Admissibility of groups over function fields of p-adic curves

B. Surendranath Reddy and V. Suresh

Abstract

Let K be a field and G a finite group. The question of ‘admissibility’ of G over K was originally posed by Schacher, who gave partial results in the case K = ℚ. In this paper, we give necessary conditions for admissibility of a finite group G over function fields of curves over complete discretely valued fields. Using this criterion, we give an example of a finite group which is not admissible over ℚ_{p}(t). We also prove a certain Hasse principle for division algebras over such fields.

Introduction

Let K be a field and G a finite group. We say that G is admissible over K if there exists a division ring D central over K and a maximal subfield L of D which is Galois over K with Galois group G. Schacher asked, given a field K, which finite groups are admissible over K and proved that if a finite group G is admissible over ℚ, then every Sylow subgroup of G is metacyclic ([Sc], 4.1). This led to the conjecture that a finite group G is admissible over ℚ if and only if every Sylow subgroup of G is meta-cyclic. This conjecture has been proved for all solvable groups ([So]) and for certain non-solvable groups of small order ([CS], [FS], [Fe1], [Fe2], [Fe3]).

Recently Harbater, Hartman and Krashen ([HHK2], 4.5) gave a characterization of admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue fields. In this paper, we consider the function fields of curves over complete discretely valued fields without any assumptions on the residue fields and prove the following

Theorem 1. Let K be a complete discretely valued field with residue field k and F = K(X) be the function field of a curve X over K. Let G be a finite group. Suppose that the order of G is coprime to char(k). If G is admissible over F then every Sylow subgroup P of G has a normal series P ⊇ P_{1} ⊇ P_{2} such that

(1) P/P_{1} and P_{2} are cyclic

(2) P_{1}/P_{2} is admissible over a finite extension of the residue field of a discrete valuation of F.

A main ingredient for the proof of the above theorem is the following Hasse principle for central simple algebras, which has independent interest.

Theorem 2. Let K be a complete discretely valued field with residue field k and F = K(X) be the function field of a curve X over K. Let A be a central simple algebra over F of degree n = ℓ^{r} for some prime ℓ and r ≥ 1. Assume that ℓ is not equal to char(k) and K contains a primitive n^{th} root of unity. Then index(A) = index(A ⊗ F_{v}) for some discrete valuation v of F.

For the proof of the above theorem, we use the patching techniques of ([HHK1]). A similar Hasse principle is proved for quadratic forms over such fields in ([CTPS], 3.1). In ([HHK3], 9.12), it is proved that if a central simple algebra A over F (F as above), is split over F_{ν} for all discrete valuations on F, then A is split over F.

There are some examples of classes of finite groups which are admissible over the rational function fields. However there was no example, in the literature, of a finite group which is not admissible over ℚ_{p}(t). Using Theorem 1, we give an example of a finite group which is not admissible over ℚ_{p}(t). We also prove admissibility of a certain class of groups over ℚ_{p}(t) using patching techniques.

Theorem 3. Let K be a p-adic field and F the function field of a curve over F. Let G be a finite group with order coprime to char(k). If every Sylow subgroup of G is a quotient of ℤ^{4}, then G is admissible over F.

In [FSS], it was proved that every abelian group on three or less generators is admissible over ℚ(t). We conclude by showing that every abelian group of order n with four or less generators is admissible over ℚ(ζ)(t), where ζ is a primitive n^{th} root of unity.

1. Some Preliminaries

In this section we recall a few basic definitions and facts about division algebras and patching techniques ([GS], [HH], [HHK1], [P], [S2], [Sc], [Sch], [Ser]).

Let K be a field and Br(K) be the Brauer group of central simple algebras over K. For an integer n ≥ 2, let _{n}Br(K) denote the n-torsion subgroup of Br(K). If A and B are two central simple algebras over K, we write A ≃ B if A and B are isomorphic as K-algebras and we write A = B if they represent the same element in Br(K). Let n be an integer coprime to char(K). Suppose E/K is a cyclic extension of degree n and σ a generator of Gal(E/K). For b ∈ K^{∗}, let (E/K, σ, b) (or simply (E, σ, b)) be the Kalgebra generated by E and y with y^{n} = b and λy = yσ(λ) for all λ ∈ E. Then (E, σ, b) is a central simple algebra over K and represents an element in _{n}Br(K). Suppose that K contains a primitive n^{th} root of unity. Then E = K(\sqrt[n]{a}) for some a ∈ K^{∗} and σ(\sqrt[n]{a}) = ζ \sqrt[n]{a} for a primitive n^{th} root of unity ζ ∈ K^{∗}. The algebra (E, σ, b) is generated by x, y with x^{n} = a, y^{n} = b and xy = ζyx and we denote the algebra (E, σ, b) by (a, b)_{n}.

Suppose that ν is a discrete valuation of K with residue field k. Let n be a natural number which is coprime to char(k). Then we have a residue homomorphism ∂_{ν} : _{n}Br(K) → H^{1}(k, ℤ/nℤ), where H^{1}(k, ℤ/nℤ) denotes the first Galois cohomology group. Suppose k contains a primitive n^{th} root of unity. By fixing a primitive n^{th} root of unity, we identify H^{1}(k, ℤ/nℤ) with k^{∗}/k^{∗n}. With this identification we have ∂_{ν}((a, b)_{n}) = \overline{\frac{a^{ν(b)}}{b^{ν(a)}}} ∈ k^{∗}/k^{∗n}, where for any c ∈ K^{∗} which is a unit at ν, \overline{c} denotes its image in k^{∗}. More generally, let E/K be a cyclic unramified inert extension of K and σ ∈ Gal(E/K) be a generator. Let π ∈ K^{∗} be a parameter at ν. Then the residue of (E/K, σ, π) is (E_{0}/k, σ_{0}), where k is the residue field of K at ν, E_{0} is the residue field of E at the unique extension of ν to E and σ_{0} is the image of σ.

Let K be a field and n an integer. Then H^{1}(K, ℤ/nℤ) classifies the equivalence classes of pairs (E, σ), where E is a cyclic Galois field extension of K of degree a divisor of n and σ a generator of the Galois group G(E/K) of E/K. Let (E, σ) be a pair representing an element in H^{1}(K, ℤ/nℤ). Let m ≥ 1 be an integer. We now describe m(E, σ) in the group H^{1}(K, ℤ/nℤ). Let d be the greatest common divisor of m and [E : K]. Let E(m) be the subfield of E fixed by σ^{[E:K]/d} and σ(m) be the restriction of σ^{m/d} to E(m). Then m(E, σ) = (E(m), σ^{m/d}). The order of (E, σ) in H^{1}(K, ℤ/nℤ) is equal to [E : K]. Since the identity element of H^{1}(K, ℤ/nℤ) is (K, id), we have m(E, σ) is trivial in H^{1}(K, ℤ/nℤ) if and only if E(m) = K if and only if [E : K] divides m.

