A P -collection C ˜ induced from C and auxiliary elements

Chapter V: The generalized P -scheme algorithm

5.7 A P -collection C ˜ induced from C and auxiliary elements

The idempotent decompositionsI_K produced in Section 5.4 define a P-scheme of double cosetsCrather than an (ordinary)P-scheme. Section 5.7–5.9 are devoted to turning it to aP-schemeC˜. In particular, this section focuses on the definition ofC˜ as aP-collection.

We assume p > deg(f) in Section 5.7–5.9. As mentioned in Section 5.1, this assumption implies that the wild inertia groupW_Q0 ⊆GofQ₀ overK₀is trivial.

Suppose the partitions inCall have locally constant ramification indices and inertia degrees (with respect to(D_Q0,I_Q0)). Then forK ∈ F andδ∈I_K, the (nonempty) set of maximal idealsPofOK satisfying

δ ≡1 (mod ¯P) where P¯ := P/pO_K

Rad( ¯O_K) ∩R_K

all have the same ramification indexe(P) and the same inertia degreef(P). We denote e(P) by e_δ and f(P) by f_δ. Note that e_δ and f_δ are coprime to p by Theorem 5.3 and the assumptionp > deg(f).

Recall that for a finite extensionK ofK₀ andi ∈ N⁺, we denote byA_K,i the ring ( ¯O_K/Rad( ¯O_K))⊗_Fq Fqⁱ. To defineC˜, we need an auxiliary collection of elements in ringsO¯K orAK,i. We call such a collection of elements anI-advice:

Definition 5.7. Suppose I = {I_K : K ∈ F } is a collection of idempotent de- compositions of the rings R_K, K ∈ F, that defines to a P-collection of double cosetsC(with respect toD_Q0), such that all the partitions inChave locally constant ramification indices and inertia degrees (with respect to(D_Q0,I_Q0)). AnI-advice {S,T }consists of the following data:

• S = {s_δ : δ ∈ I_K, e_δ > 1}, where each s_δ ∈ S is an element of O¯_K such that s_δ ∈ m− m² for all the maximal ideals m of O¯_K satisfying δ ≡ 1 (mod m/Rad( ¯O_K)).

• T = {t_δ : δ ∈ I_K, f_δ >1}, where eacht_δ ∈ T is an element of A_K,fδ such thattδ6∈mfor all the maximal idealsmofAK,f_δ satisfyingδ ≡1 (mod m), andσK,f_δ(tδ) =ξ·tδ, whereξ ∈Fq^fδ is a primitivefδth root of unity.13 AnI-advice can be computed fromIby the following lemma. Its proof is deferred to Appendix C.

Lemma 5.17. Under GRH, there exists a subroutine ComputeAdvice that given I = {I_K : K ∈ F } as in Definition 5.7, either properly refines some idempotent decompositionI_K ∈ I, or computese_δ, f_δfor K ∈ F, δ ∈I_K and anI-advice.14 Moreover, the subroutine runs in time polynomial inlogpand the size ofF.

We also need the following notations: recall that forH ∈ P, we chose an isomor- phismτ_H : K →L^H overK₀ whereK is the unique field inF isomorphic toL^H overK₀. The induced isomorphismO¯_K ∼= ¯O_LH identifies eachs_δ ∈ S (whereS is as in Definition 5.7) with an element inO¯_L^H, which we denote bys_δ,H. Similarly, we identify eacht_δ∈ T with an element inA_L^H_,fδ, denoted byt_δ,H.

Next we define aP-collectionC˜usingIand anI-advice:

Definition 5.8. Let I = {I_K : K ∈ F } be as in Definition 5.7 and {S,T } be an I-advice. LetC = {C_H : H ∈ P}be the P-collection of double cosets with respect to D_Q0 associated with I (see Section 5.4). For H ∈ P, let K be the unique field inF isomorphic toL^H overK₀, and define the partitionC˜_H ofH\G so that Hg, Hg⁰ ∈ H\G are in the same block of C˜_H iff the following conditions are satisfied:

13We regardδ∈RK ⊆O¯K/Rad( ¯OK)as an element ofAK,f_δviaδ7→δ⊗1, andξ∈Fq^fδ as an element ofAK,f_δviaξ7→1⊗ξ.

