On some modules associated with Galois orbits by Victor Alexandru (1), Marian Vâjâitu (2), Alexandru Zaharescu (3)

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1 Bull. Math. Soc. Sci. Math. Roumanie Tome 61 (109) No. 1, 2018, 3 11 On some modules associated with Galois orbits by Victor Alexandru (1), Marian Vâjâitu (2), Alexandru Zaharescu (3) Dedicated to the memory of Professor Nicolae Popescu Abstract Given a prime number p we consider C p, which is usually called the Tate field, the topological completion of the algebraic closure of the field of p-adic numbers. We introduce and study a class of modules associated with factor groups of profinite groups, especially of those which are the Galois groups of the normal closure of algebraic infinite extensions. In particular, we show that the module associated with a Galois orbit of an arbitrary element of C p is a factor of the Iwasawa algebra of a normal element of C p by an ideal which can be described. Key words and phrases: Galois orbits, Iwasawa algebra, local fields, distributions Mathematics Subject Classification: 11S99. Introduction For a prime number p, we consider Q p the field of p-adic numbers and Q p a fixed algebraic closure of Q p. Let C p be the completion of Q p with respect to the p-adic valuation and O Cp the valuation ring of C p. Let K be an infinite algebraic extension of Q p and L the normal closure of K. Let A be a closed subring of C p. Denote H = Gal(L/K) and G L = Gal(L/Q p ). We introduce and study a class of modules associated with factor groups G L /H and respectively G L /H σ, where H σ = σhσ 1 with σ G L. We associate with G L a Λ A (G L )-module and relate it with Λ A (G L /H) and respectively with a direct product of Λ A (G L /H σ ). This offers new ways to relate distributions in the sense of Mazur and Swinnerton-Dyer (see [6]), functionals, and linear maps defined on an infinite algebraic extension of Q p, with those defined on its normal closure. When x is an arbitrary element of C p, we consider O(x), the Galois orbit with respect to the Galois group of all continuous automorphisms of C p /Q p. Here the module associated with O(x) with scalars drawn from Q p, i.e. Λ Qp (O(x)), which is isomorphic as Q p -module with the space of distributions in the sense of Mazur and Swinnerton-Dyer defined on O(x) with values in Q p, is a factor of the Iwasawa algebra of a Galois orbit of a normal element of C p with scalars drawn from Q p by an ideal which can be described. A particular situation, when x is a normal element of C p, is studied in [1]. The present paper consists of four sections. The first one contains notation and some basic results. In the second section we introduce the module associated with a factor group of a profinite group with scalars drawn from a closed subring of C p, see Propositions 1 and 2. In section three we consider the special case of Galois orbits of elements of C p, which Corresponding author: Marian Vâjâitu

2 4 On some modules associated with Galois orbits are not necessarily normal. We prove that the module associated with a Galois orbit of an arbitrary element of C p with scalars drawn from Q p is a factor of the Iwasawa algebra of a Galois orbit of a normal element of C p with scalars drawn from Q p by an ideal, which can be described, see Theorem 1. In the last section we embed the module associated with the Galois group of the normal closure of an algebraic infinite extension, with scalars drawn from a closed subring of C p, into a direct product of modules associated with some factor groups, see Theorem 2. These modules associated with factor groups of a profinite group, with scalars drawn from a closed subring of C p, enjoy some nice properties and deserve further study. 1 Notation and basic results Let {X n, ϕ n } n 1 be a projective system, such that X n is a finite set and ϕ n : X n+1 X n is a surjective map, for all n 1. Denote by X = lim X n the projective limit defined by this projective system. Consider a closed subring A of C p. The map ϕ n induces a map of the corresponding A-modules A[X n+1 ] A[X n ]. The projective limit lim A[X n ] is a complete A-module which we denote by A[[X]] or Λ A (X). Definition 1. ([6]) By a distribution on {X n, ϕ n } n 1 (or on X) with values in A we mean a set µ = {µ n } n 1 of mappings: µ n : X n A such that the following compatibility relations µ n (x) = µ n+1 (y) (1.1) hold for all n 1 and all x X n. y ϕ 1 n (x) Let D(X, A) be the set of all distributions µ = {µ n } n 1 defined above. canonical isomorphism of A-modules between D(X, A) and Λ A (X). For any µ D(X, A) denote We have a µ = sup{ µ n (x) : x X n }, (1.2) n 1 the norm of µ. When µ < we say that µ is a measure. Let M(X, A) be the set of all measures on X with values in A. When X is a profinite group the space D(X, A) becomes an A-algebra, which is endowed with the convolution product of distributions. The measures M(X, A) D(X, A) correspond, via the above isomorphism, to the sequences of Λ A (X) that are uniformly bounded. Now, let p be a prime number, Q p the field of p-adic numbers, Q p a fixed algebraic closure of Q p and C p the completion of Q p with respect to the p-adic valuation. Denote by G the p-adic absolute Galois group Gal(Q p /Q p ) endowed with the Krull topology. It is known that G is canonically isomorphic to Gal cont (C p /Q p ), the group of all continuous automorphisms of C p. In what follows we shall identify these two groups. For any subset X of C p we denote by X the topological closure of X in C p. It is clear that if X is a field then X is a closed subfield of C p. Let H be a closed subgroup of G. Denote F ix(h) = {x C p : σ(x) = x for all σ H}. For an arbitrary element x C p denote H(x) = {σ G : σ(x) = x}, which is a subgroup