Let K be a complete discretely valued field with residue field k. Let L/K be a finite extension of K. Since K is complete, the discrete valuation of K extends uniquely to a discrete valuation of L and L is complete with respect to this discrete valuation. Let n be a natural number which is coprime to the characteristic of k. Let L/K be a Galois extension of degree n. Let L_{1} be the maximal unramified extension of K. Since n is coprime to char(k), the residue field L_{0} of L is same as the residue field of L_{1}. Since L/K is Galois, L_{1}/K and L_{0}/k are also Galois and there is a natural isomorphism Gal(L/K) → Gal(L_{0}/k). Let E_{0}/k be a cyclic extension of degree n. Then there exists a unique (up to isomorphism) unramified cyclic extension E of K with residue field E_{0} and a natural isomorphism Gal(E/K) → Gal(E_{0}/k). Let (E_{0}, σ_{0}) ∈ H^{1}(k, ℤ/nℤ). Then we have a unique (E, σ) ∈ H^{1}(K, ℤ/nℤ) with [E : K] = [E_{0} : k], E_{0} as the residue field of E and the image of σ equal to σ_{0}; (E, σ) is called the lift of (E_{0}, σ_{0}).

Let X be a regular integral scheme with function field F. Let n be an integer which is a unit on X. Let f ∈ F and P ∈ X be a point. If f is regular at P, then we denote its image in the residue field κ(P) at P by f(P). Let X^{1} denote the set of codimension one points of X. For each codimension one point x of X, we have discrete valuation ν_{x} on F. Let κ(x) denote the residue field at x. Since n is a unit on X, n is coprime to char(κ(x)) and we have the residue homomorphism ∂_{x} : _{n}Br(F) → H^{1}(κ(x), ℤ/nℤ). Let α ∈ _{n}Br(F). We say that α is unramified at x if ∂_{x}(α) = 0. We say that α is unramified on X if it is unramified at every codimension one point of X. Let A be a central simple algebra over F. We say tat A is unramified if its class in Br(F) is unramified. If X = Spec(B), then we say that α is unramified on B if it is unramified on X.

Let F be a field and A a central simple algebra over F. Then A ≃ M_{m}(D) for some central division algebra D over F. The degree of A is defined as \sqrt{dim_{F} A} and the index of A is defined as the degree of D. If L/K is a field extension, then index(A⊗_{F} L) divides index(A). Let B be an integral domain and F its field of fractions. Let A be an Azumaya algebra over B, then we define the index of A to be the index of A ⊗_{B} F.

Let B be a regular integral domain of dimension at most 2 which is complete with respect to a prime ideal P. Assume that B/P is a regular integral domain of dimension at most 1 (for example P is a maximal ideal). Let κ(P) be the field of fractions of B/P. Let A be a central simple algebra over the field of fractions F of B which is unramified on B. Then there exists an Azumaya algebra A over B such that A ⊗_{B} F ≃ A ([CTS], 6.13)). For an ideal I of B, we denote the algebra A ⊗_{B} B/I by A(I).

Lemma 1.1 Let A, B, A and P be as above. Then index(A) = index (A ⊗_{B/P} κ(P)).

Proof. Suppose A ≃ M_{n}(D) for some division algebra D over F. Since A is unramified on B, D is also unramified on B. Since B is a regular domain of dimension 2, there exists an Azumaya algebra D over B such that D ⊗_{B} K ≃ D ([CTS], 6.13)). Since Br(B) → Br(F) is injective ([AG], 7.2), A = D in Br(B). In particular A ⊗_{B/P} κ(P) = D ⊗_{B/P} κ(P) in Br(κ(P)). Hence index(A) = degree(D) = \sqrt{dim_{F} (D)} = \sqrt{rank_{B}(D)} = \sqrt{rank_{B/P} (D ⊗_{B} B/P)} = \sqrt{dim_{κ(P)}(D ⊗_{B/P} κ(P))} ≥ index(A ⊗_{B/P} κ(P)).

Suppose A ⊗_{B/P} κ(P) ≃ M_{m}(D_{0}) for some central division algebra D_{0} over κ(P). Since B/P is regular integral domain of dimension 1, there exists an Azumaya algebra D_{0} over B/P such that D_{0} ⊗_{B/P} κ(P) ≃ D_{0}. Since B is P -adically complete, there exists an Azumaya algebra D̃_{0} over B such that D̃_{0} ⊗_{B} B/P ≃ D_{0} and A = D̃_{0} in Br(B) ([C], [KOS]). In particular A = D̃_{0} ⊗_{B} F in Br(F). Hence index(A ⊗_{B/P} κ(P)) = degree(D_{0}) =

\sqrt{dim_{κ(P)}(D_{0})} = \sqrt{rank_{B/P} (D_{0})} = \sqrt{rank_{B}(D̃_{0})} = \sqrt{dim_{F} (D̃_{0} ⊗ F)} ≥ index(A). Thus index(A) = index(A ⊗_{B/P} κ(P)). □

Let K be a field and L a finite extension of K. Then L is called Kadequate if there is a division ring D central over K containing L as a maximal subfield. A finite group G is called K-admissible if there is a Galois extension L of K with G as the Galois group of L over K, and L is K-adequate.

A finite group G is called metacyclic if G has a normal subgroup H such that H is cyclic and G/H is cyclic.

2. A local-global principle for central simple algebras

Let K be a complete discretely valued field with residue field k. Let F be the function field of a curve over K. Let n be an integer which is coprime to the characteristic of k. Assume that K contains a primitive n^{th} root of unity. In this section we prove a certain Hasse principle for central simple algebras over F of index n. We begin with the following

Lemma 2.1. (cf. [FS], Proposition 1(3) and [JW], 5.15) Let R be a complete discrete valuated ring and K its field of fractions. Let A be a central simple algebra over K of index n which is unramified at R. Let E be an unramified cyclic extension of K of degree m and σ a generator of the Galois group of E/K. Let π be a parameter in R. Assume that mn is invertible in R. Then index(A ⊗ (E, σ, π)) = index(A ⊗ E) · [E : K].

The following two lemmas (2.2, 2.3) are well known.

Lemma 2.2. Let R be a regular ring of dimension 2 with field of fractions F. Let n be an integer which is a unit in R. Assume that F contains a primitive n^{th} root of unity. Suppose A is a central simple algebra over F which is unramified on R. Let x ∈ R be a regular prime and ν be the discrete valuation on F given by x. Suppose that R is complete with respect to (x)-adic topology. Let u ∈ R be a unit. Then index(A ⊗ F_{ν}(\sqrt[n]{u})) = index(A ⊗ F (\sqrt[n]{u})).

Proof. Let S be the integral closure of R in F (\sqrt[n]{u}). Since R is a regular ring and n, u are units in R, S is also a regular ring. Let x be a regular prime in R, then x is also a regular prime in S. Thus by replacing R by S, it is enough to show that index(A ⊗ F_{ν}) = index(A).

Since R is a two-dimensional regular ring, there is an Azumaya algebra A over R with A ⊗ F ≃ A ([CTS], 6.13)). Since R is complete with respect to (x)-adic topology, index(A) = index(A ⊗ κ(x)) (cf. 1.1), where κ(x) is the field of fractions of R/(x). Let R_{ν} be the ring of integers in F_{ν}. Since A is unramified on R, A is also unramified on R_{ν}. Since F_{ν} is complete, index(A ⊗ F_{ν}) = index(A ⊗ κ(ν)) (cf. 1.1). Since κ(ν) = κ(x), index(A) = index(A ⊗ F_{ν}). □

Lemma 2.3. Let R be a complete regular local ring with field of fractions F and residue field k. Let n be an integer which is a unit in R. Let u_{1}, ⋯, u_{r} ∈ R be units. Suppose x ∈ R is a regular prime. Let ν be the discrete valuation on F given by x. Then [F_{ν}(\sqrt[n]{u_{1}}, ⋯, \sqrt[n]{u_{r}}) : F_{ν}] = [F (\sqrt[n]{u_{1}}, ⋯, \sqrt[n]{u_{r}}) : F].