14We need to compute the ringsA_K,fδ before computing the elementst_δ ∈A_K,fδ. These rings will be computed before the call of the subroutineComputeAdvice. See Section 5.9.

1. HgD_Q0 andHg D_Q0 are in the same blockB ofC_H.

2. Letδbe the unique idempotent inI_Ksuch thatτ˜_H(δ) =δ_B(see Definition 5.4), where B ∈ C_H is as in the previous condition. If e_δ > 1, the order of the unique elementcinκ^×_Q

0 satisfying

g⁻¹s_δ,H +I =c·(^g0−1s_δ,H +I) is coprime toe_δ, whereI = (Q₀/pO_L)^e(Q0)/eδ+1.

3. Letδ ∈I_K be as in the previous condition. Let m₀ be an arbitrary maximal ideal ofA_L,fδ containing _{Rad( ¯}^Q0/pO_O^L

L). Iff_δ > 1, the order of the unique element cin(A_L,fδ/m₀)^×satisfying

g⁻¹

tδ,H +m0 =c·(^g0−1tδ,H +m0) is coprime tof_δ.

DefineC˜={C˜_H : H ∈ P}, which is aP-collection. We sayC˜is theP-collection associated withIand{S,T }.

We check thatC˜is well defined:

Lemma 5.18. TheP-collectionC˜in Definition 5.8 is well defined.

The proof of Lemma 5.18 is routine and can be found in Appendix C.

5.8 (C,D)-separatedP-collections

We continue the discussion in the previous section. Our goal is to compute I = {I_K : K ∈ F } and an I-advice {S,T } such that the associated P-collection C˜ is a strongly antisymmetric P-scheme. To achieve this goal, we introduce another property ofP-collections called(C,D)-separatedness:

Definition 5.9. Let P be a subgroup system over a finite group G, and let C = {C_H :H ∈ P}be aP-collection of double cosets with respect to a subgroupDof G. We say aP-collectionC˜= {C˜_H : H ∈ P} is(C,D)-separatedif the following conditions are satisfied:

1. All the partitionsC˜_H ∈C˜are invariant under the action ofDby inverse right translation, i.e. for allB ∈C˜_Handg ∈ D, the set^gB ={Hhg⁻¹ :Hh∈B}

is also inC˜H.

2. ForH ∈ P, the mapπ_H : H\G → H\G/D sendingHg ∈ H\Gto HgD maps each block ofC˜_H bijectively to a block ofC_H.

It is worth noting that ifC˜is(C,D)-separated, then all the partitions inC automati- cally have locally constant ramification indices and inertia degrees:

Lemma 5.19. Suppose C˜ = {C˜_H : H ∈ P} is a (C,D)-separated P-collection where P, C, D are as in Definition 5.9. LetI be a normal subgroup of D. Then all the partitions inChave locally constant ramification indices and inertia degrees with respect to(D,I).

Proof. FixH ∈ P,B ∈C_H, andB˜ ∈C˜_H such thatπ_H( ˜B) = B, whereπ_H is as in Definition 5.9. LetD⁰ be a subgroup ofD. Consider arbitraryHgD⁰, Hg⁰D⁰ ∈ B and lift them to Hg, Hg⁰ ∈ B˜ respectively. Choose h₁, . . . , h_k ∈ D⁰ such that the D⁰-orbit ofHgis{Hgh₁, . . . , Hgh_k}and the cosetsHgh_iare all distinct. We claim Hg⁰h₁, . . . , Hg⁰h_k are also distinct. Assume to the contrary thatHg⁰h_i1 =Hg⁰h_i2 holds for distinct i₁, i₂ ∈ [k]. Then Hg⁰h_i1 and Hg⁰h_i2 are in the same block of C˜_H. It follows by the first condition in Definition 5.9 that Hgh_i1 and Hgh_i2 are also in the same block. ButHgh_i1 6=Hgh_i2 and they are both mapped toHgDby πH, contradicting the second condition in Definition 5.9. This proves the claim. So the cardinality of theD⁰-orbit of anyHg ∈H\Gonly depends on the block inC_H containingHgD. In particular, this holds forD⁰ =DandD⁰ =I. The lemma then follows from Definition 5.2.

