etale topology and by showing $H_{et}^{1}(X, Z/p)\simeq$. See [KS; Th. 2.1]
$R/(F-1)R$ , where denotes the Frobenius map
for details.
Then, a natural question one might ask next is the following:
$Y$

$F$

$trightarrow t^{p}$QUESTION: Is every \’etale covering of degree of an affinevariety (or, at any rate, of the affine space) Galoisian, andtherefore an Artin-Schreier covering?
$p$$A^{n}$Srinivas and I addressed this question in [KS] and have verified that:
The answer is ’YES’ for characteristic $p=2$ or 3.

2

Nori’s Theorem and other recent results

The question we asked as to whether or not all degree-p \’etale coverings of theaffine space in characteristic are Galoisian has been more than answeredby the following theorem due to Madhav Nori (see [K] for proofs, etc.):$p$

Theorem 3 Let $X$ be an irreducible affine scheme of positive dimension andof finite type over a field of characteristic $p>0$ ; let be a connected affine
algebraic group defined over a finite fieldof $q=p^{e}$ elements. Assume
that , modulo its unipotent radical, is semisimple and simply connected.
Then, there exist \’etale Galois coverings $Yarrow X$ of $X$ with Galois group$G(Y/X)\simeq G(F_{q})=$ ( $the$ group of-rational points on ).
$G$

$k$

$F_{q}$$G$$G$$F_{q}$As an immediate corollary to this theorem we get examples showingthat the answer to the question above is negative for the characteristic $p=5$
or 7. Namely, writing
respectively for the symmetric and theandalternating groups on letters, we find:$S_{n}$$A_{n}$$n$3

$\overline{\mathcal{L}}\perp$

Example 2. Consider the groups$PSL(2,5)\simeq A_{5}$and $PSL(2,7)\simeq SL(3,2)$ .

The former contains a $subgroup\simeq A_{4}$ of index 5, while the latter has a$subgroup\simeq S_{4}$ of index 7, neither subgroups being normal. By pulling theseback, we can build within $SL_{2}(F_{5})$ and $SL_{2}(F_{7})$ non-normal subgroups ofindex 5 and 7, respectively. Now apply Nori’s theorem to these to obtain\’etale Galois coverings, over any affine variety, of degrees 120 and 336 incharacteristics 5 and 7, respectively. Finally take the intermediate coverings
corresponding to the non-normal subgroups found already. We have \’etale,non-Galois covenings of degree in characteristics $p=5,7$ .
$p$With the apparition of Nori’s Theorem and consequent EXAMPLE 2, it
looked to me as though the matter had been put to rest insofar as unramified
were concerned–until I began a year ago to receive letters
coverings offrom Jean-Pierre Serre and then from Ram Abhyankar. Serre seems to feelthat the$A^{1}$

Problem of determining which finite group can occur as Galois groups of an unramified covering of the affine line

is rather akin to its famous counterpart concerning Galois extensions of ,and he seems to be hot in pursuit of what one might regard his old baby[S]. Abhyankar, ‘inspired’ (his own word) by correspondence with Serre,
has reentered the scene and has already pushed through some tremendous
calculations aided in part by Macsyma.
To conclude this report I $shaU$ attempt to list up some salient points
that they make in their letters to me. What follows should be considered
my free quotations. Barest indications of proofs will be offered as in theirletters$Q$

Let us adopt a shorthand: In place of saying “a finite group occurs as, we simply
Galois group of an \’etale Galois covering of the affine line
state thatoccurs”.
$G$$A^{1}’$$G$

1I have been able to attach my own proofs to certain of their results, while others
remain unverified by me.

4Professor Serre’s remarks:(S1) He has calculated the Galois group of the Galois closure of Abhyankar’s example, EXAMPLE l(b) above, and after ‘some extra work’ hasfound that it is $PSL(2, p)$ , which of course has no quotients of order as$p$soon as $p\geq 5$ .