Proof. Since F_{ν} is a complete discretely valued field and n, u_{1}, ⋯, u_{r} are units in the ring of integers, [F_{ν}(\sqrt[n]{u_{1}}, ⋯, \sqrt[n]{u_{r}}) : F_{ν}] = [κ(ν)(\sqrt[n]{\overline{u}_{1}}, ⋯, \sqrt[n]{\overline{u}_{r}}) : κ(ν)]. Since κ(ν) is the field of fractions of the complete local ring R/(x) and the the residue field of R/(x) is k, we have [κ(ν)(\sqrt[n]{\overline{u}_{1}}, ⋯, \sqrt[n]{\overline{u}_{r}}) : κ(ν)] = [k(\sqrt[n]{\overline{u_{1}}}, ⋯, \sqrt[n]{\overline{u_{r}}}) : k]. Since R is complete and u_{1}, ⋯, u_{r} are units, we also have [F (\sqrt[n]{u_{1}}, ⋯, \sqrt[n]{u_{r}}) : F] = [k(\sqrt[n]{\overline{u}_{1}}, ⋯, \sqrt[n]{\overline{u}_{r}}) : k]. Hence [F_{ν}(\sqrt[n]{u_{1}}, ⋯, \sqrt[n]{u_{r}}) : F_{ν}] = [F (\sqrt[n]{u_{1}}, ⋯, \sqrt[n]{u_{r}}) : F]. □

Proposition 2.4. Let R be a 2-dimensional complete regular local ring with maximal ideal m = (x, y). Let F be the field of fraction of R and k the residue field of R. Let n be an integer coprime to char(k). Assume that F contains a primitive n^{th} root of unity. Let A be a central simple algebra over F of degree n. Suppose that A is unramified on R except possibly at x and y. Then index(A) = index(A ⊗ F_{ν}) for the discrete valuation ν of F given by the prime ideals either (x) or (y) of R.

Proof. Suppose that A is unramified on R. Let ν be the discrete valuation of F given by x. Since A is unramified on R, by (2.2), we have index(A⊗F_{ν}) = index(A).

Suppose that A is ramified on R only at the prime ideal (x). Let ν be the discrete valuation on F given by the prime ideal (x) of R. By ([S1]), we have A = A′ ⊗ (u, x) for some unit u in R and a central simple algebra A′ over F which is unramified on R. Since F_{ν} is complete and u is a unit at ν, there is a unique extension of ν to F_{ν}(\sqrt[n]{u}) such that F_{ν}(\sqrt[n]{u}) is complete with the residue field κ(ν)(\sqrt[n]{\overline{u}}), where \overline{u} is the image of u in κ(ν). Since A′ is unramified on R, we have

index(A ⊗ F_{ν}) = index(A′ ⊗ (u, x) ⊗ F_{ν})
= index(A′ ⊗ F_{ν}(\sqrt[n]{u})) · [F_{ν}(\sqrt[n]{u}) : F_{ν}] (by (2.1))
= index(A′ ⊗ F (\sqrt[n]{u})) · [F_{ν}(\sqrt[n]{u}) : F_{ν}] (by (2.2))
= index(A′ ⊗ F (\sqrt[n]{u})) · [F (\sqrt[n]{u}) : F] (by (2.3)).

Since A = A′ ⊗ (u, x), the index of A divides index(A′ ⊗ F (\sqrt[n]{u})) · [F (\sqrt[n]{u}) : F] = index(A ⊗ F_{ν}). Thus index(A) = index(A ⊗ F_{ν}).

Assume that A is ramified on R at both the primes (x) and (y). Then by ([S1]), either A = A′ ⊗ (u_{1}, x) ⊗ (u_{2}, y) or A = A′ ⊗ (uy^{r}, x) where u_{1}, u_{2}, u are units in R, r coprime with n and A′ unramified on R.

Suppose that A = A′ ⊗ (u_{1}, x) ⊗ (u_{2}, y) for some units u_{1}, u_{2} ∈ R and A′ unramified on R. Let ν be the discrete valuation on F given by y. Then by (2.1), we have index(A ⊗ F_{ν}) = index(A′ ⊗ (u_{1}, x) ⊗ F_{ν}(\sqrt[n]{u_{2}})) · [F_{ν}(\sqrt[n]{u_{2}}) : F_{ν}]. By (2.2), we have [F_{ν}(\sqrt[n]{u_{2}}) : F_{ν}] = [F (\sqrt[n]{u_{2}}) : F]. We now compute index(A′ ⊗ (u_{1}, x) ⊗ F_{ν}(\sqrt[n]{u_{2}})).

Since A′ ⊗ (u_{1}, x) is unramified at ν and u_{2} is a unit at ν, index(A′ ⊗ (u_{1}, x)⊗F_{ν}(\sqrt[n]{u_{2}})) = index(A′⊗(u_{1}, x)⊗κ(ν)(\sqrt[n]{\overline{u}_{2}})) (cf. 1.1), where \overline{u_{2}} is the image of u_{2} in κ(ν). Since κ(ν) is the field of fractions of R/(y), the image of x in κ(ν) gives a discrete valuation μ on the residue field κ(ν) with κ(μ) = k and κ(ν) is complete with respect to μ. Since \overline{u_{2}} is a unit at μ, κ(ν)(\sqrt[n]{\overline{u_{2}}}) is a complete discrete valuated field with \overline{x} as parameter and residue field k(\sqrt[n]{\overline{u_{2}}}). Thus, we have

index(A′ ⊗ (u_{1}, x) ⊗ F_{ν}(\sqrt[n]{u_{2}})) = index(A′ ⊗ (u_{1}, x) ⊗ κ(ν)(\sqrt[n]{\overline{u}_{2}}))(by (1.1)) = index(A′ ⊗ κ(ν)(\sqrt[n]{\overline{u_{2}}}, \sqrt[n]{\overline{u_{1}}})) · [κ(ν)(\sqrt[n]{\overline{u_{2}}}, \sqrt[n]{\overline{u_{1}}}) : κ(ν)(\sqrt[n]{\overline{u_{2}}})] (by (2.1)) = index(A′ ⊗ F_{ν}(\sqrt[n]{u_{2}}, \sqrt[n]{u_{1}})) · [F_{ν}(\sqrt[n]{u_{2}}, \sqrt[n]{u_{1}}) : F_{ν}(\sqrt[n]{u_{2}})] (by (1.1)) = index(A′ ⊗ F (\sqrt[n]{u_{2}}, \sqrt[n]{u_{1}})) · [F (\sqrt[n]{u_{2}}, \sqrt[n]{u_{1}}) : F (\sqrt[n]{u_{2}})] (by (2.2), (2.3)).

Hence

index(A ⊗ F_{ν}) = index(A′ ⊗ (u_{1}, x) ⊗ F_{ν}(\sqrt[n]{u_{2}})) · [F_{ν}(\sqrt[n]{u_{2}}) : F_{ν}](by (2.1)) = index(A′ ⊗ F (\sqrt[n]{u_{2}}, \sqrt[n]{u_{1}})) · [F (\sqrt[n]{u_{2}}, \sqrt[n]{u_{1}}) : F (\sqrt[n]{u_{2}})] · [F (\sqrt[n]{u_{2}}) : F] = index(A′ ⊗ F (\sqrt[n]{u_{2}}, \sqrt[n]{u_{1}})) · [F (\sqrt[n]{u_{2}}, \sqrt[n]{u_{1}}) : F].

