Let L/K be a totally ramified, finite abehan extension of local fields, let DL and D be the valuation rings, and let G be the Galois group. We consider the powers ...
manuscripta math.
73, 2 8 9 -
311
(1991)
manuscripta mathematica 9 Springer-Verlag 1991
ON GALOIS ISOMORPHISMS BETWEEN IDEALS IN EXTENSIONS OF LOCAL FIELDS
Nigel Byott
Let L/K be a totally ramified, finite abehan extension of local fields, let DL and D be the valuation rings, and let G be the Galois group. We consider the powers ~3L" of the maximal ideal of DL as modules over the group ring DG. We show that, if G has order p'~ (with p the residue field characteristic), if G is not cyclic (or if G has order p), and if a certain mild hypothesis on the ramification of L/K holds, then ~3L" and ~3L" are isomorphic iff r - r' mod p'~. We also give a generalisation of this result to certain extensions not of p-power degree, and show that, in the case p = 2, the hypotheses that G is abehan and not cyclic can be removed.
1
Introduction and statement
of results
Let K be the field of fractions of a complete discrete valuation ring O of characteristic 0 and residual characteristic p > 0. Throughout this paper we regard D and hence K as fixed. We write ~3 for the maximal ideal of O and e for the absolute ramification index of K (so pO = ~ ) . L/K will always denote a finite Galois field extension of K, and G its Galois group. We write DL for the valuation ring of L, and q3z for the maximal ideal of Oz.
289
BYOTT DL can be regarded as a (left) module over the group ring DG, and more generally so can each fractional DL-ideal ~3L'. It is well-known that ~L is free as an DG-module if and only if L / K is at most tamely ramified, and Ullom [16] has shown that every fractional ideal is then free over ~G. In this paper, we assume that L / K is totally and wildly ramified, and investigate when two fractional ideals ~3L" and ~3Lr' are isomorphic as ~G-modules. Clearly a sufficient condition for this is that r = r' (rood n), where n is the degree of the extension L/K. Our main result is that, if L / K satisfies certain hypotheses, this congruence condition is also necessary. Recall that the ramification groups Gi of L / K are defined by
G~=
G ifi= -1 { ~ E G : ( ~ r - - 1 ) D L _ C ~ 3 L ~+1} if i_>0.
L / K is totally ramified if and only if Go = G, and is then wildly ramified if and only if G1 # {1}. If L / K is wildly ramified, we write t(L/K) for the first positive ramification number, i.e. t(L/K) = max{t : Gt = G1}. If G has order p'~k, where (p, k) = 1, then t(L/K) < e k p / ( p - 1) (see Proposition 3(iv) below):- in particular, if G is a p-group then t(L/K) < ep/(p - 1). Our main result is the following:
Theorem
1 Let L / K be a totally ramified abelian extension of degree p'~
with t(L/Z)
2, assume that G is not cyclic. Then, for any integers r and r', the fractional DL-ideals q3L" and q3ff' are isomorphic as DG-modules if and only -
(rood p = ) .
The proof of this involves an analysis of the case where L / K is cyclic of degree p, using the calculations of Ferton [6], and a reduction of the general case to this special case using the notion of factor equivalence, which has recently been studied extensively by Fr6hlich ([9], [10]). We will also use standard ramification theory for local fields, as presented for instance in [14].
29O
BYOTT
The necessity of the hypothesis on t ( L / K ) is shown by the following example:- let K contain a primitive pth root of unity, and take L = K(Pv/'~), where w generates the maximal ideal V of D. Then t ( L / Z ) = e p / ( p - 1). It is easily seen that every fractional 9L-ideal is a free module over the maximal order in K G , and hence that any two such ideals are isomorphic as ~Gmodules. We shall show that the conclusion of Theorem 1 is in fact false for any cyclic extension of degree p with t(L/g)
>
ep p-1 - -
- 1,
so that, for extensions of degree p, Theorem 1 is best possible. If we remove the hypothesis that L / K be of p-power degree, our methods can still be applied to give some partial results in certain cases. To illustrate this, we will prove the following generalisation of the non-cyclic case of Theorem 1:
T h e o r e m 2 Let L / K be a totally ramified abelian eztension of degree p'~k, where (p, k) = 1, and suppose that ekp t(L/K) < - -
p-1
-
k,
that G is not cyclic, and that p'~ - 1 k < - -p-1 If ~3L~ and ~3L" are isomorphic as DG-modules, then either r - r' (mod p'~k) or, interchanging r and r' if necessary, r = ap TM and r' = ap TM + 1 (mod p~k) f o r s o m e a ~ 0 (mod k).