The following lemma provides a criterion for a(C,D)-separatedP-collection to be a strongly antisymmetricP-scheme.

Lemma 5.20. Let P be a subgroup system over a finite group G, and let C = {C_H : H ∈ P}be aP-scheme of double cosets with respect to D ⊆ G. Suppose C˜ = {C˜_H : H ∈ P} is a compatible, invariant, (C,D)-separated P-collection.

Then it is actually a P-scheme. Moreover, if C is antisymmetric (resp. strongly antisymmetric), so isC˜.

Proof. For the first claim, we just need to showC˜is regular. Consider H, H⁰ ∈ P withH⊆H⁰. Letπ_H :H\G→H\G/Dbe the map sendingHg ∈H\GtoHgD,

and defineπ_H⁰ similarly. Then the following diagram commutes.

H\G H⁰\G

H\G/D H⁰\G/D

π_H,H0

πH π_H0

π^D

H,H0

For B ∈ C˜_H and B⁰ ∈ C˜_H⁰ containing π_H,H⁰(B), we need to show the map π_H,H⁰|_B : B → B⁰ has the constant degree, i.e., the cardinality ofπ_H,H⁻¹ 0(y)∩B is independent ofy ∈B⁰. AsC˜is(C,D)-separated, the mapπ_H sendsBbijectively to π_H(B)∈C_H, and similarlyπ_H⁰ sendsB⁰ bijectively toπ_H⁰(B⁰)∈C_H⁰. The claim then follows from regularity ofC.

Note that the conjugations also commute with the maps π_H, i.e.,π_hHh⁻¹ ◦c_H,h = c^D_H,h◦π_H for H ∈ P andh ∈ G. Assume C˜is not strongly antisymmetric. Then there exists a nontrivial permutation τ of a blockB ∈ C˜_H for some H ∈ P that arises as a composition of mapsσ_i : Bi−1 → B_i, i = 1. . . , k whereB_i is a block of C˜_Hi, H_i ∈ P, and σ_i is of the form c_Hi−1,h|_Bi−1 (whereh ∈ G), π_Hi−1,Hi|_Bi−1, or (π_Hi,Hi−1|_Bi)⁻¹ (see Definition 2.7). As the maps π_Hi|_Bi : B_i → π_Hi(B_i) are bijective and commute with projections and conjugations, we seeτ⁰ :=σ_k⁰ ◦ · · · ◦σ₁⁰ is a nontrivial permutation ofπ_H(B) ∈ C_H, where each mapσ_i⁰ := π_Hi|_Bi ◦σ_i ◦ (πHi−1|Bi−1)⁻¹is of the formc^D_Hi−1,h|Bi−1,π_H^D_i−1,Hi|Bi−1, or(π_H^D_i,Hi−1|Bi)⁻¹. SoC is not strongly antisymmetric. The proof of antisymmetry is the same except that we only consider mapsσ_i that are conjugations.

We need to compute I = {I_K : K ∈ F } and an I-advice {S,T } such that the associated P-collectionC˜is (C,D_Q0)-separated. The following lemma states that for P-collections arising from Definition 5.8, the first condition of(C,D_Q0)- separatedness is in fact automatic.

Lemma 5.21. LetI,{S,T },C andC˜be as in Definition 5.8. Then all the partitions inC˜are invariant under the action ofD_Q0 by inverse right translation.

To prove it, we need the following observation.

Lemma 5.22. Let m₀ be a maximal ideal of A_L,fδ containing _{Rad( ¯}^Q0/pO_O^L

L). For all x ∈ A_L,fδ, ω ∈ I_P, and σ ∈ D_Q0 such that the image of σ in Gal(κ_Q0/O¯_K0) is the Frobenius automorphismx 7→x^q overF^q, it holds that^ωx ≡x (mod m0)and

σx≡σ_L,fδ(x) (mod m0).

