Division Algebras and Maximal Orders for Given Invariants

Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000

Division algebras and maximal orders for given invariants

Gebhard B¨ockle, Dami´anGvirtz

Abstract Brauer classes of a global field can be represented by cyclic algebras. Effective constructions of such algebras and a maximal order therein are given for Fq(t), excluding cases of wild ramification. As part of the construction, we also obtain a new description of subfields of cyclotomic function fields.

1. Introduction

Let F be a global function ﬁeld and S a ﬁnite set of places of F . For each v ∈ S let sv = nv/rv be a reduced fraction in Q such that for their classes in Q/Z we have X sv ≡ 0 mod Z. (1.1) v∈S

We extend (sv)v to a sequence of invariants ranging over all places of F , by setting sv = 0/1 whenever v does not lie in S. Denote by Br(K) the Brauer group of a ﬁeld K. For any place v of F , denote by Fv the completion of F at v. Then as a consequence of Hasse’s Main Theorem on the Theory of Algebras one has a short exact sequence

(α )7→P inv [A]7→[A⊗F Fv ]v ` v v v 1 / Br(F ) / v Br(Fv) / Q/Z / 0, (1.2) for suitably defined local isomorphisms invv : Br(Fv) → Q/Z. Hence given any sequence (sv)v as in the previous paragraph, there exists a unique class in Br(F ) with this set of invariants, i.e., up to isomorphism, a unique division algebra D with this set of invariants. Moreover the theorem of Grunwald-Wang implies that D can be written in the form of a cyclic algebra. For the rest of the paper, let F be the field Fq(t) where Fq denotes the finite field of q elements and q is a power of a prime p. The aim of this paper is to effectively construct a cyclic algebra, starting only with its invariants (with the restriction that s∞ = 0), and once this has been achieved, to find a maximal Fq[t]-order in it. We content ourselves with finding one and not all of the (by the Jordan-Zassenhaus theorem) up to conjugation finitely many maximal orders. Our approach will work whenever q is prime to the denominators of any invariant. Besides, we also provide a Kummer theory way of calculating a subfield of a cyclotomic function field, which to our knowledge is new and computationally performs better than more naive approaches. Our method can be adapted to F = Q, as we have checked, and with some refinements probably also to all global function fields. We focus on F = Fq(t) for two reasons: (i) Our interest in having access to explicit models of maximal Fq[t]-orders Λ for a fixed set of invariants stems from planned experimental investigations of certain function field automorphic forms for F = Fq(t), namely harmonic cochains on Bruhat-Tits buildings that are equivariant for a suitable action of Λ. (ii) We wish to focus on the division algebra aspect and not aspects related to Dedekind domains for general global function fields.

2000 Mathematics Subject Classiﬁcation 00000. Page 2 of 16 GEBHARD BOCKLE,¨ DAMIAN´ GVIRTZ

We thank Nils Schwinning for formulating the basis of this paper in his Master’s thesis. There he implemented an algorithm for computing a cyclic algebra from given invariants in the number ﬁeld case that was suggested by G. B., as was the analysis of the eﬀectiveness of the algorithm. G. B. was supported by the DFG within the SPP1489 and FG1920. Magma code for all constructions is available from https://github.com/dgvirtz/divalg.

2. Central simple algebras This section reviews well-known basic results on central simple algebras. Basic references are [6, 10].

2.1. Cyclic algebras Let K be a ﬁeld and L/K be a Galois extension with cyclic Galois group of order d with a chosen generator σ. Pick an element a ∈ K∗.

Definition 1. The (non-commutative) cyclic algebra attached to L/K, σ and a is (L/K, σ, a) := L[τ, τ −1]/(τ d − a)L[τ, τ −1]. where multiplication in the Laurent polynomial ring L[τ, τ −1] = L Lτ n is deﬁned by n∈Z n m n n+m ατ · βτ = ασ (β)τ for α, β ∈ L, n, m ∈ Z.

It is an L-vector space with basis {τ i | i = 0, . . . , d − 1}. It is also a K-algebra, by identifying K with the central subﬁeld Kτ 0. From [10, Thms. 30.3 and 29.6] we infer that (L/K, σ, a) j is a central simple algebra over K, and that {αiτ | i, j = 0, . . . , d − 1} is a K-basis for every K-basis α0, . . . , αd−1 of L. We recall some basic properties of cyclic algebras for K and L ﬁxed.

Theorem 2 [10, Thm. 30.4]. The following hold: (i)( L/K, σ, a) =∼ (L/K, σs, as) for any s ∈ Z with gcd(s, d) = 1. ∼ ∗ (ii)( L/K, σ, a) = Mn(K) if and only if a ∈ NormL/K (L ).

The following result is useful, for instance, when passing from K to its completion.

Theorem 3 [10, Thm. 30.8]. Let K0 ⊃ K be any ﬁeld extension; denote by L0 a splitting ﬁeld over L of the minimal polynomial over K of any primitive element for L/K, so that 0 0 n H := Gal(L /K ) is naturally a subgroup of Gal(L/K). Let k = min{n ∈ N>0 | σ ∈ H}. Then 0 0 0 k K ⊗K (L/K, σ, a) ∼ (L /K , σ , a).

In particular, (L/K, σ, a) ∈ Br(L/K), i.e., (L/K, σ, a) ⊗K L is trivial in Br(L).

2.2. The Hasse invariant of a local algebra In this subsection, K denotes a complete discretely valued field with respect to a valuation v = vK , with ring of integers O and finite residue field k. By π we denote a uniformizer. For the definition of the Hasse invariant, consult [10, Thms. 14.3, 14.5 and footnote p. 148]. ALGORITHMS FOR DIVISION ALGEBRAS Page 3 of 16

Theorem 4 [10, Thm. 31.5]. Let W and W 0 be unramiﬁed extensions of K of degrees d and d0. Denote by σ, σ0 the respective Frobenius automorphisms, and let s, s0 be in Z such that s/d = s0/d0. Then 0 (W/K, σ, πs) ∼ (W 0/K, σ0, πs ).

The following result is basic to determine the invariant of a cyclic algebra over a local ﬁeld.