Factor equivalence yields no information about cyclic extensions, and, at least in the simple formulation given in [9], is only applicable to abelian extensions. One can, however, extend Theorem 1 to non-abelian extensions and to cyclic p-extensions in the case p = 2. Indeed, without using factor equivalence, we will prove the following supplementary result to Theorem 1:
291
BYOTT T h e o r e m 3 Let L / K be a totally ramified extension of degree p'~, with
t ( L / K ) -- 1 p-1
L / K is cyclic and m > 2.
Note that v is determined by the ramification groups of L / K ([14] IV Proposition 4). We end this section with some conditions on the associated order 9IL/K of L / K which ensure that Oz is self-dual. Recall that
~aL/K = {~ 6 K G
: a ~ L C_ ~L}
(and more generally, the associated order of azly fractional idea/~3 L" is defined by replacing both occurrences of DL by ~3L" in this definition). For any element a = ~ % g of KG, we write & = ~ agg -1. e6G
292
BYOTT 1 Let L / K be an abelian extension. (i) If PAL/K = ~L/K and PaL/K ~- (9~L/K)* as DG-modules, then DL is self-dual. 5i) If faL/K is a Hopf order in the Hopf algebra K G (cf. [15]), then DL is self-dual.
Proposition
Proof. (i) Since OL* ~- ~L/K -1, the associated order Of ~L/K -1 is faL/K, which coincides with PaL/K. Moreover, as ~L/K ~ (PaL/K)*, pan/x is a (weakly) selfdual order in the sense of [7] w By Theorem 10 of that paper, DL and ~L/K -1 are both free over PaL/K, and so are isomorphic. (ii) is a special case of (i): if 9aL/K is a Hopf order then by [13] (PaL~K)* ~PaL~K, since D is a principal ideal domain; and ~L/K = 9-tL/K since a ~-* & is the antipode of the HolSf algebra KG.
2
Extensions
of
degree
p
In this section, we will prove Theorem 1 for extensions of degree p, and will also discuss almost maximally ramified extensions of degree p. We make use of the calculations of Ferton [6], and largely follow her notation. Thus let L / K be a totally ramified cyclic extension of degree p, and let t = t ( L / K ) . Then 1 < t < e p / ( p - 1), and (t,p) = 1 unless t = ep/(p - 1) ([14] IV w Ex.3). Any such value of t can occur (Ioc. cit. Ex.5). Lett=pa0+awith0
ep -p-1
p-1
_1.
then
(1)
Now let w (resp. ~') be a generator of ~3 (rasp. ~3L), and fix a generator tr of the Galois group G of L / K . Set f = ~r- 1 6 DG. Since ( f + 1)p = ~v = 1, we then have
f,'=-~
P n=l
293
f".
(2)
BYOTT
I r a # 0 then ~ra is a normal basis for L / K , i.e. L = KG~r ~ ([3] Proposition 3). F o r a + l - p < r < a l e t = {a E KG:a
~r~ E
~L'}"
Then ~ is a fractional oG-ideal containing OG, and ~ ~ ~L" as oGmodules. From [6] Proposition 3 we have the following explicit description of the lattices g~.: P r o p o s i t i o n 2 I f a ~t 0 and a + 1 - p < r < a then the elements fl
(0 < i < p - 1)
.
f o r m an D-basis of tl,, where
(For any real number y, [y] denotes the largest integer not exceeding y.) We can now prove Theorem 1 for extensions of degree p: L e m m a 1 Let L / K that
be a totally ramified cyclic eztension of degree p, such t ( L / K ) p, comparing coefficients of 1 = f0, and noting that v0 = v~ = 0, we see t h a t Co E O • where D x denotes the group of units of O. Without loss of generality, we suppose Co = 1. Then g' has a basis consisting of the elements fi 03ui
=03
~-~i f i 03 i
+
p-l-i
E
cj
j=l
fi+j vi +t,{ 03 ~
p-1
+
E
j=p-i
fi+j C3
03r,i + ~
"
(3)
We will show that fi+j
03,,,+------~}E g' whenever p - i < j < p - 1.