Proof. By bilinearity, we may assume x = a⊗b where a ∈ O¯_L/Rad( ¯O_L) and b ∈ Fq^fδ. Asω ∈ I_P, it holds that^ωa ≡ a (mod _{Rad( ¯}^Q0/pO_O^L

L))and hence ^ω(a⊗b) ≡

ωa⊗b≡a⊗b (mod m₀). Similarly, we have^σa≡a^q (mod _{Rad( ¯}^Q0/pO_O^L

L))by definition and hence^σ(a⊗b)≡^σa⊗b≡a^q⊗b ≡σ_L,fδ(a⊗b) (mod m₀).

Now we are ready to prove Lemma 5.21.

Proof of Lemma 5.21. ConsiderH ∈ P andHg, Hg⁰ in the same block ofC˜_H. Fix h ∈ D_Q0. We prove Hgh⁻¹, Hg⁰h⁻¹ are also in the same block by verifying the three conditions in Definition 5.8. LetBbe the block ofC_H containing bothHgD_Q0 andHg⁰D_Q0. The first condition in Definition 5.8 obviously holds forHgh⁻¹ and Hg⁰h⁻¹ sinceHgh⁻¹D_Q0 =HgD_Q0 ∈B andHg⁰h⁻¹D_Q0 =Hg⁰D_Q0 ∈B. LetK be the field inF isomorphic toL^H overK₀. Letδ be the idempotent inI_K satisfyingτ˜_H(δ) =δ_B(see Definition 5.4). Supposee_δ >1. By Definition 5.8, the order of the unique elementcinκ^×_Q

0 satisfying

g⁻¹

sδ,H +I =c·(^g0−1sδ,H +I)

is coprime toe_δ, where I = (Q₀/pO_L)^e(Q0)/eδ+1. We have^hI =I sinceh∈ D_Q0. Therefore

hg⁻¹s_δ,H +I =^hc·(^hg0−1s_δ,H +I), where^hc∈κ^×_Q

0 has the same order asc. So the second condition in Definition 5.8 is satisfied byHgh⁻¹andHg⁰h⁻¹.

Now supposef_δ >1. Letm₀ be a maximal ideal ofA_L,fδ containing _{Rad( ¯}^Q0/pO_O^L

L). By Definition 5.8, the order of the unique elementcin(A_L,fδ/m₀)^×satisfying

g⁻¹t_δ,H +m₀ =c·(^g0−1t_δ,H +m₀)

is coprime to f_δ. Fix σ ∈ D_Q0 whose image in Gal(κ_Q0/O¯_K0) is the Frobenius automorphismx7→x^qoverF^q. Chooseω ∈ IQ0 andi∈Zsuch thath =ωσⁱ. By Lemma 5.22, we have

hg⁻¹t_δ,H ≡^ωσ

i

(^g−1t_δ,H)≡^σ

i

(^g−1t_δ,H)≡σ_L,fⁱ

δ(^g−1t_δ,H)≡^g−1 σⁱ_L,f

δ(t_δ,H)

≡^g−1(ξⁱ·t_δ,H)≡ξⁱ·^g−1t_δ,H (mod m0),

whereξ is the primitivef_δth root of unity satisfyingσ_K,fδ(t_δ) =ξ·t_δas in Defini- tion 5.7. The same argument shows^hg0−1t_δ,H ≡ ξⁱ ·^g0−1t_δ,H (mod m₀). It follows that

hg⁻¹

tδ,H +m0 =c·(^hg0−1tδ,H +m0).

So the third condition in Definition 5.8 is also satisfied byHgh andHg h . We also show thatP-collections arising from Definition 5.8 always satisfy a weaken- ing of the second condition of(C,D_Q0)-separatedness, where bijectivity is replaced by injectivity:

Lemma 5.23. LetI,{S,T },C andC˜be as in Definition 5.8. Then forH ∈ P, the mapπH : H\G→H\G/DQ0 sendingHg ∈H\GtoHgDQ0 maps each block of C˜_H injectively to a block ofC_H.