Theorem 5. Let W/K be an unramified extension of degree d. Denote by σ ∈ Gal(W/K) the Frobenius automorphism, i.e. the unique automorphism that on residue fields is given by x 7→ x#k. Let a be in K∗ and let D be the division algebra equivalent to (W/K, σ, a) in Br(K). Then the Hasse invariant† satisfies v (a) inv D = K . K d

00 Proof. Lacking a reference, we give a proof of this simple fact: Define d := gcd(vK (a), d) and d0 := d/d00, and denote by W 0 the unique subfield of W with degree d0 over K, and by σ0 ∈ Gal(W 0/K) the Frobenius automorphism on W 0. Then by Theorem 4, The- 00 ∗ 0 0 vK (a)/d orem 2 and NW/K (W ) ⊂ OK , we have (W/K, σ, a) ∼ (W /K, σ , π ), and D := 00 0 0 vK (a)/d 0 (W /K, σ , π ) is a division algebra over K of degree d . Because vK (a)/d = 00 0 r (vK (a)/d )/d , we shall assume from now on that a = π for some r ∈ Z prime to d, so that in particular (W/K, σ, a) is a division algebra. 0 0 r0 n Choose integers r , n such that rr + nd = 1, and define πD := τ π . Observe that σ(π) = π 0 d since π ∈ K, so that τ and π commute. From the choices of r , n we thus see that πD = π. Let now q := #k and denote by ω be a primitive qd-th root of unity in W . Then 0 −1 r0 n −n −r0 r0 qr πDωπD = τ π ωπ τ = σ (ω) = ω .

0 r vK (a) Since rr ≡ 1 (mod d), we compute invK D = d = d .

2.3. Discriminants Suppose that K is a global ﬁeld that is the quotient ﬁeld of a Dedekind domain A, and that −1 d −1 d a lies A. Then OL[τ, τ ]/(τ − a) is an A-order of L[τ, τ ]/(τ − a). The main purpose of this subsection is the computation of the discriminant of this order.

Lemma 6. Let D be the cyclic algebra (L/K, σ, a). Denote by TrD/K the reduced trace of Pd−1 i D over K and by TrL/K the trace of L over K. Then for x = i=0 αiτ ∈ D one has

TrD/K (x) = TrL/K (α0).

Proof. The reduced trace of x is the trace of left multiplication by x on D, considered as a ‡ i right L-vector space. A basis of this vector space is given by {τ }i=0,...,d−1. On this i i −i j xτ = τ σ (α0) + terms in τ , j 6= i. P −i Hence TrD/K (x) = i=0,...,d−1 σ (α0) = TrL/K (α0).

0 † −1 qr 0 Recall that invK D is deﬁned in [10, footnote p. 148] as r/d provided that πDωπD = ω for some r with 0 d d rr ≡ 1 (mod d), where ω a primitive q − 1-th root of unity in D and πD ∈ D with πD a uniformizer of K ⊂ D ‡ This can be seen via the explicit isomorphism D ⊗K L → EndL(D), a ⊗ λ 7→ (z 7→ azλ). Page 4 of 16 GEBHARD BOCKLE,¨ DAMIAN´ GVIRTZ

Corollary 7. Let OL denote the maximal order of L over A. Denote by Λ the A-order P i OLτ of D = (L/K, σ, a). Then the discriminant of Λ (which depends on the choice of σ) is given by disc(Λ) = ad(d−1) disc(L/K)d.

Proof. Since the formation of the discriminant commutes with localization, it suffices to prove the asserted formula after localizing A at sufficiently many b ∈ A \{0} such that Spec A = ∪b Spec A[1/b]. Since A arises from a global field, one can find b such that the rings A[1/b] are principal ideal domains. I.e., it suffices to prove the corollary in the case, where OL possesses j an A-basis b1, . . . , bd. Then an A-basis of Λ is given by biτ , where i, j range over 0, . . . , d − 1 – note that we regard all indices modulo d. Now we compute  0, if j + j0 6= 0, d j j0 j j+j0  j 0 TrD/K biτ bi0 τ = TrD/K biσ (bi0 )τ = TrL/K (biσ (bi0 )), if j + j = 0 j 0  a TrL/K (biσ (bi0 )), if j + j = d. Thus the discriminant of Λ is given by

d−1 0 Y j d(d−1) d det TrL/K (biσ (bi0 )) det a TrL/K (biσ (bi0 )) = a disc(L/K) . i,i0 i,i0 j=1

3. Cyclic extensions of the ﬁeld F

In this section we recall parts of the theory of abelian extensions for the ﬁeld F = Fq(t). References are [4, Ch. 3] and [11, Ch. 12]. By ∞ we denote the place of vanishing of 1/t.

3.1. Tamely ramiﬁed abelian extensions of Fq(t)

Denote by A the ring Fq[t]. The Carlitz-module is the unique ring homomorphism

φ: A → A[τ]: a 7→ φa, q characterized by φα = α, for α ∈ Fq, and t 7→ φt = t + τ, and where τa = a τ in A[τ] for a ∈ i qi A. Substituting τ by x for i ≥ 0 provides one with a polynomial φa(x) ∈ A[x]. For any a ∈ A \ Fq we denote by λa ∈ F a primitive a-torsion point of φ, i.e., φa(λa) = 0 and φa is completely split over F (λa). If v(a) = 0, we also write λa for its image in Fv. ∗ The field F (λa) is an abelian extension of F with Galois group isomorphic to (A/a) , and an isomorphism is given by ∗ ¯ (A/a) −→ Gal(F (λa)/F ), b 7−→ σb : λa 7→ φb(λa) . (3.1) It is ramified exactly at those finite places of F defined by the finite prime divisors of a.A function field analog of the Kronecker-Weber theorem states that any finite abelian extension of F which is totally split above ∞, except for an initial tamely ramified extension of degree dividing q − 1, is contained in F (λa) for some a ∈ A \ Fq. It will be useful to gather information about the fields Fv(λa). Let v be a finite place of F . Denote by hv ∈ Fq[t] the monic irreducible polynomial with v(hv) = 1.