(4)
Admitting this for the moment, we must have v~ - vi > 0 for each i, since the elements (3) lie in ft. We have already observed, however, t h a t v~ < vi, so v~ = vi for 0 < i < p - 1. It then follows from the definitions of vi and v' that r = r'. Thus it ordy remains to prove (4). Writing i + j = p + k and using (2), we have
fi+j
p-z(
P
n=l
~
) fk+, 03vi +t'di "
If k + n > p then, using (2) again and noting t h a t the binomial coefficients are divisible by p, we have
p) fk+,, p2 7 C_DG C_~t'. ,,+~,--'--wE ,~+~,"'---'DG 71"
03
J
oJ
295
.~
BYOTT On the other hand, if k + n < p then -'n
Oa
~
.~
k+,L - ~ s -
j
7/.
0.~ :.+r,
and we must show that e + v kI+ , ~ - v i - v j _ !> 0 w h e n e v e r i + j = p + k > p a n d
l0
296
> -p,
BYOTT Thus (5) holds in all cases, which completes the proof of (4) and thus of the Lemma.
In [11], Jacobinski defines the notion of an almost maximally ramified extension. In the case of a cychc extension of degree p, the definition reduces to the following condition: L / K is almost maximally ramified r
t >
e___p__p_ 1. p-1
Although it is unnecessary for the proofs of the three theorems, we will now consider extensions with this property, thereby showing that Theorem 1 is best possible for extensions of degree p: L e m m a 2 Let L / K be an almost maximally ramified cyclic extension of degree p. Then DL is self-dual as an DG-module. In particular, if t + 1 > ep/(p - 1) then the conclusion of Theorem 1 is false for L / K . Proof. We first consider the maximally ramified case t = ep/(p - 1). Here, K contains a primitive pth root of 1 ([12] Theorem 3), and L = K(Pv ~ ) for some generator w of ~3. As noted after the statement of Theorem 1, it follows that every fractional DL-ideal is free over the maximal order in K G , and hence is isomorphic to every other fractional DL-ideal. For any value of t, we have ~L)K = ~3L- ' , where, by [14] IV Proposition 4, v = ( p - 1 ) ( t + l ) = - a - 1 (modp). Ift+l=ep/(p-1) thenv=0 ,.. ~ - 1 (rood p), so certainly DL = L/K, whilst if t + 1 > ep/(p - 1) then by (1), t=(ep-a)/(p-1)withO e p / ( p - 1) there m a y be other isomorphisms between ideals in addition to t h a t given by L e m m a 2. For example, in the absolutely unramified case e = 1, any ideal ~3L" must have as its associated order either the group ring DG or the maximal order 1
9~=DG+- ~ . P o'6G
These are b o t h self-dual orders in the sense of [7], so ~L" is free over its associated order, and hence isomorphic to either D G or ~Y~.Thus there must be m a n y isomorpkisms between the ideals DL, ~ L , . 9 ~L p-I" (The author is indebted to D.Burns for this observation.) In the situation of L e m m a 2, DL need not be free over its associated order. Indeed, Bertrandias, Bertrandias and Ferton [2] give a criterion for DL to be free when L / K is almost maximally ramified:- writing t / p = [ao; a l , . . . , a,] for the continued fraction expansion of t / p , DL is free over its associated order if and only if n < 4. (The same is also true for almost maximally ramified cychc extensions of degree pro, m > 2:- see [1]). Thus if L / K is almost maximally ramified and cyclic of degree p, with n > 5, then DL is self-dual but not free over its associated order. The first example of this is
299
BYOTT
p = 13, a = 8. The question of the existence of abelian extensions with this property was raised in [5] (p148). Some elementary abelian examples of degree p~ are constructed in [4]; these have t + 1 = ep/(p - 1).