On the other hand we have index(A) = index(A′ ⊗ (u_{1}, x) ⊗ (u_{2}, y)) divides index(A′ ⊗F (\sqrt[n]{u_{1}}, \sqrt[n]{u_{2}}))·[F (\sqrt[n]{u_{1}}, \sqrt[n]{u_{2}}) : F]. In particular index(A⊗F_{ν}) = index(A).

Assume that A = A′ ⊗ (uy^{r}, x) for some unit u ∈ R, r coprime to n and A′ unramified on R. Let ν be the discrete valuation on F given by the prime ideal (x) of R. By (2.1), we have index(A ⊗ F_{ν}) = index(A′ ⊗ F_{ν}(\sqrt[n]{uy^{r}})) · [F_{ν}(\sqrt[n]{uy^{r}}) : F_{ν}]. Since y is a regular prime in R and r is coprime to n, it follows that [F (\sqrt[n]{uy^{r}}) : F] = [F_{ν}(\sqrt[n]{uy^{r}}) : F_{ν}] = n. Since F_{ν} is a complete discrete valuated field and A′ is unramified, we have index(A′ ⊗F_{ν}(\sqrt[n]{uy^{r}})) = index(A′ ⊗ κ(ν)(\sqrt[n]{\overline{uy^{r}}})) (cf. 1.1). Since κ(ν) is a complete discrete valuated field with \overline{y} as a parameter and residue field k, κ(ν)(\sqrt[n]{\overline{uy^{r}}}) is also a complete discrete valuated field with residue field k. Since A′ is unramified on R, we have index(A′⊗κ(ν)(\sqrt[n]{\overline{uy^{r}}})) = index(A′⊗k) (cf. (1.1)). Since R is complete, by (1.1), we have index(A′) = index(A′ ⊗ k) = index(A′ ⊗ κ(ν)(\sqrt[n]{\overline{uy^{r}}})) = index(A′ ⊗ F_{ν}(\sqrt[n]{uy^{r}})). Therefore, we have

index(A ⊗ F_{ν}) = index(A′ ⊗ (uy^{r}, x) ⊗ F_{ν})
= index(A′ ⊗ F_{ν}(\sqrt[n]{uy^{r}})) · [F_{ν}(\sqrt[n]{uy^{r}}) : F_{ν}]
= index(A′) · n.

Since A = A′ ⊗ (uy^{r}, x), index(A) ≤ index(A′) · n = index(A ⊗ F_{ν}). Thus index(A) = index(A ⊗ F_{ν}). □

Proposition 2.5. Let R be a 2-dimensional regular ring with field of fraction F and n an integer which is a unit in R. Assume that F contains a primitive n^{th} root of unity. Suppose that R is complete with respect to (s)- adic topology for some prime element s ∈ R with R/(s) a Dedekind domain. Let ν be the discrete valuation on F given by s. Let A be a central simple algebra over F of degree n which is unramified on R except at ν. Further assume that the residue of A at (s) is given by a unit a in R/(s). Then index(A) = index(A ⊗ F_{ν}).

Proof. Suppose that A is unramified on R. Since R is (s)-adically complete and R/(s) is a regular domain of dimension 1 with field of fractions κ(ν), index(A) = index(A ⊗_{R/(s)} κ(ν)) = index(A ⊗ F_{ν}) (cf. (1.1)).

Suppose that A is ramified on R. Then by the assumption on A, A is ramified on R only at the prime ideal (s) of R and the residue of A at (s) is given by a unit a in R/(s). Let u ∈ R with image a ∈ R/(s). Since R is (s)-adically complete and the image a of u modulo (s) is a unit, u is a unit in R. The cyclic algebra (u, s) is ramified on R only at (s) and the residue of (u, s) at (s) is (a). Since A is ramified only at the prime ideal (s) of R, and (a) is the residue of A at s, we have A = A′ ⊗ (u, s) for some central simple algebra A′ over F which is unramified on R. By (2.1), we have index(A ⊗ F_{ν}) = index(A′ ⊗ F_{ν}(\sqrt[n]{u})) · [F_{ν}(\sqrt[n]{u}) : F_{ν}]. Since R is (s)- adically complete and R/(s) is integrally closed domain with u a unit, we have [F (\sqrt[n]{u}) : F] = [F_{ν}(\sqrt[n]{u}) : F_{ν}]. Since A′ is unramified on R and R is (s)-adically complete, by (2.2), we have index(A′ ⊗ F_{ν}(\sqrt[n]{u})) = index(A′ ⊗ F (\sqrt[n]{u})). Hence we have index(A ⊗ F_{ν}) = index(A′ ⊗ F (\sqrt[n]{u})) · [F (\sqrt[n]{u}) : F]. In particular index(A ⊗ F_{ν}) = index(A). □

Theorem 2.6. Let T be a complete discrete valuation ring with fraction field K and residue field k. Let X be a regular, projective, geometrically integral curve over K and F = K(X) the function field of X. Let l be prime not equal to char(k) and A a central simple algebra over F of index n = l^{d} for some d ≥ 1. Assume that K contains a primitive n^{th} root of unity. Then index(A) = index(A ⊗ F_{ν}) for some discrete valuation ν of F.

Proof. Let A be a central simple algebra over F. We choose a regular proper model X /T of X/K such that the support of the ramification divisor A and the components of the special fibre of X /T are a union of regular curves with normal crossings. Let Y = X ×_{T} k denote the special fibre.

For each irreducible curve C in the support of ramification divisor of A, let a_{C} ∈ κ(C)^{∗} be such that the residue of A at C is (a_{C}) ∈ H^{1}(κ(C)^{∗}, ℤ/nℤ).

Let S be a finite set of closed points of the special fibre containing all singular points of Y and all those points of irreducible curves C where a_{C} is not a unit.

For each P ∈ S, let F_{P} be the field of fractions of the completion R̂_{P} of the local ring R_{P} of X at P. Let t be a uniformizing parameter for T. For each irreducible component U of Y \ S, let R_{U} be the ring of elements in F which are regular on U. It is a regular ring. Let R̂_{U} be the completion of R_{U} at (t) and F_{U} the field of fractions of R̂_{U}. As in ([CTPS], proof of 3.1), we choose S such that for every irreducible component U of Y \ S, t = u.s^{r} for some integer r ≥ 1, a unit u ∈ R_{U} and R_{U}/s = R̂_{U}/s is a Dedekind domain with field of fractions k(U). In particular, the t-adic completion R̂_{U} coincides with the s-adic completion of R_{U}.

Let U be the set of irreducible components of X \ S. By ([HHK1], 5.1), we have index(A) = lcm of index(A ⊗ F_{ζ}), ζ ∈ U ∪ S. Since index(A ⊗ F_{ζ}) is a power of the prime l for all ζ ∈ P ∪ P, we have index(A) = index(A ⊗ F_{ζ}) for some ζ ∈ U ∪ S. In particular index(A) equal to either index(A ⊗ F_{U}) for some irreducible components U of Y \ S or index(A ⊗ F_{P}) for some P ∈ S.