3 V. Alexandru, M. Vâjâitu, A. Zaharescu 5 of G, not necessarily normal. One has F ix(h(x)) = Q p [x], the closure of the ring Q p [x] in C p. We say that x is a topological generic element of Q p [x]. By [2], any closed subfield K of C p has a topological generic element, i.e. there exists x K such that K = Q p [x]. It is proved in [3] that for any element y of C p, the ring Q p [y] and the field Q p (y) have the same topological closure in C p, that is, Qp [y] = Q p (y). By this result, the topological closure in C p of a ring of the form Q p [y] is always a field. Now for an arbitrary element x of C p let O(x) = {σ(x) : σ G} be the orbit of x. It is clear that the map σ σ(x) from G to O(x) is continuous, and it defines a homeomorphism from G/H(x) to O(x), see [2]. Then O(x) is a compact and totally disconnected subspace of C p. Moreover, the group G acts continuously on O(x): if σ G, τ(x) O(x) then σ τ(x) = (στ)(x). Definition 2. ([1]) An element x C p is called normal if the extension Q p (x)/q p is normal, i.e. Gal cont (C p / Q p (x)) = H(x) is a normal subgroup of Gal cont (C p /Q p ). The notions of trace and trace series associated with an element x of C p were introduced and investigated in [4,7,9]. Given an x C p, the trace T r(x) is defined by the equality T r(x) = tdπ x (t), (1.3) O(x) provided that the integral with respect to the Haar distribution π x on the right side of (1.3) is well defined. This is the case, for example, when π x is bounded, i.e. when π x is a measure. The integral is also well defined when x is a Lipschitz element, see [4, Theorem 4.4]. Precisely, an element x C p is Lipschitz if one has lim ε 0 ε N(x, ε) = 0, where N(x, ε) is the number of open balls of radius ε which cover O(x) and stands for the p-adic absolute value. An element x C p is called p-bounded if there exists an s N such that p s does not divide the number N(x, ε), for any ε > 0. It is clear that a p-bounded element of C p is also Lipschitz. The trace function F (x, z) is defined by F (x, z) = O(x) 1 1 zt dπ x(t), for all those z C p for which the integral is well defined. This is an analytic object that embodies a significant amount of algebraic data. The Taylor series expansion F (x, z) = T r(x n ) z n n=0 is usually called the trace series associated with x. 2 Modules associated with factor groups Let G be a profinite group, G = lim G n with G n finite groups such that G n = G/H n, where {H n } n 1 is a family of normal subgroup of G of finite index such that H n+1 H n, for any