3
Factor e q u i v a l e n c e
We now recall from [9] the notion of factor equivalence between OG-lattices spanning the same KG-module. We will then derive a necessary and sufficient condition for two fractional Oz-ideals to be factor equivalent. Let r be a finite abehan group, let F t denote its group of (complexvalued) abehan characters, and let S(P t) denote the lattice of subgroups of r t. There is an inclusion-reversing bijection between the lattice of subgroups of F and S(Ft), given by associating to each A 1. Let G be its Galois group. Then:-
(i) G1 has order p'~, and G is a semidirect product of G1 by a cyclic group C of order k;
304
BYOTT (ii) if C is normal in G then, writing M = L c for the fixed field of C, 1 t(L/K); t ( M / K ) = -~ (iii) if k = 1 or if G is abelian then G has a normal subgroup H of index p whose fixed field F = L H satisfies 1 t(L/K); t ( F / K ) = -~ (iv) if C is normal in G then t ( L / K )
k -
1.
(9)
Thus if b' = b we must have v = O, and so s' = s, giving r' = r. On the other hand, if b' - b > 1 then, since v > 1 - p, (8) and (9) can only be satisfied
307
BYOTT
simultaneously by b' - b = 1, v = 1 - p. This value of v forces sl = p - 1, 8~' = 0 for all i, so s = p m _ 1 + (b ' - 1)p TM, s' = b'p TM. Thus, putting a = b', we have r = ap "~, r' = ap TM + 1. It only remains to show that a ~ 0 ( m o d k ) . Take H and F as in Proposition 3(iii), so F / K is cyclic of degree p with Galois group Q = G / H , and t(F/Z)
a if and only if s~ _> a, completing the induction.
309
BYOTT
References [1]
Bertrandias, F.: Sur les extensions cycliques de degr6 p'~ d'un corps local. Acta Arith. 34, 361-377 (1979)
[2]
Bertrandias, F., Bertrandias, J-P. and Ferton, M-J,: Sur l'anneau des entiers d'une extension cyclique de degr6 premier d'un corps local. C. R. Acad. Sci. Paris 274, A1388-A1391 (1972)
[3]
Bertrandias, F. and Ferton, M-J.: Sur l'anneau des entiers d'une extension cyclique de degr6 premier d'un corps local. C. R. Acad. Sci. Paris 274, A1330-A1333 (1972)
[4]
Byott, N.P.: Some self-dual local rings of integers not free over their associated orders, to appear in Math. Proc. Camb. Phil. Soc. 1991
[5]
Cassou-Nogu~s, Ph. and Taylor, M.J.: Elliptic functions and rings of integers. (Progress in Mathematics 66) Boston-Basel-Stuttgart: Birkhs 1987
[6] M.-J.Ferton, M-J.: Sur les id~aux d'une extension cyclique de degr6 premier d'un corps local. C. R. Acad. Sci. Paris 276, A1483-A1486 (1973) [7] FrShlich, A.: Invariants for modules over commutative separable orders. Quart. J. Math. Oxford (2) 16, 193-232 (1965) [8] FrShlich, A.: Local fields, in: Cassels, J.W.S. and FrShlich, A. Algebraic number theory. London: Academic Press 1967
(eds.):
[9] FrShlich, A.: Module defect and faetorisability. Illinois J. Math. 32, 407-421 (1988) [10] FrShlich, A.: L-values at zero and multiplicative Galois module structure (also Galois Gauss sums and additive Galois module structure). J. reine angew. Math. 397, 42-99 (1989) [11] Jacobinski, H.: [Iber die Hauptordnung eines KSrpers als Gruppenmodul. J. reine a~ugew. Math. 213, 151-164 (1963)
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BYOTT
[12] MacKenzie, R.E. and Whaples, G.: Artin-Schreier equations in characteristic zero. Am. J. Math. 78,473-485 (1956) [13] Larson, R.G. and Sweedler, M.E.: An associative orthogonal bilinear form for Hopf Mgebras. Am. J. Math. 91, 75-94 (1969). [14] Serre, J-P.: Local Fields. (Graduate Texts in Mathematics 67) BerlinHeidelberg-New York: Springer 1979. [15] Taylor, M.J.: Hopf structure and the Kummer theory of formal groups. J. reine angew. Math. 375/376, 1-11 (1987) [16] Ullom, S.: Integral normal bases in Galois extensions of local fields. Nagoya Math. J. 39, 141-148 (1970) Nigel Byott, New College, Oxford OX1 3BN, U.K.
(Received December 3, 1990; in revised form June 28, 1991)
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