Suppose that index(A) = index(A ⊗ F_{P}) for some P ∈ S. The local ring R̂_{P} is a complete regular local ring of dimension 2 with maximal ideal (x, y) such that A is ramified on R̂_{P} at most at x and y. By (2.4), we have index(A) = index(A ⊗ F_{P}_{ν}) for the discrete valuation of F_{P} given by either (x) or (y). Since the restriction ν_{0} of ν to F is non-trivial, ν_{0} is a discrete valuation on F and index(A) = index(A ⊗ F_{ν_{0}}).

Suppose that index(A) = index(A ⊗ F_{U}) for some irreducible component U of X \ S. Then, by the choice of U, A is unramified on R̂_{U} except at (s) and the residue at (s) is given by a unit in R_{U}/(s). Let ν be the discrete valuation on F_{U} given by (s). Since R̂_{U} is (s)-adically complete, by (2.5), index(A) = index(A ⊗ F_{U}_{ν}). Since the restriction of ν to F is also given by the ideal (s) in R_{U}, ν is non-trivial on F. In particular F_{ν} ⊂ F_{U}_{ν} and index(A) = index(A ⊗ F_{ν}). □.

Remark 2.7. Let F and A be as in the above theorem. Then by (2.6), it follows that there exists a discrete valuation ν of F such that index(A) = index(A ⊗ F_{ν}). From the proof of (2.6) it follows that this discrete valuation comes from a codimension one point of a regular proper model of F. In particular, the residue field κ(ν) is either a finite extension of K or a function field of a curve over a finite extension of k.

3. Necessary conditions for Admissibility

In this section, we give a necessary condition for a finite group to be admissible over function fields of curves over complete discretely valued fields.

Let K be a complete discretely valued field with residue field k and π a parameter. Let D be a central division algebra over K of degree n and L a maximal subfield of D. Suppose that n is coprime to char(k). Let (E_{0}, σ_{0}) ∈ H^{1}(k, ℤ/nℤ) be the residue of D and (E, σ) ∈ H^{1}(K, ℤ/nℤ) the lift of (E_{0}, σ_{0}) (cf. §1). We fix an algebraic closure \overline{K} of K and also an algebraic closure \overline{k} of k. All the finite extensions of K and k are considered as subfields of \overline{K} and \overline{k} respectively. Let F = L ∩ E. Let L_{1} be the maximal unramified extension of F contained in L. Since E/K is unramified, L∩E = L_{1} ∩ E = F.

Lemma 3.1. Let K, D, L, L_{1}, (E_{0}, σ_{0}), (E, σ), and F be as above. Then D ⊗_{K} F ⊗_{F} (E/F, σ^{[F:K]}, π)^{op} is unramified at the discrete valuation of F.

Proof. Since the residue of D is (E_{0}, σ_{0}), D ⊗ (E, σ, π)^{op} is unramified at the discrete valuation of K. In particular, D ⊗_{K} (E, σ, π) ⊗ F is unramified at the discrete valuation of F. Since F ⊂ E, we have (E, σ, π) ⊗_{K} F = (E/F, σ^{[F:K]}, π) in Br(F). Thus (D⊗_{K} F)⊗_{F} (E/F, σ^{[F:K]}, π)^{op} is unramified at the discrete valuation of F. □

Lemma 3.2. Let K, D, L, L_{1}, (E_{0}, σ_{0}), (E, σ) and F be as above. Then [E : F] divides [L : L_{1}].

Proof. We have the following commutative diagram (cf., [S1], 10.4)

\xymatrix{_{n}Br(K) \ar[r] \ar[d]_{res} & H^{1}(k, ℤ/nℤ) \ar[d]^{e.res} \\ _{n}Br(L) \ar[r] & H^{1}(L_{0}, ℤ/nℤ)} ,

where L_{0} is the residue field of L and e the ramification index of L/K. From the above commutative diagram, the residue of D ⊗ L is the restriction of e(E_{0}, σ_{0}) to L_{0}. Since L is a maximal subfield of D, D ⊗ L is a split algebra. In particular, the residue of D⊗L is trivial. Hence the restriction of e(E_{0}, σ_{0}) to L_{0} is trivial and [E : F] divides e (cf. §1). Since [E : F] = [E_{0} : F_{0}] and e = [L : L_{1}], [E : F] divides [L : L_{1}]. □

Lemma 3.3. Let K, D, L, L_{1}, (E_{0}, σ_{0}), (E, σ) and F be as above. Then index(D ⊗_{K} E) = [L_{1}E : E].

Proof. Since (E/F, σ^{[F:K]}, π)^{op} ⊗ E is split, we have D ⊗_{K} E = D ⊗_{K} F ⊗_{F} (E/F, σ^{[F:K]}, π)^{op} ⊗_{F} E. By (3.1), D ⊗_{K} F ⊗_{F} (E/F, σ^{[F:K]}, π)^{op} is unramified at the discrete valuation of F. Hence D ⊗_{K} E is unramified at the discrete valuation of E. Since L is a maximal subfield of D, D ⊗_{K} L is split. Since L/L_{1} is totally ramified, LE/L_{1}E is totally ramified. In particular, the residue field of LE and L_{1}E are equal. Since D ⊗_{K} E is unramified and (D ⊗_{K} E) ⊗_{E} LE is plit, (D ⊗_{K} E) ⊗_{E} L_{1}E is split (cf. 1.1). In particular index(D ⊗_{K} E) ≤ [L_{1}E : E].

Since (D ⊗_{K} F) ⊗_{F} (E/F, σ^{[F:K]}, π)^{op} is unramified and π is a parameter in E, by (2.1), we have

index(D ⊗_{K} F) = index((D ⊗_{K} F) ⊗_{F} (E/F, σ^{[F:K]}, π)^{op} ⊗_{F} (E/F, σ^{[F:K]}, π)) = index((D ⊗_{K} F) ⊗_{F} (E/F, σ^{[F:K]}, π)^{op} ⊗_{F} E) · [E : F] = index(D ⊗_{K} E) · [E : F]

On the other hand, since F ⊂ L and L is a maximal subfield of D, index(D⊗_{K} F) = [L : F]. Hence index(D ⊗_{K} E) · [E : F] = [L : F] = [L : L_{1}] · [L_{1} : F]. By (3.2), we have [E : F] divides [L : L_{1}]. Hence index(D ⊗_{K} E) ≥ [L_{1} : F] ≥ [L_{1}E : E]. Therefore index(D ⊗_{K} E) = [L_{1}E : E]. □

Lemma 3.4. Let K be a complete discretely valued field with residue field k and P a p-group, p a prime. Suppose that p is coprime to char(k). If P is admissible over K, then P has a normal series P ⊇ P_{1} ⊇ P_{2} such that

(1) P/P_{1} and P_{2} are cyclic

(2) P_{1}/P_{2} is admissible over some finite extension of k.

Proof Suppose that P is admissible over K. Then there exists a Galois extension L/K and a division ring D central over K which contains L as maximal subfield such that P = G(L/K). Let L_{0} be the residue field of L. Let ∂(D)=(E_{0}, σ_{0}) ∈ H^{1}(k, ℤ/nℤ) be the residue of D and (E, σ) ∈ H^{1}(K, ℤ/nℤ) be the lift of (E_{0}, σ_{0}). Let L_{1} be the maximal unramified extension of K contained in L. Since E is unramified extension of K, we have L ∩ E = L_{1} ∩ E. Let F = L ∩ E. Since E/K is cyclic, F/K is also cyclic.