Proof. ConsiderH ∈ P andg ∈Gandh∈ D_Q0 such thatHg 6=Hgh⁻¹. We want to prove thatHgandHgh⁻¹are in different blocks ofC˜_H.

LetB be the block ofC_H containingHgD_Q0 =Hgh⁻¹D_Q0. LetK be the field in F isomorphic toL^H overK₀. Letδbe the idempotent inI_K satisfyingτ˜_H(δ) =δ_B (see Definition 5.4). Fixσ ∈ D_Q0 whose image inGal(κ_Q0/O¯_K0)is the Frobenius automorphismx7→x^q overFq.

As we assumep >deg(f), the wild inertia groupW_Q0 ⊆GofQ0overK₀is trivial.

SoI_Q0 is a cyclic group of ordere(Q₀). Fix a generatorωofI_Q0. By Theorem 5.3 and Definition 5.2, we knowe_δis the smallest positive integerksatisfyingHgω^−k= Hg, and f_δ is the smallest positive integer k satisfying Hgσ^−kI_Q0 = HgI_Q0. So there exist unique i ∈ {0, . . . , f_δ − 1} and j ∈ {0, . . . , e_δ − 1} such that Hgh⁻¹ = Hgσ⁻ⁱω^−j. AsHg 6= Hgh⁻¹, we have (i, j)6= (0,0). By replacing h withω^jσⁱ if necessary, we may assumeh=ω^jσⁱ.

First assume i 6= 0. Then fδ > 1. Let m0 be a maximal ideal ofAL,f_δ containing

Q0/pO_L

Rad( ¯O_L). As shown in the proof of Lemma 5.21, we have

hg⁻¹t_δ,H ≡ξⁱ·^g−1t_δ,H (mod m₀),

whereξis a primitivef_δth root of unity. The order ofξⁱ isf_δ/gcd(f_δ, i)>1and is a divisor of f_δ. So the third condition in Definition 5.8 is not satisfied byHg and Hgh⁻¹. It follows thatHg andHgh⁻¹ are in different blocks ofC˜_H, as desired.

Now assumei= 0andj 6= 0. Theneδ >1. Letme=Q0/pOLandk=e(Q0)/eδ. As shown in the proof of Lemma 5.18, we have ^g−1s_δ,H ∈ m^k_e −m^k+1_e . Choose π_L ∈ m_e−m²_e. We have a group homomorphism I_Q0 → κ^×_Q

0 sending g ∈ I_Q0 to the unique elementc_g ∈ κ^×_Q

0 satisfying^gπ_L+m²_e =c_g(π_L+m²_e). This map is

injective since its kernel is W_Q0 = {e}. In particular, we know c_ω is a primitive e(Q₀)th root of unity inκ^×_Q

0. Choosec∈κ^×_Q

0 such that

g⁻¹s_δ,H +m^k+1_e =c(π_L^k+m^k+1_e ),

which exists since^g−1s_δ,H andπ_L^k are both inm^k_e−m^k+1_e . Then we have

hg⁻¹s_δ,H +m^k+1_e =^ω

j

(^g−1s_δ,H +m^k+1_e ) = ^ω

j

(c(π_L^k+m^k+1_e ))

=c·c^jk_ω ·(π_L^k +m^k+1_e ) =c^jk_ω ·(^g−1s_δ,H +m^k+1_e ).

The order ofc^jk_ω ∈ κ^×_Q

0 ise(Q₀)/gcd(e(Q₀), jk) = e_δ/gcd(e_δ, j) >1, which is a divisor ofe_δ. So the second condition in Definition 5.8 is not satisfied byHg and Hgh⁻¹. It follows thatHgandHgh⁻¹are in different blocks ofC˜H, as desired.

In the next section, we give subroutines that refine the idempotent decompositions I_K so that C˜is eventually a compatible, invariant, (C,D)-separated P-collection, and hence a strongly antisymmetricP-scheme.