Projection morphisms and Hecke correspon- correspon-dences

Let S_F,^r_K be a fixed Drinfeld modular variety. For each g ∈ GL_r(A^fF) and all compact open subgroups K^′ ⊂g⁻¹Kg of GLr(A^f_F), we have a well-defined map

S_F,^r_K′(C∞) → S_F,^r_K(C∞)

[(ω, h)] 7→ [(ω, hg⁻¹)]. (2.1.1) Theorem 2.1.1. This map is induced by a unique finite morphism πg :S_F,^r_K′ → S_F,^r_K defined over F of degree [g⁻¹Kg :K^′·(K ∩F^∗)].

Proof. In the case that K and K^′ are contained in GL_r( ˆA), we already showed the existence of a morphismπ_gwhich is described by (2.1.1) onC∞-valued points in the proof of Theorem 1.1.2. IfKandK^′ are arbitrary withK^′ ⊂g⁻¹Kg, there is an s ∈GL_r(A^f_F) with

sK^′s⁻¹ ⊂sg⁻¹Kgs⁻¹ ⊂GL_r( ˆA).

By our definition in the proof of Theorem 1.1.2, we have S_F,^r_K′ = S_F,s^r _K′s−1, where under the identifications of C∞-valued points introduced in the proof of Theorem 1.1.2

[(ω, h)]∈S_F,^r_K′(C_∞)←→[(ω, hs⁻¹)]∈S_F,s^r _K′s⁻¹(C_∞).

Similarly, we haveS_F,^r_K=S_F,sg^r ₋1Kgs⁻¹ with

[(ω, h)]∈S_F,K^r (C_∞)←→[(ω, hgs⁻¹)]∈S_F,sg^r −1Kgs⁻¹(C_∞).

Using these identifications, we can define the morphism πg : S_F,^r_K′ → S_F,^r_K as π₁ : S_F,s^r _K′s−1 → S_F,sg^r _−1Kgs−1(C∞). Since the latter morphism π₁ is given by [(ω, h)]7→[(ω, h)] onC_∞-valued points, by the above identificationsπ_g is indeed described by (2.1.1) on C∞-valued points. So we have shown the existence of the morphism π_g defined over F. It is uniquely determined by (2.1.1) because C_∞ is algebraically closed.

To show finiteness of the morphism π_g and the statement about its degree, we use Proposition 0.3.1 and 0.3.2. By the above definition of a general mor-phism π_g, it is enough to show these statements for morphisms of the form π₁ :S_F,^r_K′ →S_F,^r_K with K^′ ⊂ K ⊂GL_r( ˆA).

We first assume that K^′ = K(I) is a principal congruence subgroup. Then π1 is the canonical projection

S_F,^r_K(I)→S_F,^r_K(I)/K

by the construction in the proof of Theorem 1.1.2. By the discussion at the end of Section 0.3, in this caseπ₁ is finite of degree |ρ(K)|= [K: ker(ρ)] for

ρ: K −→ Aut(S_F,^r_K(I))

g 7−→ π_g .

So we have to show that ker(ρ) = K(I)·(K ∩F^∗). For g = kλ with k ∈ K(I) and λ∈ K ∩F^∗, we have

πg([(ω, h)]) = [(ω, hλ⁻¹k⁻¹)] = [(ω◦λ⁻¹, h)] = [(ω, h)]

for all [(ω, h)]∈S_F,^r_K(I)(C∞), hence K(I)·(K ∩F^∗)⊂ker(ρ).

Conversely, assume that g ∈ ker(ρ). By Lemma 1.1.6, there is a geometric point p = [(ω, h)] ∈ S_F,K(I)^r (C_∞) with End(p) = F. For this point, we have [(ω, h)] = [(ω, hg⁻¹)], hence there are c∈C^∗_∞,T ∈GL_r(F) and k ∈ K(I) with

ω = c·ω◦T⁻¹, h = T hg⁻¹k.