(S2) Towards Abhyankar’s conjecture stated at the beginning of this ar$PSL(2, q)$ ,ticle, he has shown that all ChevaUey groups of rank 1 occur,$PSU(3, q^{2})$ , and the Suzuki and Ree groups in char. 2 and 3, whereis apower of . The construction, according to him, uses ‘the Deligne-Lusztigcurves’.
(S3) Apropos of EXAMPLE 2 above, Serre offers a novel construction ofof any characteristic $p\geq 5$ ; Let $E$ be
an \’etale, non-Galoisian covering ofa supersingular elliptic curve in char. , and express its nowhere-vanishingdifferential of the first kind as $\omega=df$ . A proper choice of such an $f$ willproduce an \’etale covering$\dot{i},e.,$$q$$p$$A^{1}$$p$$\omega$$f$: $E-$ { $unique$ pole of$f$}$arrow A^{1}$.

but also for any irre(S4) Every finite p-group occurs–not only for
ducible affine scheme $X$ of positive dimension in char. . As a consequence,
for any unipotent algebraic group defined over a finite field, $G(F_{q})$ occurs
for X. (cf. $[K$ ; intro., p.640].) The proof uses two propositions about theprofinite group which, by definition, is the Galois group of the maximal
unramified p-extension of $X$ : Namely, $H^{2}(\Gamma, Z/p)=0$ and $H^{1}(\Gamma, Z/p)$ isinfinite-dimensional.
$A^{1}$$p$$G$$\Gamma$

Professor Abhyankar’s remarks.
(A1) Pushing (S1) further, he has calculated Galois groups of the Galois
closures of the following unramified coveringsof the affine line:$Y_{t}$$Y_{t}$$:=(y^{p+t}-xy^{p}+1=0)arrow A^{1}=$ (the x-axis),

where is a positive integer not divisible by . This type of covering isa special case of Abhyankar’s own example [ ; Prop. 1, p.831], andis
precisely EXAMPLE l(b) treated in (S1). It has been shown that, for $t>1$ ,the Galois group foris the alternating group.
(A2) In a similar vein, he has examined the following situation:
$t$

$p$

$A$

$Y_{1}$$A_{p+t}$$Y_{t}$

5

23

Fix the integers $n\geq p=char(k),$ $0\leq r<n,$ $0\leq t<n,$ any integer,
and choose nonzero $a\in k$ . Let $Y=Y(n, r, s, t;a)$ be the norphism$s$$Y:=(y^{n}+y^{r}-ax^{s}y^{t}=0)arrow A^{1}=$ ( $the$ x-axis).

-theorists about multiply-transitive groups andWith the aid of several
that of Macsyma, he has found a large number of cases in which the $Y$
gives an \’etale covering of the affine line, and proceeded to compute thecorresponding Galois groups. omitting to cite these cases individually, letme just state the consequences:$grou^{b}\tilde{p}$(a) For $aHn\geq p>2$ , the alternating group$A_{n}$(b) For $aUn\geq p=2$ , the symmetric group$S_{n}$REF EREN C Eoccurs.
occurs.

$S$[A] S. S. Abhyankar, ‘Coverings of algebraic curves”, American Journal79(1957), 825-856.
Mathematics,of[H] D. Harbater, ‘Ordinary and supersingular covers in characteristic
Pacific Journal of Mathematics, 113(1984), 349-363.

$p’$

,

[KS] T. Kambayashi, V. Srinivas, ‘On \’etale coverings of the affine space“
IN: Algebraic Geometry–Proceedings of Ann Arbor Conference, Lecture
Notes in Math., vol. 1008(1983), Springer-Verlag, New York.

[K] T. Kambayashi, ‘Nori’s construction of Galois coverings in positive
characteristics“, IN: Algebraic and Topological Theories–to the memory ofDr. Takehiko Miyata, 640-647, 1985, Kinokuniya, Tokyo.
[S] J.-P. Serre, “Sur la topologie des vari\’et\’es algebriques en caract\’eristique ’ IN: Symposium Internacional de la Topologia Algebraica, 24-53,
1956, Univ. Autonom. de Mexico.
$p$6