Theorem 8. The extension Fv(λa)/Fv is unramiﬁed if and only if hv - a. In this situation, [Fv(λa): Fv] = ord(A/a)∗ (hv). ALGORITHMS FOR DIVISION ALGEBRAS Page 5 of 16

We can also describe the ramiﬁcation above ∞ of the ﬁelds F (λa):

Theorem 9 [11, Thm. 12.14], [3, Lemma 1.4]. For any monic a ∈ A \ Fq we have: deg a (i) The polynomial φa(x)/x splits over F∞ into (q − 1)/(q − 1) irreducible factors of degree q − 1. (ii) For any root λa of φa(x)/x in a ﬁnite extension of F∞, the extension F∞(λa)/F∞ is totally tamely ramiﬁed of degree q − 1. (iii) Suppose a is irreducible. Then for any extension of F∞ as in (b) one has

∗ Z 1 1 NormF∞(λa)/F∞ (F∞(λa) ) = t (1 + t Fq[[ t ]])

3.2. Subfields of a cyclotomic extension The last theme, we wish to cover in this subsection is that of primitive elements of the rings of integers of subfields of cyclotomic field, a problem that is computationally considerably harder than the number field analogues by Leopoldt and Lettl, e.g. [8]. Let k = Fq and E = F (λa) be the cyclotomic extension for an irreducible polynomial a ∈ k[t] of degree d. We follow the Kummer-theoretical approach of [2] to find a primitive element of the subfield L of E of degree r over F for any divisor r of qd − 1: 0 0 Qd Let k = Fqd . The polynomial a splits completely in k [t], and we write a = i=1 ai as a product of distinct linear factors where we regard the indices as elements in Z/(d). We order 0 0 0 0 the ai so that under the natural action of Gal(k /k) = Gal(k (t)/k(t)) on k (t) = k F , the q Frobenius automorphism F : x 7→ x maps each ai to ai+1. Let u1, . . . , ud be r-th roots of 0 0 0 −a1,..., −ad and set L = k F [u1, . . . , ud]. Because L/F is cyclic of degree r and k contains all r-th roots of unity, by Kummer theory the field k0L is contained in L0. We display the relevant fields in the following diagram:

L0 k0L E k0F L F = k(t)

0 Writing Ur for the r-th roots of unity in k , we have an isomorphism

d ' 0 0 (Ur) −→ Gal(L /k F ) from Kummer theory, deﬁned by havingα ¯ = (αi)1=1,...,r act on u = (ui)1=1,...,r via

α¯(u) = (αiui)1=1,...,r.

0 Furthermore, the Frobenius automorphism F can be extended to Φ ∈ Gal(L /F ) via ui 7→ ui+1, to yield an automorphism of order d. For j ∈ Z/(d) one computes j α¯(Φ (ui)) =α ¯(ui+j) = αi+jui+j

0 Thusα ¯Φj =α ¯0Φj only if j = j0, α¯ =α ¯0. It follows by a size argument that the set of d · rd j 0 0 d automorphismsα ¯Φ forms the Galois group Gal(L /F ). More precisely, Gal(L /F ) = (Ur) o Z/(d), where the generator Φ of the right factor sendsα ¯ in the left factor to (Fα1,..., Fαd). Qd−1 qd−1−j Deﬁne δi = j=0 ui+j for i ∈ Z/(d). Then

d−1 d−1 Y qd−1−j Y qd−1−j Φ(δi) = ui+j+1 = δi+1, α¯(δi) = (αi+jui+j) =α ˆiδi j=0 j=0

Qd−1 qd−1−j whereα ˆi = j=0 αi+j . Page 6 of 16 GEBHARD BOCKLE,¨ DAMIAN´ GVIRTZ

Pd−1 qi 0∗ 0 Setting vβ = i=0 β δi for β ∈ k we want to compute Gal(L /F (vβ)). It follows from q αî =α î+1 that d−1 d−1 d−1 d−1 X qi X qi+1 X qi X qi qi Φ(vβ) = Φ(β δi) = β δi+1 = vβ, α¯(vβ) = αîβ δi = αˆ0 β δi = vαˆ0β i=0 i=0 i=0 i=0

An automorphismα ¯Φ leaves F (vβ) ﬁxed if and only ifα ˆ0 = 1. Hence for such anα ¯Φ, the 0 d−1 ﬁrst d − 1 components ofα ¯ determine the last. This means that # Gal(L /F (vβ)) = d · r and

# Gal(F (vβ)/F ) = r

Lemma 10. One has F (vβ) = L.

0 q 0 0 d Proof. We write ai = (t − βi) for suitable βi ∈ k with βi = βi+1. Define u1, . . . , ud as q − 0 0 0 0 0 0 0 1-st roots of −a1,..., −ad. With E = k F [u1, . . . , ud] we can analogously define Φ , δi and vβ and get Gal(E0/F ) = (k0∗)d o Z/(d). The restriction Gal(E0/F ) → Gal(L0/F ) is given by 0j r0 r0 j (α0, . . . , αd−1)Φ 7−→ (α0 , . . . , αd−1)Φ , (3.2) 0 d 0 r0 r0 where r = (q − 1)/r. We defineα ¯ = (α0 , . . . , αd−1). A direct computation found in [2] reveals that φ (v0 ) = v0 q + T v0 = v0 . Inductively this t β β β β−1β gives φ (v0 ) = v0 . Therefore φ (v0 ) = 0 and because the v0 are distinct, they have to be f β f(β−1)β h β β 0 exactly the h-torsion points of the Carlitz module. One concludes F (vβ) = E. As in the preceding construction we know that 0 j 0 Gal(E /E) = {α¯Φ ∈ Gal(E /F ) | αˆ0 = 1} Using the above restriction formula (3.2), we deduce from the previous paragraph 0 j 0 0 Gal(E /F (vβ)) = {α¯Φ ∈ Gal(E /F )) | αˆ0 = 1} 0 0 0 Obviouslyα ˆ0 = 1 impliesα ˆ0 = 1, so Gal(E /E) ⊂ Gal(E /F (vβ)) or equivalently E ⊃ F (vβ). From [L : F ] = [F (vβ): F ] = r we find L = F (vβ).