The first equality implies that ω(F^r) = c ·ω(F^r), hence c ∈ F^∗ because of End(p) = F. The matrix T ∈ GL_r(F) is therefore equal to the scalar matrix c∈F^∗. By the second equality, we conclude that g =kT and T ∈ K because g andkboth lie inK. Hence,g ∈ K(I)·(K∩F^∗) and indeed ker(ρ) =K(I)·(K∩F^∗).

This concludes the proof for the caseK^′ =K(I).

For general subgroups K^′ ⊂ K ⊂GLr( ˆA), choose a proper ideal I of A with K(I) ⊂ K^′. Then we have the following commutative diagram of projection maps:

In the following, we call the morphismsπ_g projection morphisms of Drinfeld modular varieties. In the case g = 1 we also call them canonical projections of Drinfeld modular varieties. For two elementsg, g^′ ∈GL_r(A^fF) and two subgroups K^′ ⊂g⁻¹Kg, K^′′ ⊂g^′−1K^′g^′, by the description on C∞-valued points, we have

π_gg′ =π_g◦π_g′. (2.1.2) Definition 2.1.2. A compact open subgroup K ⊂ GL_r(A^fF) is called amply small if there is a proper ideal I of A and a g ∈ GL_r(A^fF) such that gKg⁻¹ is contained in the principal congruence subgroup K(I)⊂GL_r( ˆA).

Proposition 2.1.3. Let K ⊂GL_r(A^f_F) be amply small,g ∈GL_r(A^f_F) and K^′ ⊂ g⁻¹Kg. Then the morphism π_g : S_F,^r_K′ → S_F,^r_K is ´etale. Furthermore, if K^′ is a normal subgroup of g⁻¹Kg, it is an ´etale Galois cover over F with group g⁻¹Kg/K^′ where the automorphism of the cover corresponding to a coset [x] ∈ g⁻¹Kg/K^′ is given by π_x :S_F,^r_K′ →S_F,^r_K′.

Proof. We first reduce ourselves to the caseg = 1 andK^′ ⊂ K ⊂ K(I)⊂GL_r( ˆA) for some proper ideal I of A. For K,K^′ and g arbitrary with K amply small, let

h∈GLr(A^f_F) andI a proper ideal ofA such thath⁻¹Kh⊂ K(I)⊂GLr( ˆA). By the relation (2.1.2), we have the commutative diagram

S_F,h^r _{−1gK′g−1h}^πg−1_∼^h //

Case (i): LetK^′ be a principal congruence subgroupK(J) modulo a proper ideal J of A, i.e., K^′ =K(J)▹K ⊂ K(I).

Then, by our definition in the proof of Theorem 1.1.2, π₁ :S_F,^r_K(J) →S_F,^r_K is the canonical projection

S_F,^r_K(J)−→S_F,^r_K(J)/K.

We show that K/K(J) acts freely on the closed points of S_F,^r_K(J). By Proposi-tion 0.3.2, this implies that this projecProposi-tion is an ´etale morphism. By the modular interpretation ofS_F,^r_K(J) in the proof of Theorem 1.1.2, it is enough to show that the action of K/K(J) on isomorphism classes of Drinfeld A-modules over C∞

together with J-level structure is free.

Indeed, assume that a coset [k] ∈ K/K(J) stabilizes the isomorphism class of the Drinfeld moduleφ overC∞ associated to a lattice Λ⊂C∞ together with J-level structure α : (J⁻¹/A)^r →^∼ J⁻¹·Λ/Λ. By our definition of the action of GL_r( ˆA) on Drinfeld modules withJ-level structure in the proof of Theorem 1.1.2, the image of (φ, α) under k is (φ, α◦k⁻¹). So there is an automorphism of φ

commutes. Since c is an automorphism of φ, it is an element of the group of units of the endomorphism ring End(φ). By Theorem 4.9 (2) in [9] the latter

is an order in a finite extension F^′ of F in which there is exactly one place ∞^′ above∞. In particular, End(φ) is contained in the integral closure A^′ ofAinF^′ which is equal to all elements of F^′ regular outside ∞^′. This implies c ∈ A^′∗, the identity, hence by the commutativity of the diagram c is the identity on I⁻¹·Λ/Λ. Therefore, for all x∈I⁻¹·Λ, we have (c−1)·x=cx−x∈Λ. This contradicts (c−1)·Λ = Λ becauseI⁻¹·Λ (Λ.