Let P_{1} be the Galois group of L/F. Since F/K is cyclic, P_{1} is a normal subgroup of P and P/P_{1} is cyclic.

Let P_{2} be the Galois group of L/L_{1}. Then P_{2} is a subgroup of P_{1}. Since L/L_{1} is a totally ramified Galois extension of degree coprime to char(k), P_{2} is cyclic ([Se], Cor.2 and Cor.4 of Ch.IV, §2). Since L_{1}/F is a Galois extension (cf. §1), P_{2} is a normal subgroup of P_{1}. The residue field F_{0} of F is the same as the intersection of L_{0} and E_{0}. We now show that P_{1}/P_{2} is admissible over E_{0}.

Let D′ be the division algebra with center E which is Brauer equivalent to D ⊗_{K} E. Since D′ ⊗_{E} L_{1}E = (D ⊗_{K} E)⊗_{E} L_{1}E and (D ⊗_{K} E)⊗_{E} L_{1}E is split (cf. proof of (3.3)), D′ ⊗_{E} L_{1}E is also split. Since, by (3.3) degree(D′) = [L_{1}E : E], L_{1}E is a maximal subfield of D′. By (3.1), D′ = D ⊗_{K} E is unramified at the discrete valuation of E. Let \overline{D′} be the image of D′ over the residue field E_{0} of E. Since E is complete, \overline{D′} is central division algebra over E_{0} and L_{0}E_{0} is a maximal subfield of \overline{D′}. Hence Gal(L_{0}E_{0}/E_{0}) is admissible over E_{0}. Since the residue field of L_{1}E is L_{0}E_{0} and L_{1}E/E is an unramified Galois extension, we have Gal(L_{0}E_{0}/E_{0}) ≃ Gal(L_{1}E/E). Since L_{1}/K and E/K are Galois and F = L_{1} ∩ E, we have Gal(L_{1}E/E) ≃ Gal(L_{1}/F). Since P_{1}/P_{2} ≃ Gal(L_{1}/F) ≃ Gal(L_{0}E_{0}/E_{0}), P_{1}/P_{2} is admissible over E_{0}. □

The above lemma immediately gives the following

Proposition 3.5. Let K be a complete discretely valued field with residue filed k and G be a finite group. Suppose that the order of G is coprime to char(k). If G is admissible over K then every Sylow subgroup P of G has a normal sereis P ⊇ P_{1} ⊇ P_{2} such that

(1) P/P_{1} and P_{2} are cyclic

(2) P_{1}/P_{2} is admissible over some finite extension of the residue field of K.

Proof. Let G be an admissible group over K. Then there is a field extension L/K and a division algebra D central over K containing L as a maximal subfield with Gal(L/K) = G. Let P be a Sylow subgroup of G. Let L^{P} be the fixed of P. Then L^{P} is a complete discretely valued field. Let D′ be the commutant of L^{P} in D. Then D′ is a central division algebra over L^{P} and G(L/L^{P}) = P is admissible over L^{P}. Since L^{P} is also a complete discrete valued field, the result follows by (3.4). □

Corollary 3.6. Let K be a complete discretely valued field with residue filed k either a local field or a global field. Let G be a finite group such that char(k) coprime to the order of G. If G is admissible over K then every Sylow subgroup P of G has a normal series P ⊇ P_{1} ⊇ P_{2} such that

(1) P/P_{1} and P_{2} are cyclic

(2) P_{1}/P_{2} is metacyclic.

Proof. Every finite extension of the residue field is either a local field or a global field. The corollary follows from (3.5) and ([Sc]). □

Theorem 3.7. Let K be a complete discretely valued field with residue field k. Let F be the function field of a curve over K. Let G be a finite group with order coprime to char(k). If G is admissible over F then every Sylow subgroup P of G has a normal series P ⊇ P_{1} ⊇ P_{2} such that

(1) P/P_{1} and P_{2} are cyclic

(2) P_{1}/P_{2} is admissible over some finite extension of the residue field at a discrete valuation of F.

Proof First we reduce to a Sylow subgroup as in (3.5). Let G be an admissible group over F. Then there is a field extension L/F and a division algebra D central over F containing L as a maximal subfield with Gal(L/F) = G. Let P be a Sylow subgroup of G. Let L^{P} be the fixed of P. Then L^{P} is a complete discretely valued field. Let D′ be the commutant of L^{P} in D. Then D′ is a central division algebra over L^{P} and G(L/L^{P}) = P is admissible over L^{P}. Since L^{P} is a finte extension of F, L^{P} is also a function field of a curve over a finite extension of K. Since the degree of D′ is a power of prime and coprime to the char(k), by (2.6), there exists a discrete valuation ν of L^{P} such that D′ ⊗_{L^{P}} L_{ν}^{P} is division. Since L ⊗_{L^{P}} L_{ν}^{P} is a maximal subfield of D′ ⊗_{L^{P}} L_{ν}^{P} and P = Gal(L ⊗_{L^{P}} L_{ν}^{P} /L_{ν}^{P}), the result follows from (3.5). □

The following is immediate from (3.7) and (3.6).

Corollary 3.8. Let K be a local adic field and F the function field of a curve over K. Let n be a natural number which is coprime to the characteristic of the residue field of K and G a finite group of order n. If G is admissible over F then every Sylow subgroup P of G has a normal series P ⊇ P_{1} ⊇ P_{2} such that

(1) P/P_{1} and P_{2} are cyclic

(2) P_{1}/P_{2} metacyclic.

The following is proved in ([HHK2], 4.5).

Corollary 3.9. Let K be a complete discretely valued field with residue field algebraically closed. Let F be the function field of a curve over K. Let G be a finite group of order n such that the characteristic of the residue field of K is coprime to n. If G is admissible over F then every Sylow subgroup P of G is metacyclic.

Proof. By (3.7), every Sylow subgroup P of G has a normal series P ⊇ P_{1} ⊇ P_{2} such that

(1) P/P_{1} and P_{2} are cyclic

(2) P_{1}/P_{2} is admissible over some finite extension of the residue field of K.

Let k be the residue field of K. By (2.7) and the proof of (3.7), the residue field at the discrete valuation given in (2) is either a finite extension K or the function field of a curve over k. Since k is algebraically closed, these residue fields have cohomological dimension at most one and there are no non-trivial division algebras over such fields. Hence P_{1} = P_{2}. □

We now give an example of a finite group which is not ℚ_{p}(t)-admissible.

Example 3.10. Let ℓ and p be two distinct primes. Let P = (ℤ/lℤ)^{5}. We claim that this group is not admissible over ℚ_{p}(t). Suppose that P is admissible over ℚ_{p}(t). Then by (3.8), there is a normal series P ⊇ P_{1} ⊇ P_{2} such that

(1)P/P_{1} and P_{2} are cyclic

(2)P_{1}/P_{2} is metacyclic

Since P_{2} and P/P_{1} are cyclic their orders will be at most l. This implies that |P_{1}/P_{2}| ≥ l^{3}. Since every element of P_{1}/P_{2} has order at most l and |P_{1}/P_{2}| ≥ l^{3}, P_{1}/P_{2} cannot be metacyclic. □

Remark 3.11. Let F be a field and p a prime. Suppose that for every finite extension E of F there is a set Ω_{E} of discrete valuations of E such that given a central division algebra D over E of degree a power of p, there exists a discrete valuation ν ∈ Ω_{E} such that D ⊗ E_{ν} is division. Let G be a finite group. The proof of (3.7) gives us the following: If G is admissible over F, then every p-Sylow subgroup of G has filtration as in (3.7).