Lemma 11. Keeping the above notation, the local Frobenius at an unramiﬁed prime g ∈ k[t] r0qd−e of degree e, with g monic, is given by vβ 7→ vxβ where x = g(βe−1) .

q Proof. Split g into linear factors g = (t − b0) ... (t − be−1) with bi = bi+1. The Frobenius is given by taking the qe-th power modulo (g). One computes:

q r0 qe q qe−1 r0 δi = ai δi+1, δi = (ai+e−1ai+e−2 . . . ai ) δi+e

Reducing modulo g by substituting t 7→ b0 the base term can be simpliﬁed: q qe−1 q qe−1 (ai+e−1ai+e−2 . . . ai ) = (βi+e−1 − t)(βi+e−1 − t ) ... (βi+e−1 − t ) ≡ (βi+e−1 − b0)(βi+e−1 − b1) ... (βi+e−1 − be−1) (mod g) qi = g(βi+e−1) = g(βe−1) Finally we can compute: d−1 d−1 qe X qi+e r0qi X r0qd−e qi+e vβ ≡ β (g(βe−1)) δi+e (mod g) = (β(g(βe−1)) ) δi+e = vxβ i=0 i=0 ALGORITHMS FOR DIVISION ALGEBRAS Page 7 of 16

Remark 12. Of course, there are other approaches to get a primitive element of L: In [1, Lemma 11] a normal integral basis of OE is given in terms of universal Gauss-Thakur sums. It can be combined with [2, Lemma 4], which states that taking the trace over L of a normal integral basis yields an integral basis of OL. Alternatively the norm of a primitive h-torsion element can be taken ([5, Theorem 4.3]). These two alternative approaches need at least O(qd) multiplications in the big cyclotomic ﬁeld E as opposed to O(r) in the Kummer extension L0 as described above. Implementations of all three constructions conﬁrm the better performance of the approach introduced here.

4. The basic algorithm

Let S be a finite set of finite places of F , and for each v ∈ S, let sv = nv/rv be a reduced fraction, and define r = lcm(rv : v ∈ S). Then the theorem of Grunwald-Wang guarantees the existence of a cyclic field L/F of degree r and of a cyclic algebra of the form D = (L/F, σ,¯ a) whose local invariant satisfy invv(D) = sv. For the simple algorithm below, we shall make the following further restriction:

Assumption 13. None of the rv is divisible by p, and ∞ ∈/ S.

More concretely, the simple algorithm wishes to ﬁnd a subextension L of E = F (λw) for a place w of F , i.e., L/F is of prime conductor, such that (1) r = [L : F ], and (2) for all v ∈ S we have rv|[Lv : Fv] =: dv. We ﬁx a generator σ of G := Gal(E/F ) and denote by H ≤ G the subgroup corresponding to L. We display the situation in the following diagram

E Ev

r dv H=hσ i Hv =hσv i ∗ G=(A/hw) ∼ ∼ L Gv = /(fv ) L =Z/(qw−1) Z v G=hσ¯i Gv =hσ¯v i ∼ ∼ =Z/(r) =Z/(dv ) FFv, where σv denotes the Frobenius endomorphism at v. A bar on top of an endomorphism is used to denote the restriction of the endomorphism to the intermediate ﬁeld L or Lv.

4.1. Global conditions deg(w) deg(w) We deﬁne qw = q . Then the global condition is r|qw − 1, i.e., that q ≡ 1 (mod r). Recall that we assume that p does not divide r here, so that q is a unit in Z/(r).

4.2. Local conditions

For the local degree of Ev/Fv (it depends on v and w, so we add both parameters into our notation) we have

fv(w) = ord(A/w)∗ (hv). Hence the local decomposition group is

(qw−1)/fv (w) Gv = hσ i = hσvi.

The decomposition group of Lv/Fv is now given by HGv/H. Since everything takes place inside the cyclic group G the subgroup HGv is cyclic, and for the order dv of the quotient HGv/H Page 8 of 16 GEBHARD BOCKLE,¨ DAMIAN´ GVIRTZ we deduce q − 1 q − 1 q − 1 d = lcm w , f (w) / w = f (w)/gcd w , f (w) . v r v r v r v

The condition we require is rv|dv.

4.3. Statement of the algorithm After a number of simple manipulations we obtain the following algorithm:

Algorithm 14. Input: an integer m ≥ 2, pairwise distinct irreducible elements h1, . . . , hm forming the set S, reduced fractions sv = nv/rv for v ∈ S. Output: a monic irreducible polynomial hw. P (i) Check whether v sv ≡ 0 mod Z. If not, stop procedure. (ii) Check whether the hv are irreducible and pairwise distinct. If not, stop procedure. (iii) Divide each hv by its leading term, so that it is monic. (iv) Check whether the sv are reduced fractions – if not, reduce them. (v) Compute r = lcm(rv : v ∈ S). If p|r, then stop. ∗ (vi) Compute d0 = ord(Z/(r)) (q). (vii) Start for loop over j = 1, 2, 3,.... (viii) Start for loop over all monic irreducible polynomials hw of degree d0j. (ix) If h ∈ S, then go to next irreducible. If all irreducibles of degree d0j are tested, increase j. (x) Compute for v ∈ S the quantity fv(w) = ord(A/w)∗(hv) and check if it is divisible by rv. If not go to next irreducible. If all irreducibles of degree d0j are tested, increase j. qw−1 fv (w) (xi) Check for v ∈ S whether gcd( , fv(w)) divides . If test fails for one v, go to r rv next irreducible. If all irreducibles of degree d0j are tested, increase j. (xii) Return hw and end the loops.

While the algorithm provides a cyclotomic extension which is suﬃcient to construct a cyclic algebra with the required invariants, the later explicit construction of a maximal Fq[t]-order will need to replace the condition rv|dv by the stronger condition of equality rv = dv. This amounts to changing the divisibility checks (xi) in the algorithm by tests for equality. We will refer to this altered algorithm as Algorithm 14 with strong conditions while the original algorithm will be referred to as Algorithm 14 with weak conditions.