4 6 On some modules associated with Galois orbits n 1 and n 1 H n = {e}, the neutral element of G. Let H be a closed subgroup of G, which is not necessarily normal. Let H n := H n H be the subgroup of G that is generated by the product of H n with H. We have that H n is an open and closed subgroup of G of finite index and n 1 H n = H. For the sake of simplicity we denote by G/H = {σ (mod H) : σ G} the left cosets of H in G and G n = G/H n the left cosets of H n in G. One has the following result, which is a generalization of a classical result for normal subgroups [8]. Proposition 1. Let G = lim G/H n be a profinite group where {H n } n 1 is a family of normal subgroups of G of finite index such that H n+1 H n for any n 1, and n 1 H n = {e}, the neutral element of G. Let H be a closed subgroup of G, which is not necessarily normal, and H n := H n H the subgroup of G that is generated by the product of H n with H. There is a homeomorphism between G/H and lim G/H n. Proof. Because H H n+1 H n G we have the following commutative diagram lim G n f π n Gn ϕ n G/H fn+1 G n+1 where π n, ϕ n, f n+1 are canonical maps and f is the unique map obtained via the universal property of projective limit such that π n f = ϕ n f n+1 = f n, for any n 1. The map f is a bijection. Indeed, let us suppose that f(σ (mod H)) = f(τ (mod H)). By this one has f n (σ (mod H)) = f n (τ (mod H)), which means that σ ( ) ( ) mod H n = τ mod Hn, for any n 1. Because n 1 H n = H, clear σ (mod H) = τ (mod H), which means that f is an injective map. Now, let x = (x 1, x 2,..., x n,... ) lim G n. There is σ n G such that ( ) f n (σ n (mod H)) = σ n mod Hn = xn, for any n 1. By using the commutativity of the above diagram we have that σ n+k ( mod Hn ) = σn ( mod Hn ), (2.1) for any n, k 1. From hypothesis G is profinite, so compact, and by this there exists a subsequence {σ nm } m 1 of the sequence {σ n } n 1 that converges to an element σ G. Because H n is a closed subgroup of G with respect to Krull s topology, by using (2.1), one has σ ( ) ( ) mod H n = σn mod Hn, (2.2) for any n 1, and by (2.2) one obtains f(σ (mod H)) = x, which means that f is a surjective map. The continuity of f follows from [8, Proposition 1.1.1] and the proof is done. We have a natural action of G n on A[G n ]. Indeed, let ˆτ G n, τ G, and σ G n, which is the left coset of σ G in G n. By defining ˆτ σ = τσ one has that A[G n ] is an A[G n ]-finite generated module, which is free over A. The canonical morphisms A[G n+1 ] A[G n ] verify the compatibility relations of a projective system. By definition, one considers that Λ A (G/H) = A[[G/H]] = lim A[G n ]

5 V. Alexandru, M. Vâjâitu, A. Zaharescu 7 is the module of G/H with scalars drawn from A. We can see that Λ A (G/H) = lim U A[G/UH], where the inverse limit is taken over all open subgroups U of G, so this module does not depend on the choice of {H n } n 1 and it becomes an Λ A (G)-module. Now, let A be a closed subring of C p which is compact or is not contained in O Cp. In the second situation A is a field which contains Q p, see [3, Theorem 7] and [3, Lemma 6]. We have the following commutative diagramm A[G n ] A[G n ] 0 A[G n+1 ] A[G n+1 ] 0 and, by [8, Lemma 1.1.5] and [5, Proposition 9.1], one has the following exact sequence Λ A (G) Λ A (G/H) 0. The first arrow in the above sequence is continuous, see also [8]. We collect the above results in the following proposition. Proposition 2. Let A be a closed subring of C p, G a profinite group and H a closed subgroup of G, which is not necessarily normal. If A is compact or is not contained in O Cp then Λ A (G/H) is a factor of Λ A (G). When H is a normal divisor of G then Λ A (G/H) becomes an algebra. In particular, when G/H is a compact p-adic Lie group and A Z p, the module Λ A (G/H) turns out to be the Iwasawa algebra of G/H. In the next section we will focus on the following important example: G is the p-adic absolute Galois group and H = H(x) = {σ G : σ(x) = x} is the stabilizer of an arbitrary element x of C p, which is a subgroup of G, not necessarily normal. 3 The case of Galois orbits It is known that the p-adic absolute Galois group G is profinite so we have a tower of fields Q p = L 0 L 1 L 2 L n Q p such that L n /Q p is finite and normal with Galois group G n. If we denote H n = Gal(Q p /L n ) then G/H n G n. Let x be an arbitrary element of C p, which is not necessarily normal. By denoting H = Gal(Q p / Q p [x] Q p ), which is not necessarily a normal subgroup of G, one has that H H(x) = {σ Gal cont (C p /Q p ) : σ(x) = x}, via [1, Proposition 1]. Let H n = H n H be the subgroup of G that is generated by the product of H n with H and G n = G/H n the left cosets of H n in G. Let K n = F ix(h n ) be the field fixed by H n and K n the normal closure of K n. Denote G n = Gal(K n/q p ), H n = Gal(Q p /K n) and K x = n 1 K n. In fact K x is the Galois closure of the subfield of Q p fixed by H. Denote G Kx = Gal(K x /Q p ). For any n 1, we have a morphism Φ n : Q p [G n] Q p [G n ], which is defined canonically, such that