Remark 3.12. Let G be a finite group satisfying the conditions of (3.5). Then every homomorphic image of G also satisfy the same conditions. However there is an example of a group G with a homomorphic image H such that G is admissible over the complet discrete valuation field ℚ((t)) but H is not admissible ([FS]). Hence the conditions given in (3.5) for a group to be admissible are not sufficient.

4. A class of Admissible groups over ℚ_{p}(t)

Let K be a discretely valued field with residue field k. Let F be the function field of a curve over K. Let n be an integer which is coprime to the characteristic of k. Suppose that K contains a primitive n^{th} root of unity. Then in ([HHK2], 4.4) it is proved that every finite group of order n with every Sylow subgroup product of at most two cyclic groups is admissible over F. They used the patching techniques to prove this result. In this section we prove a similar result for groups with every Sylow subgroup is a product of at most 4 cyclic groups, with an additional assumption on the residue field k. We begin with the following Lemma 4.1. Let R be a regular local ring of dimension two with residue field k and field of fraction F. Let n_{1} and n_{2} be natural numbers which are coprime to the char(k). Assume that F contains a primitive (n_{1}n_{2})^{th} root of unity and there is an element in k^{∗}/k^{∗n_{2}} of order n_{2}. Then there is a central division algebra D over F of degree n_{1}n_{2}.

Proof. Let m be the maximal ideal of R. Since R is a regular local ring of dimension two, we have m = (t, s). By the assumption on k, there is an element λ_{0} ∈ k^{∗} such that its order in k^{∗}/k^{∗n_{2}} is n_{2}. Let λ ∈ R which maps to λ_{0}. Let a ∈ R be a unit with a^{n_{1}} ≠ 1. Let ξ_{1} be a primitive n_{1}^{th} root of unity and ξ_{2} a primitive n_{2}^{th} root of unity.

Let

D_{1} = (\frac{s}{s − t}, \frac{s − t^{2}}{s − a^{n_{1}}t^{2}} )_{n_{1}}

and

D_{2} = (\frac{s}{s − t^{2}} , \frac{s − λt^{2}}{s − t^{2}} )_{n_{2}}.

Let D = D_{1} ⊗_{F} D_{2}. Then the degree of D is n_{1}n_{2}. We now show that D is a division algebra.

Let S = R[x]/(s − t^{2}x). Then the field of fractions of S is isomorphic to F. We have

D_{1} = (\frac{tx}{tx − 1}, \frac{x − 1}{x − a^{n_{1}}} )_{n_{1}}

and

D_{2} = (\frac{x}{x − 1}, \frac{x − λ}{x − 1})_{n_{2}}.

The ideal (t) of S is a prime ideal and gives a discrete valuation ν on F. Let F_{ν} be the completion of F at ν. To show that D_{1} ⊗_{F} D_{2} is a division algebra, it is enough to show that D_{1} ⊗_{F} D_{2} ⊗_{F} F_{ν} is a division algebra. Since tx/tx − 1 is a parameter at ν, by (2.1), we have

index(D_{1}⊗_{F} D_{2}⊗_{F} F_{ν}) = index(D_{2}⊗_{F} F_{ν}( \sqrt[n_{1}]{\frac{x − 1}{x − a^{n_{1}}}} ))·[F_{ν}( \sqrt[n_{1}]{\frac{x − 1}{x − a^{n_{1}}}} ) : F_{ν}].

Since \frac{x−1}{x−a^{n_{1}}} is a unit at ν and the residue field κ(ν) at ν is k(x), by the

assumption on a, we have [F_{ν}( \sqrt[n_{1}]{\frac{x−1}{x−a^{n_{1}}}} ) : F_{ν}] = n_{1}. Since D_{2} is unramified

at ν, the index of D_{2} ⊗_{F} F_{ν}( \sqrt[n_{1}]{\frac{x−1}{x−a^{n_{1}}}} ) is equal to the index of its image

(\frac{x}{x−1}, \frac{x−λ}{x−1})_{n_{2}} over the residue field k(x)( \sqrt[n_{1}]{\frac{x−1}{x−a^{n_{1}}}} ). Let θ be the discrete

valuation on k(x) given by (x). Let υ be the extension of θ to k(x)( \sqrt[n_{1}]{\frac{x−1}{x−a^{n_{1}}}} ).

Then the residue field of υ is k. The residue of (\frac{x}{x−1}, \frac{x−λ}{x−1})_{n_{2}} at υ is the class of λ_{0}. Since the order of λ_{0} in k^{∗}/k^{∗n_{2}} is n_{2}, the index of D_{2} is n_{2}. Hence the index of D_{1} ⊗_{F} D_{2} ⊗ F_{ν} is n_{1}n_{2}. Since the degree of D_{1} ⊗_{F} D_{2} is n_{1}n_{2}, D_{1} ⊗_{F} D_{2} is a division algebra. □

Theorem 4.2. Let K be a discretely valued field with residue field k and F the function field of a curve over K. Let n be an integer which is coprime to the characteristic of k. Suppose that K contains a primitive n^{th} root of unity. Assume that for every finite extension L of k, there is an element in L^{∗}/L^{∗n} of order n. If G is a finite group of order n with every Sylow subgroup is a quotient of ℤ^{4}, then G is admissible over F.

Proof. Let R be the ring of integers in K. Let X be a regular proper two dimensional scheme over R with function field F and the reduced special fibre is a union of regular curves with normal crossings. Let p_{1}, ⋯, p_{r} be the prime factors of n. Let Q_{1}, ⋯, Q_{r} be regular closed points on the special fibre of X. Let R_{Q_{i}} be the regular local ring at Q_{i}, R̂_{Q_{i}} be the completion of R_{Q_{i}} at the maximal ideal and F_{Q_{i}} the field of fractions of R̂_{Q_{i}}. Let t_{i} ∈ R_{Q_{i}} be a prime defining the irreducible component of the special fibre of X containing Q_{i}. Let P_{i} be a p_{i}-Sylow subgroup of G. By ([HHK2], 4.2), it is enough to show that there exists a central division algebra D_{i} over F_{Q_{i}} and maximal subfield L_{i} of D_{i} with Gal(L_{i}/F_{Q_{i}}) ≃ P_{i} and L_{i} ⊗ F̂_{Q_{i}} a split algebra.

For a given i, 1 ≤ i ≤ r, let Q = Q_{i}, t = t_{i} and P = P_{i}. Since the residue field κ(Q) of R̂_{Q} is a finite extension of k, by the assumption on k, there is an element in κ(Q)^{∗}/κ(Q)^{∗n} of order m for any m dividing n. Since P is a quotient of ℤ^{4}, P ≃ C_{n_{1}} × C_{n_{2}} × C_{n_{3}} × C_{n_{4}}. Since R̂_{Q} is regular local ring of dimension 2 and t is a regular prime, we have m_{Q} = (t, s). Let ξ_{1} be a primitive n_{1}n_{2}^{th} root of unity and ξ_{2} a primitive n_{3}n_{4}^{th} root of unity.