4.4. Construction of L, σ¯ and a hσr i We now assume that the search for w was successful and set L = E , so thatσ ¯ = σ|L is a generator of Gal(L/F ). uv Then uv := r/dv is the smallest positive integer such thatσ ¯ ∈ Gv. In particular, both uv ∼ σ¯ , and the Frobenius automorphismσ ¯v at v are generators of Gv = Z/(dv). Therefore the following deﬁnition makes sense:

0 0 ∗ uv uv Definition 15. Deﬁne uv ∈ (Z/dvZ) , such that σ¯ =σ ¯v .

0 uv This means thatσ ¯uv acts on the residue ﬁeld at v as α 7→ αqv . 00 0 Let uv ∈ Z an inverse to uv modulo dv. Then for a ∈ F we have

0 Thm. 2(a) 00 Thm. 3 uv u (L/F, σ,¯ a) ⊗F Fv = (Lv/Fv, σ¯v , a) ∼ (Lv/Fv, σ¯v, a v ). ALGORITHMS FOR DIVISION ALGEBRAS Page 9 of 16

In particular 00 uv v(a) invv(L/F, σ,¯ a) ≡ (mod Z). dv 00 uv v(a) Locally at v we want to have invv D = nv/rv, i.e. nv/rv ≡ (mod ). Since dv is a dv Z multiple of rv, we obtain dv 00 nv ≡ uv v(a) (mod dv). rv Solving for v(a) yields

0 dv v(a) ≡ uvnv (mod dv). (4.1) rv

The above argument holds for all ﬁnite places. We deﬁne gv to be the integer such that 0 gv ≡ uvnv (mod rv) and 0 ≤ gv ≤ rv − 1. (4.2)

Q gv dv /rv We deﬁne a = v hv .

Theorem 16. With notation as above, D = (L/F, σ,¯ a) is a central division algebra over F with invv D = sv for all places v of F .

Proof. The statement is clear for the invariants at all finite places different from w. Because P sv = 0, it remains to show s∞ = 0. By Theorem 3, we have k (L/F, σ,¯ a) ⊗F F∞ = (L∞/F∞, σ¯ , a), k where k > 0 is the smallest integer such thatσ ¯ lies in Gal(L∞/F∞). But now observe that a ∗ ∗ − deg a is a norm from F∞(λw) to F∞ by definition of a∞ and Theorem 9(c), because t a is a ∗ 1-unit in F∞ and hence a is a norm from F∞(λw) .

5. Termination of the algorithm Theorem 17. Algorithm 14 terminates with weak and strong conditions.

We shall prove the theorem, by showing that the set of primes w for which the algorithm terminates is a Cebotarovˇ set of positive density.

5.1. Weak conditions

Q ei We introduce a notation for the prime factorization of r, namely we denote it by r = i pi for pairwise distinct prime numbers pi and integer exponents ei ≥ 1. The conditions on the sought for w are then as follows: (a) w∈ / S. (b) ord /(r)∗ (q)| deg(w). Z ¯ (c) fv(w) := ordA/(w)∗ (hv) is a multiple of rv, or in other words:

∀i : ordpi (rv) ≤ ordpi (fv(w)). qw−1 (d) ∀v ∈ S, ∀pi: min ordpi r , ordpi (fv(w)) ≤ ordpi (fv(w)) − ordpi (rv). This in turn is equivalent to

∀v ∈ S, ∀pi : if pi|rv then ordpi (qw − 1) − ordpi (r/rv) ≤ ordpi (fv(w)). Page 10 of 16 GEBHARD BOCKLE,¨ DAMIAN´ GVIRTZ

Lemma 18. Suppose condition (a) holds. Then condition (b) is equivalent to Frobw = 1 in Gal(F (ζr)/F ) for ζr a primitive r-th root of unity.

Proof. Using that w should not ramify in F (ζr) and the Frobenius generates the local Galois group, one gets a chain of equivalences

Frobw = 1 ∈ Gal(F (ζr)w/Fw) ⇔ [F (ζr)w : Fw] = 1 ⇔ [Fw(ζr): Fw] = 1 ⇔ r|qw − 1, where Fw denotes the residue class ﬁeld of F at w.

Moreover under (d), condition (c) is superﬂuous, i.e., the join of the two becomes:

(e) ∀v ∈ S, ∀pi : if pi|rv then ordpi (qw − 1) − ordpi (r/rv) ≤ ordpi (fv(w)).

We deﬁne ei,v := ordpi (r/rv) + 1 whenever pi|rv, and note that under this condition we have ei,v ≤ ei since ordpi (rv) ≥ 1.

Lemma 19. Let Fw denote the residue ﬁeld of F (ζr) at w and hv the reduction of hv in that ﬁeld. Assuming that Frobw = 1 in Gal(F (ζr)/F ) and 0 ≤ ei,v ≤ ei, the following assertions are equivalent: ei,v√ p (i) Frobw = 1 in Gal(F (ζr, i hv)/F ), ei,v p i p ∗ (ii) hv ∈ Fw,

(iii) ordpi (fv(w)) ≤ ordpi (qw − 1) − ei,v .

ei,v ei Note that adjoining any pi -th root of unity is a Kummer extension of F (ζr) since pi divides r.

Proof. As in the proof of the previous lemma

ei,v p p Frobw = 1 ∈ Gal(F (ζr, i hv)/F ) e ei,v i,vq p p i p i ∗ ⇔ Frobw = 1 ∈ Gal(F (ζr, hv)/F (ζr)) ⇔ hv ∈ Fw where the ﬁrst equivalence uses that Frobw = 1 in Gal(F (ζr)/F ). e p i,v ∗ For (ii) ⇒ (iii), let h = x i ∈ . Then ord ∗ x|q − 1, so v Fw Fw w e p i,v ord (f (w)) = ord (ord ∗ (x i )) = e + ord (ord ∗ x) ≤ e + ord (q − 1). pi v pi Fw i,v pi Fw i,v pi w ∗ s For (iii) ⇒ (ii), let x be a generator of Fw and x = hv. Then because

ord (ord ∗ h ) = ord (f (w)) ≤ ord (q − 1) − e pi Fw v pi v pi w i,v e i,v ei,v ei,v p p i s/pi we know that ordpi (s) ≥ ei,v, hence pi |s and hv is given by x .