6 8 On some modules associated with Galois orbits the following diagram of Q p -vector spaces Q p [G n] π n Φ n Qp [G n ] 0 π n Q p [G n+1] Φ n+1 Qp [G n+1 ] 0 is commutative, where π n and π n are canonical projections. By [5, Proposition 9.1] one obtains the following exact sequence Λ Qp (G Kx ) Λ Qp (O(x)) 0, (3.1) where the first arrow is continuous, see [8]. From the proof of [1, Theorem 2] one has Hom(K n, Q p ) Q p [G n], which is an isomorphism of Q p -vector spaces, so Λ Qp (G Kx ) Hom(K x, Q p ). By (3.1) we have the following exact sequence Hom(K x, Q p ) where Φ = lim Φ n, via the above isomorphism. Denote where ker Φ n = { ˆσ G n Φ Λ Qp (O(x)) 0, (3.2) I = ker Φ lim ker Φ n, (3.3) aˆσ ˆσ : ˆσ G n, σ=τ aˆσ = 0 for any τ G n From [1, Theorem 2] it follows that there is a normal element y of K x such that Λ Qp (O Kx (y)) Hom(K x, Q p ). Summing up, one has the following result. Theorem 1. For any element x of C p there is a minimal Galois extension K x of Q p, which is the Galois closure of the subfield of Q p fixed by the stabilizer of x, such that the following sequence Φ Hom(K x, Q p ) Λ Qp (O(x)) 0 is exact, where Φ is defined in (3.2). Moreover, there is a normal element y of Kx such that Λ Qp (O(x)) Λ Q p (O Kx (y)) I where I = ker Φ is the ideal described in (3.3). Remark 1. In other words, the above result says that the module associated with a Galois orbit O(x) of an arbitrary element x of C p with scalars drawn from Q p, which is isomorphic as Q p -module with the space of distributions in the sense of Mazur and Swinnerton-Dyer defined on O(x) with values in Q p, is a factor of the Iwasawa algebra of a Galois orbit of a normal element of C p with scalars drawn from Q p by an ideal which can be described. Also, a finite field extension of Q p can be used instead of the base field in Theorem 1. }.

7 V. Alexandru, M. Vâjâitu, A. Zaharescu 9 4 A complementary result We preserve the same notation and definitions as in the previous sections. Now, let Q p K be an infinite algebraic extension and Q p K 1 K 2 K n K a tower of fields such that Q p K n is a finite extension for any n 1 and, moreover, K = n 1 K n. We denote by L the normal closure of K and by L n the normal closure of K n. One has Q p L 1 L 2 L n L and L = n 1 L n. We use the following notation: G L = Gal(L/Q p ) Gal cont ( L/Q p ), H = Gal(L/K), H n = Gal(L/K n ), U n = Gal(L/L n ). We denote by N GL (H) the normalizer of H in G L and by S H a system of representatives for the left cosets of N GL (H) in G L. For any σ S H we define H σ = σhσ 1. Also, we denote by N GL (H n ) the normalizer of H n in G L and by S Hn a system of representatives for the left cosets of N GL (H n ) in G L. One has U n = σ SHn H σ n. The map F : G L σ S H (G L /H σ ), F (g) = (g (mod H σ )) σ SH, g G L, (4.1) is injective. Indeed, let g 1, g 2 G L be such that F (g 1 ) = F (g 2 ). Then (g 1 (mod H σ )) σ SH = (g 2 (mod H σ )) σ SH, so g2 1 g 1 σ SH H σ = {e}, the neutral element of G L. This is happening because L is the normal closure of K, which is the composition of the fields σ(k), σ S H. Also, the following diagram G L π σ F σ S H (G L /H σ ) π σ (4.2) G L /H σ Id G L /H σ is commutative, where π σ is the canonical map, π σ is the projection onto the σ component of the direct product, and Id stands for the identity map. In the same way as in (4.1) and (4.2) we have the following map F n : G L /U n σ S H G L /(U n H) σ, F n (g (mod U n )) = (g (mod (U n H) σ )) σ SH, g G L, which is injective because σ SH (U n H) σ = U n. Moreover, the following diagram (4.3) G L /U n π σ,n F n σ S H G L /(U n H) σ π σ,n (4.4) G L /(U n H) σ Id G L /(U n H) σ, is commutative, where π σ,n is the canonical map, π σ,n is the projection onto the σ component of the direct product, and Id stands for the identity map. By considering a closed subring A of C p, we obtain from (4.3), in a canonical way, an injective map F n : A[G L /U n ] σ S H A[G L /(U n H) σ ]. (4.5)