Let

D_{1} = (\frac{s}{s − t}, \frac{s − t^{2}}{s − a^{n_{1}n_{1}}t^{2}} )_{n_{1}n_{2}}

and

D_{2} = (\frac{s}{s − t^{2}} , \frac{s − λt^{2}}{s − t^{2}} )_{n_{3}n_{4}}.

for suitable a and λ as in (4.1). Let D = D_{1} ⊗_{F} D_{2}. Then, by (4.1), D is a division algebra over F_{Q}.

In particular D_{1} and D_{2} are division algebras over F_{Q}. The cyclic algebra D_{1} is generated by x_{1} and y_{1} with relations

x_{1}^{n_{1}n_{2}} = \frac{s}{s − t}, y_{1}^{n_{1}n_{2}} = \frac{s − t^{2}}{s − a^{n_{1}n_{2}}t^{2}} and x_{1}y_{1} = ξ_{1}y_{1}x_{1}.

Similarly D_{2} is generated by x_{2} and y_{2} with relations

x_{2}^{n_{3}n_{4}} = \frac{s}{s − t^{2}} , y_{2}^{n_{3}n_{4}} = \frac{s − λt^{2}}{s − t^{2}} and x_{2}y_{2} = ξ_{2}y_{2}x_{2}.

Let L_{1} be the subalgebra of D_{1} generated by x_{1}^{n_{1}} and y_{1}^{n_{2}}. Let L_{2} be the subalgebra of D_{2} generated by x_{2}^{n_{3}} and y_{2}^{n_{4}}. Then L = L_{1} ⊗ L_{2} is a maximal subfield of D_{1} ⊗ D_{2}, Gal(L/F) = C_{n_{1}} × C_{n_{2}} × C_{n_{3}} × C_{n_{4}} and L ⊗ F̂ is a split algebra (cf., ([HHK2]) proof of 4.4, ). □

Corollary 4.3. Let K be a local field and k its residue field. Let F be the function field of a curve over K. Let n be an integer which is coprime to the characteristic of k. Suppose that K contains a primitive n^{th} root of unity. If G is a group of order n with every Sylow subgroup a quotient of ℤ^{4}, then G is admissible over F.

Proof. Since k is a finite field, for any finite extension L of k and for any natural number n coprime to the characteristic of L, we have L^{∗}/L^{∗n} is cyclic group of order n. Hence k satisfies the condition of (4.2). □.

Let K be a complete discretely valued field with residue field k. Let G be a finite abelian group of order n which is a product of at most four cyclic groups. Suppose that n is coprime to char(k) and the order of k^{∗}/k^{∗n} is at least n. If K contains a primitive n^{th} root of unity, then by (4.3), G is admissible over K(t). We now prove that a similar result without the completeness assumption on K.

We begin with the following

Lemma 4.4. Let R be a discrete valuation ring and π ∈ R a parameter. Let K be the field of fractions of R and k the residue filed of R. Let F = K(t) be the rational function field in one variable over K. Let n be a natural number which is coprime to char(k). Assume that K contains a primitive n^{th} root of unity and there exists a λ_{0} ∈ k^{∗} such that [k(\sqrt[n]{λ_{0}}) : k] = n. Then (t, π − λt)_{n} ⊗_{F} (t + 1, π)_{m} is division over F for any λ ∈ R which maps to λ_{0}.

Proof Since π is a parameter in R, the localisation R[t]_{(π)} of R[t] at the prime ideal (π) is a discrete valuation ring. Let ν be the discrete valuation on F given the discrete valuation ring R[t]_{(π)} and F_{ν} be the completion of F at ν. Since the residue field at ν is k(t), we have [F_{ν}( \sqrt[m]{t + 1}) : F_{ν}] = m. Let w be the extension of ν to F_{ν}( \sqrt[m]{t + 1}). To show that (t, π − λt)_{n} ⊗_{F} (t + 1, π)_{m} is division, it is enough to show that (t, π − λt)_{n} ⊗_{F} (t + 1, π)_{m} ⊗_{F} F_{ν} is division. Since F_{ν} is complete and [F_{ν}( \sqrt[m]{t + 1}) : F_{ν}] = m, by (2.1), it is enough to show that (t, π − λt)_{n} ⊗ F_{ν}( \sqrt[m]{t + 1}) is division. Since t and π − λt are units at ν, the algebra (t, π −λt)_{n} ⊗F_{ν}( \sqrt[m]{t + 1}) is unramified at w. Thus it is enough to show that its image (t, −λ_{0}t)_{n} = (t, λ_{0})_{n} is division over the residue filed k(t)( \sqrt[m]{t + 1}). Let γ be the discrete valuation on k(t) given by t and γ̃ be the extension of γ to k(t)( \sqrt[m]{t + 1}). Since t + 1 is an m^{th} power in the completion of k(t) at γ, the completion of k(t)( \sqrt[m]{t + 1}) at γ̃ is k((t)). It is enough show that (t, λ_{0})_{n} is division over k((t)). Since λ_{0} ∈ k^{∗} is an element of order n, (t, λ_{0})_{n} is division over k((t)). □

Theorem 4.5. Let K be a field with a discrete valuation (not necessarily complete) and k its residue field. Let G = ℤ/l_{1}ℤ × ℤ/l_{2}ℤ × ℤ/l_{3}ℤ × ℤ/l_{4}ℤ. Suppose that the n = l_{1}l_{2}l_{3}l_{4} is coprime to char(k), K contains a primitive n^{th} root of unity and there is a λ_{0} ∈ k such that [k( \sqrt[l_{1}l_{2}]{λ_{0}} : k] = l_{1}l_{2}. Then G is admissible over K(t).

Proof. Let R be the ring of integers in K and λ ∈ R mapping to λ_{0} in k. Let π ∈ R be a parameter. Let D_{1} = (t, π − λt)_{l_{1}l_{2}} and D_{2} = (t + 1, π)_{l_{3}l_{4}}. Then, by (4.4), D_{1} ⊗ D_{2} is a division algebra over K(t). Let L_{1} = K(t)(y, z) where y^{l_{1}} = t, z^{l_{2}} = π−λt. Then L_{1} is a maximal subfield of D_{1}. Let L_{2} = K(t)(r, s) where y^{l_{3}} = t + 1, z^{l_{4}} = π. Then L_{2} is a maximal subfield of D_{2}. Therefore L_{1} ⊗_{K(t)} L_{2} is a maximal subfield of D_{1} ⊗ D_{2}. Since Gal(L_{1} ⊗_{K(t)} L_{2}/K(t)) = G, G is admissible over K(t). □

Corollary 4.6. Let p be a prim and l_{1}, l_{2}, l_{3}, l_{4} be natural numbers which are coprime to p. Let G = ℤ/l_{1}ℤ × ℤ/l_{2}ℤ × ℤ/l_{3}ℤ × ℤ/l_{4}ℤ. Let n = l_{1}l_{2}l_{3}l_{4} and ζ a primitive n^{th} root of unity. Then G is admisible over ℚ(ζ)(t).

Remark 4.7. For p, l_{1}, l_{2}, l_{3}, l_{4}, G as in (4.6), G is admissible over ℚ_{p}(t) by (4.3). However (4.5) gives an explicite constriction of a division algebra over ℚ_{p}(t) and a maximal subfield with Galois group G.

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Department of Mathematics and Statistics University of Hyderabad Gahcibowli Hyderabad - 500046 India

and

Department of Mathematics and Computer Science Emory University Atlanta, GA 30322 U.S.A.