We deduce the following result, of which Theorem 17 with weak conditions is an immediate corollary.

Proposition 20. The density of primes w in F satisfying (a)–(d) is m 1 Y Y 1 · 1 − e . φ(r) i,vj j=1 pi i:pi|rvj ALGORITHMS FOR DIVISION ALGEBRAS Page 11 of 16

Proof. Deﬁne E := Q pei,v for all v ∈ S and consider the tower of ﬁelds v i:pi|rv i

00 1/Ev1 1/Evm 0 F := F (ζr)(hv1 , . . . , hvm ) ⊃ F := F (ζr) ⊃ F

0 0 0 Then (1) Gal(F /F ) = Z/φ(r)Z, (2) the extension F /F is unramified, (3) The extensions Fj := 1/E 0 vj 0 F (hvj )/F are proper, linearly disjoint and ramified precisely at vj and totally ramified at 00 0 00 0 ∼ Qm vj, (4) F /F is abelian with Galois group Gal(F /F ) = j=1 Z/Evj Z, (5) the extension F 00/F is Galois and its group is a semidirect product Gal(F 00/F 0) o Gal(F 0/F ). 0 0 To see properness in (3), note in the function field case that F /F is unramified but Fj/F 0 ramifies at hvj ; because p is not allowed to divide rv, the extension F /F is Galois. Linear disjointness follows because the extensions have distinct ramification places. It follows that w is a place satisfying conditions (a)–(d) if and only if Frobw is trivial in 1/E 0 0 vj 0 Gal(F /F ) and Frobw is non-trivial in all primary factors of Gal(F (hvj )/F ). From the Cebotarovˇ density theorem, e.g. [9, Sec. VII.13], we get the required density.

5.2. Strong conditions We have to replace the ﬁrst inequality in condition (d) with equality and call this (d’). As above we make (c) superﬂuous and set ei,v := ordpi (r/rv) + 1 for pi|rv. Using Lemma 19, we restate (d’).

(e’) If pi|rv, then

ei,v −1 ei,v p p p p Frobw = 1 ∈ Gal(F (ζr, i hv)/F ) ∧ Frobw 6= 1 ∈ Gal(F (ζr, i hv)/F ).

e√ p i (e”) If pi - rv, then Frobw = 1 ∈ Gal(F (ζr, i hv)/F ). ei,v ei The factors in Theorem 20 change to (pi − 1)/pi for pi|rv and 1/pi for pi - rv. We deduce the following result, of which Theorem 17 with strong conditions is an immediate corollary.

Proposition 21. The density of primes w in F satisfying the strong conditions is

m 1 Y rvj Y 1 · 1 − . φ(r) r p j=1 i i:pi|rvj

6. Constructing a maximal Fq[t]-order The present section describes how to obtain a maximal order for D = (L/F, σ, a) constructed by the algorithm with strong conditions, i.e., in the case where rv = dv for all v ∈ S; in this section σ denotes a generator of Gal(L/F ), e.g. theσ ¯ from Subsection 4.4, and by order we will always mean Fq[t]-order. First, observe that we are dealing with a local question:

Theorem 22 [10, 11.6]. An order Λ is maximal if and only if for all places v 6= ∞ of F the completion Λv is maximal in Dv.

r−1 i We are going to describe how to maximize the order Λ = ⊕i=0 OLτ at the diﬀerent places of F . The globally maximal order is the linear hull of all the locally maximized orders. Page 12 of 16 GEBHARD BOCKLE,¨ DAMIAN´ GVIRTZ

6.1. The discriminant of a maximal order An important tool to decide whether an order is maximal inside a central simple algebra is its discriminant. Here we collect the relevant facts needed for the construction of a maximal Lr−1 i order in D starting with Λ = i=0 OLτ . The following result is immediate from [10, Thm. 32.1]:

Proposition 23. Suppose D has local invariants sv at the ﬁnite set of places S of K with rv corresponding primes hv and set r = lcm(rv : v ∈ S). Then for a maximal order Λ of D one r− r r Q rv has disc(Λ) = v∈S hv .

r−1 i Corollary 24. Let D be as in Theorem 16 and let Λ be the order ⊕i=0 OLτ . Then (i) The order Λ is maximal at all places outside S ∪ {w}. r r(r−1) r r(r−1) (ii) The discriminant of Λ at w is disc(L/K) = p or disc(L/K) = hw . r(r−1)·gv dv /rv (iii) The discriminant of Λ is hv at v ∈ S. (iv) The order Λ is maximal at v ∈ S if and only if rv = dv = r and gv = 1.

Proof. The ﬁrst three parts are immediate from the computation of disc(Λ) in Corollary 7 and the standard formula for discriminants of abelian extensions of Fq(t) in terms of conductors of the deﬁning characters; see [12]. For the last part we need to compare (iii) with the formula in Proposition 23. This gives the r− r r gv dv /rv ·r(r−1) ! rv dv 1 1 condition hv = hv . It is equivalent to gv (1 − ) = 1 − . From this (iv) rv r rv is immediate.

6.2. The ramiﬁed place w

L/F is totally ramiﬁed at w and we can write Dw = (Lw/Fw, σ, a), where σ par abus de notation denotes the unique extension of σ onto Lw. Because invw D = 0 we know that an ∼ isomorphism Dw = Mr(Fw) exists. Moreover this isomorphism can be described explicitly in two steps:

Theorem 25 [10, Proof of Thm. 30.4]. (i) There exists an element f ∈ O∗ such that Norm f = a. It induces an iso- Lw Lw/Fw morphism (Lw/Fw, σ, 1) → (Lw/Fw, σ, a) by letting Lw ﬁxed and τ 7→ (fτ). In fact, ∼ the existence of a solution to the norm equation is equivalent to (Lw/Fw, σ, 1) = (Lw/Fw, σ, a). ∼ (ii) An isomorphism (Lw/Fw, σ, 1) → EndFw (Lw) = Mr(Fw) is given by τ 7→ σ, x 7→ xLw

for x ∈ Lw, where xLw denotes multiplication by x. After choosing a basis e =

(e1, . . . , er) of OLw over Ow, and writing xLw e = eTx, σe = eP for unique matrices Tx,P in Mr(Fw), the isomorphism (Lw/Fw, σ, 1) → Mr(Fw) is given by τ 7→ P , x 7→ Tx.