8 10 On some modules associated with Galois orbits By passing to inverse limit, one has the following result. Theorem 2. Let K be an infinite algebraic extension of Q p and A a closed subring of C p. Let L be the normal closure of K. Denote H = Gal(L/K) and G L = Gal(L/Q p ). Also, denote by N GL (H) the normalizer of H in G L and let S H be a system of representatives for the left cosets of N GL (H) in G L. For any σ S H we define H σ = σhσ 1. Then, there exists a monomorphism F : Λ A (G L ) σ S H Λ A (G L /H σ ) of Λ A (G L )-modules, such that the following diagram Λ A (G L ) λ σ F σ S H Λ A (G L /H σ ) λσ (4.6) Λ A (G L /H σ ) Id Λ A (G L /H σ ), is commutative, where λ σ is the canonical map, λ σ is the projection onto the σ component of the direct product, and Id stands for the identity map. Remark 2. Theorem 2, Theorem 1, and Proposition 2 provide a way to relate distributions in the sense of Mazur and Swinnerton-Dyer, functionals, and linear maps defined on an infinite algebraic extension K of Q p with those defined on the normal closure of K. Questions. We end this paper with a few questions which arise naturally, and which we pose to interested readers. Let Q p K be an infinite algebraic extension and let L be the normal closure of K. Denote G L = Gal(L/Q p ). 1) Let ϕ and ψ be two linear maps on L whose restrictions to σ(k) coincide, for each σ G L. Does it follow that ϕ = ψ? 2) If the restriction of ϕ to any σ(k) is continuous, does it follow that ϕ is continuous? 3) An interesting problem would be to study cases when ϕ(σx) = ϕ(x), for any σ G, where x is a generic element of K. A special example is provided by the trace function, which satisfies this property. 4) Can any Q p -functional on K be extended to a Q p -functional on L? 5) If T r K/Qp is continuous, does it follow that T r L/Qp is continuous? References [1] V. Alexandru, N. Popescu, M. Vâjâitu, A. Zaharescu, On the Iwasawa algebra associated to a normal element of C p, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 120, No. 1(2010), [2] V. Alexandru, N. Popescu, A. Zaharescu, On closed subfields of C p, J. Number Theory 68, 2(1998), [3] V. Alexandru, N. Popescu, A. Zaharescu, The generating degree of C p, Canad. Math. Bull. 44(2001), no. 1, 3-11.

9 V. Alexandru, M. Vâjâitu, A. Zaharescu 11 [4] V. Alexandru, N. Popescu, A. Zaharescu, Trace on C p, J. Number Theory 88 (2001), no. 1, [5] R. Hartshorne, Algebraic Geometry, Springer, [6] B. Mazur, P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), [7] N. Popescu, M. Vâjâitu, A. Zaharescu, On the existence of trace for elements of C p, Algebras and Representation Theory, (2006) 9; [8] L. Ribes, P. Zalesskii, Profinite groups, Springer-Verlag, [9] M. Vâjâitu, A. Zaharescu, Trace functions and Galois invariant p-adic measures, Publ. Math., 50 (2006), Received: Accepted: (1) Department of Mathematics, University of Bucharest, 14 Academiei Street, RO Bucharest, Romania vralexandru@yahoo.com (2) Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit 5, P.O.Box RO Bucharest, Romania Marian.Vajaitu@imar.ro (3) Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit 5, P.O.Box RO Bucharest, Romania, and Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street Urbana, IL, 61801, USA zaharesc@illinois.edu