Hence, we have c = 1 and, by the commutative diagram above, k⁻¹ : (J⁻¹/A)^r →(J⁻¹/A)^r is the identity, i.e., k ∈ K(J) and [k] = 1∈ K/K(J).

So we have shown thatπ₁ :S_F,^r_K(J) →S_F,^r_K=S_F,^r_K(J)/Kis an ´etale cover. The groupK/K(J) injects into the automorphism group overF of this cover via [k]7→

π_k. Furthermore, by Corollary 3.13 in [26], an element of the automorphism group overF of this cover is uniquely determined by the image of one geometric point. Since the cover is of degree [K : K(J)] by Theorem 2.1.1 (note that K ∩F^∗ = {1} because ofK ⊂ K(I)), there are only [K :K(J)] possibilities for the image of one geometric point. Hence, the automorphism group overF of the cover must be equal to K/K(J) and the automorphism corresponding to a coset [k]∈ K/K(J) is given byπ_k. The cover is Galois with group K/K(J) (over F) because this group acts simply transitively on the geometric fibers.

Case (ii): LetK^′ be an arbitrary normal subgroup ofK, i.e.,K^′▹K ⊂ K(I). diagram, the varietyS_F,^r_Kis the quotient ofS_F,^r_K′ under this action. Furthermore, this action is free on the closed points of S_F,^r_K′ because K/K(J) acts freely on the closed points of S_F,K(J)^r . Therefore, we conclude by the same arguments

as above that π1 : S_F,^r_K′ → S_F,^r_K is an ´etale cover with group K/K^′ where the non-singular variety as explained in step (i) of the proof of Theorem 1.1.2.

Proposition 17.3.3.1 in EGA IV [19] says that ifX →Y is a flat, surjective morphism of schemes and X is regular, then Y is also regular. Therefore, S_F,^r_K and S_F,K^r ′ are both non-singular varieties.

By Proposition 10.4 in [22], a morphismf :X →Y of non-singular varieties of the same dimension over an algebraically closed field is ´etale if and only if, for every closed point x ∈ X, the induced map Tx → T_f(x) on Zariski tangent spaces is an isomorphism. We can apply this criterion because S_F,^r_K(J), S_F,^r_K and S_F,^r_K′ are all non-singular. Since the morphisms π₁ : S_F,^r_K(J) → S_F,^r_K′ and π1 :S_F,^r_K(J) →S_F,^r_K are ´etale, the commutativity of the above diagram therefore implies thatπ₁ :S_F,^r_K′ →S_F,^r_K is ´etale.

Corollary 2.1.4. If K ⊂ GL_r(A^f_F) is amply small, then the Drinfeld modular variety S_F,^r_K is non-singular.

Proof. See case (iii) of the above proof of Proposition 2.1.3.

Definition 2.1.5 (Hecke correspondence). For g ∈ GL_r(A^fF) and Kg := K ∩

is called the Hecke correspondence T_g associated tog. For subvarietiesZ ⊂S_F,^r_K we define

T_g(Z) :=π_g(π₁⁻¹(Z)).

Note that T_g(Z) is a subvariety of S_F,^r_K for any subvarietyZ ⊂S_F,^r_K because π_g is finite and hence proper. By Theorem 2.1.1, the degree of the morphism π₁ equals

deg(π₁) = [K: (K ∩g⁻¹Kg)·(K ∩F^∗)] = [K:K ∩g⁻¹Kg].

It is called thedegree degT_g of the Hecke correspondence T_g.

2.2 Inclusions and Drinfeld modular