In general, Mr(Fw) has many maximal orders, namely Mr(Ow) and all its conjugates ([10, 17.3]). Luckily, under the above isomorphism we have Λw ⊂ Mr(Ow). To see this, ﬁrst note Λw remains unchanged under the ﬁrst isomorphism as f ∈ O∗ . Under the second isomorphism Lw the action of Λw on Lw lies in the stabilizer of OLw which is equal to Mr(Ow) with the choice of an integral basis.

6.2.1. Solving the norm equation Now we describe how to find the solution f of the norm equation by a limit procedure: Choose a uniformizer π at w, so that p = OLw π is the ALGORITHMS FOR DIVISION ALGEBRAS Page 13 of 16 maximal ideal of O . The unit group U = O − p has the filtration U 0 = U , U i = Lw Lw Lw w Lw Lw Lw 1 + pi, i ≥ 1. We recall that U 0 /U 1 ∼ ∗ and U i /U i+1 ∼ pi /pi+1 ∼ , where the second Lw Lw = Fw Lw Lw = w w = Fw isomorphism depends on π. The same definitions can be made for Fw. The higher ramification groups Gi of Gw = Gal(Lw/Fw) form a decreasing sequence of normal subgroups for i ≥ −1. As proven in [13, Cor. IV.2.2,3] the higher quotients Gi/Gi+1 vanish for p - r (being direct products of groups of order p). Thus Gw = G0,Gi = 1, i ≥ 1 and with ψ(n) = rn we get

Theorem 26 [13, Prop. V.6.8, Cor. V.6.2]. (i) For every integer n ≥ 0 one has N(U ψ(n)) ⊂ U n ,N(U ψ(n)+1) ⊂ U n+1, Lw Kw Lw Kw

where N = NormLw/Fw . By passing to the quotients this yields ∗ ∗ N0 : Fw → Fw,Nn : Fw → Fw for n ≥ 1. r (ii) The map N0 is given by ξ 7→ ξ . ∗ (iii) The map Nn is surjective and given by ξ 7→ βnξ for some βn ∈ Fw.

Using this description of the norm (βn can be determined experimentally) we can construct Pn rk fn = k=0 akπ that approximate a solution f up to order rn for all n ≥ 0.

0 0 6.2.2. Necessary precision Let Λw be the maximal order Mr(Ow), so that Λw ⊂ Λw. Since both are finitely generated Ow modules of the same rank, there exists an integer s ≥ 0 such s 0 that π Λw ⊂ Λw. 0 0 0 0 −s P In terms of bases (e1, . . . , er2 ) and (e1, . . . , er2 ) of Λw and Λw we write ei = π j αijej s 0 −s P for suitable αij ∈ Ow. Now supposea îj ≡ aij (mod π ): Then defininge î := π j αîjej , 0 0 0 a set of generators of Λw is given by{eˆ1,..., eˆr2 , e1, . . . , er2 }. One concludes that it is enough to solve the norm equation up to order s − 1.

Theorem 27. The integer s can be chosen as r.

Proof. Consider the r × r-Vandermonde matrix with columns (πi, σ(πi), . . . , σr−1(πi))t r for i = 0, . . . , r − 1. Acting from the right on (λ1, . . . , λr) ∈ Fw it gives the image of 0 1 r−1 P i 0 −r (π , π , . . . , π ) under i λiτ . The condition Λw ⊂ π Λw is equivalent to showing that all P λ τ i which send (π0, π1, . . . , πr−1) to Or lie in π−rΛ , i.e., the inverse V −1 sends O i i Lw w Lw −r to π OLw . Thus, we only need to know the denominators of the inverse of a Vandermonde matrix. i Q i k According to [7, Exerc. 40], they are given by σ (π) 1≤k≤r,k6=i(σ (π) − σ (π)). Because of G0 = Gw,G1 = 1 all factors have valuation 1 and the total valuation is r.

Lemma 28. It suﬃces to solve the norm equation up to order 0.

Proof. Recall that the solution f was constructed in order jumps of size r.

0 0 6.2.3. Fixing the denominator The approximated generatorse ˆ1,..., eˆr2 may have denominators at places diﬀerent from w. To make sure that we still have a multiplicatively closed Page 14 of 16 GEBHARD BOCKLE,¨ DAMIAN´ GVIRTZ order, multiply the generators with the prime-to-w part, so that the order Λ00 they generate 0 still equals Λw at w but is contained in the original order Λv at all v 6= w. Because this inclusion in Λv may be proper, the global order sought-after (which is maximal 00 at w but Λv at other places v) is the linear hull of Λ and Λ.

6.3. Unramiﬁed places v ∈ S

For convenience, fix the notation d := dv, u := r/d throughout this section and assume d = rv (strong conditions). Fix prolongations w0, . . . , wu−1 of v to L s.t. σ acts on the wi’s by sending each to the next one and wu−1 to w0. In other words σ sends each component of Lu−1 M := L ⊗ Fv = i=0 Li to the next one, where Li := Lwi . We construct an order maximal at r−1 i v containing Λ = ⊕i=0 OLτ . Locally, D is given as: u−1 ! u−1 r M r M u u d j D ⊗ Fv = M[τ]/(τ − a) = Li [τ]/(τ − a) = Li[τ ]/((τ ) − a) τ i=0 i,j=0 u u d ∼ ∼ Defining Di := Li[τ ]/((τ ) − a), one knows that Di = Dj for all 0 ≤ i, j ≤ u − 1 and M = Mu(D0) (especially the Di have the same local invariant). Multiplication of elements in the right hand side of the above equation works via u s u s 0 i 6= j σ : Di → Di+1, b(τ ) 7→ σ(b)(τ ) , αβ = (6.1) α ·Di β i = j for b ∈ Li, α ∈ Di, β ∈ Dj. For d = rv (strong conditions) the algebra Dj actually is a skewfield, cf. [10, 31.6].

0 d−1 i Theorem 29 [10, 13.3]. Let x ∈ Dj have valuation 1/d. Then Λv = ⊕i=0 OLj x is a maximal order in Dj.

u u d Now v(τ ) = v((τ ) )/d = v(a)/d, hence for a uniformizer h of v in Fv v(hm(τ u)n) = m + nv(a)/d.

Since (v(a), d) = 1 the extended Euclidean algorithm yields m, n ∈ Z with md + nv(a) = 1 and m u n h (τ ) is the required element. Hence, in Dj we get the maximal order: m u n ODj := OLj [h (τ ) ].

Next, we will give an explicit description of a maximal order in (M/Fv, σ, a) as stabilizer of a suitable lattice.

Lu−1 i Lemma 30. V := i=0 Diτ is a left (M/Fv, σ, a)-submodule of (M/Fv, σ, a).

Proof. V is obviously closed under addition. It remains to show that left multiplication with “scalars” sends V to V . So let u−1 u−1 X j X i a = αiτ ∈ (M/Fv, σ, a), x = βiτ ∈ V i,j=0 i=0 Then a straightforward calculation shows u−1 u−1 u−1 u−1 X j X k X j j+k (6.1) X j i a · x = αiτ βkτ = αiσ (βk)τ = αiσ (βk)τ ∈ V i,j=0 k=0 i,j,k=0 i,j,k=0,j+k=i ALGORITHMS FOR DIVISION ALGEBRAS Page 15 of 16

Lu−1 i We now deﬁne a maximal lattice in V by ΛV := i=0 ODi τ . Furthermore deﬁne three lattices in M[τ]/(τ r − a):

u−1 i u−1 u−1 u−1 u−1 M M j M M j M M u j R0 := ODi τ ⊆ R1 := ODi τ ⊆ R2 := OLi [τ ]τ i=0 j=i−u+1 i=0 j=0 i=0 j=0 m u n Note that R1 is the v-adic closure of OL[τ, h (τ ) ] and R2 the v-adic closure of OL[τ].

Theorem 31. R0 equals the stabilizer of ΛV (and is therefore a maximal order).

j k Proof. Let i, k run from 0 to u − 1 and j from i − u + 1 to i. Note ﬁrst that ODi τ ODk τ = j j+k j+k ODi σ (ODk )τ = ODi ODj+k τ . By (6.1) the last term is non-zero only if i = j + k, in which case it lies in ΛV . This proves ‘⊆’. We now show the converse inclusion ‘⊇’: P j Let α = i,j αijτ with αij ∈ Di for all i = 0, . . . , u − 1. If α lies in the stabilizer, then it k maps test elements x = βkτ with βk ∈ ODk to ΛV :

X i−k i ! αx = αi,i−kσ (βk)τ ∈ ΛV i

From this we deduce that αi,i−k ∈ ODi for all k = 0, . . . , u − 1. Therefore α ∈ R0.

The explicit description of the maximal order R0 can be lifted to that of a global order maximal at v. Before stating this result, we need a short lemma:

Lemma 32. Choose integers et for t = 0, . . . , u − 1 such that d−1 m u n M et u t ODi = OLi [h (τ ) ] = OLi h (τ ) . t=0

Ld et u t Furthermore set ed := −v(a), so that ODi = t=1 OLi h (τ ) . Then et−1 − et ∈ {0, 1} for all 0 < t ≤ d.

Proof. het (τ u)t has valuation between 0 and 1 − 1/d, so

et−1 u t−1 et u t u 1 > |v(h (τ ) ) − v(h (τ ) )| = |et−1 − et − v(τ )|

u (4.2) Noting that 0 ≤ v(τ ) = v(a)/d = (gid/ri)/d = gi/ri < 1, the claim follows.

We now rewrite the maximal order R0: u−1 i d−1 u−1 u−1 d−1 M M M es u s j M M M es u s j−u R0 = OLi h (τ ) τ = R1 + OLi h (τ ) τ i=0 j=i−u+1 s=0 i=0 j=i+1 s=0 u−1 u−1 d−1 u−1 u−1 d−1 M M M es u s−1 j M M M es+1 u s j = R1 + OLi h (τ ) τ = R1 + OLi h (τ ) τ i=0 j=i+1 s=0 i=0 j=i+1 s=0

¯ Let (bij)j=0,...,d−1 be a basis of ki/k (the residue ﬁeld of Li over that of Fv) for all i = ˜ 0, . . . , u − 1. Choose lifts bij to OLi and bij to OL. Note that OL/(h) = ⊕iOLi /(h), so that bij Page 16 of 16 ALGORITHMS FOR DIVISION ALGEBRAS

˜ can be thought of as a ﬁrst order approximations of the tuple (0,..., 0, bij, 0,..., 0). Then u−1 u−1 d−1 d−1 u−1 u−1 d−1 d−1 X X X X ˜ es+1 u s j X X X X es+1 u s j R0 = R1 + Avbith (τ ) τ = R1 + Avbith (τ ) τ . i=0 j=i+1 s=0 t=0 i=0 j=i+1 s=0 t=0 The last equation follows because Lem. 32 ˜ es+1 u s j M es+1+1 u s j M es u s j Av(bit − bit)h (τ ) τ ⊂ OLi h (τ ) τ ⊂ OLi h (τ ) τ ⊂ R1.

The above formula for R0 provides a global lift of R0. It is the order u−1 u−1 d−1 d−1 m u n X X X X es+1 u s j OL[τ, h (τ ) ] + Abith (τ ) τ . i=0 j=i+1 s=0 t=0 Fixing the denominator at other places is not necessary.

6.4. Unramiﬁed places v∈ / S

For these places, any order containing Λ = OL[τ] will be maximal since the discriminant commutes with localization and disc(Λ) = ad(d−1) disc(L/K) is a local unit.

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Gebhard Böckle DamiánGvirtz UniversitätHeidelberg The London School of Geometry and InterdisziplinäresZentrum für Number Theory wissenschaftliches Rechnen (IWR) Department of Mathematics Im Neuenheimer Feld 368 University College London 69120 Heidelberg Gower Street Germany LONDON WC1E 6BT [email protected] United Kingdom [email protected]