4 γ А GL2pAqззззγ ¢ B ¢

MODULAR CURVE OF LEVEL Γ^∆₁pnq

SHIN HATTORI

Abstract. Letpbe a rational prime and q a power of p. Let n be a non-constant monic polynomial in Fqrts which has a prime factor of degree prime toq1. In this paper, we define a Drinfeld modular curveY₁^∆pnqoverAr1{nsand study the structure around cusps of its compactification X₁^∆pnq, in a parallel way to Katz- Mazur’s work on classical modular curves. Using them, we also define a Hodge bundle over X₁^∆pnq such that Drinfeld modular forms of level Γ₁pnq, weight k and some type are identified with global sections of itsk-th tensor power.

1. Introduction

Let p be a rational prime and q a power of p. Put A Fqrts, K₈ Fqpp1{tqqand letC8be thep1{tq-adic completion of an algebraic closure of K₈. We denote by Ω the Drinfeld upper half planeC8zK₈, which has a natural structure of a rigid analytic variety over K₈. Let n and℘ be monic polynomials inAsuch that℘ is irreducible of degree d¡0 and prime to n. We put

Γ₁pnq

"

γ PGL₂pAq γ

0 1

modn

*

and Γ₁₁pnq Γ₁pnqXSL₂pAq. LetK be the℘-adic completion ofF^qptq, which is a complete discrete valuation field with uniformizer ℘.

For any k P Z and l P Z{pq1q, a Drinfeld modular form of level Γ₁pnq, weight k and type l is a rigid analytic function f : Ω Ñ C8

satisfying f

az b cz d

padbcq^lpcz dq^kfpzq for any zP Ω,

a b c d

PΓ₁pnq and a certain holomorphy condition at cusps. It is a function field ana- logue of the notion of elliptic modular form of level Γ₁pNqand weightk.

As in the latter case, for any non-constant n, Drinfeld modular forms of level Γ₁pnq, weight k and type l can be considered as global sections

Date: October 2, 2019.

1

of the k-th tensor power of a natural line bundle ¯ω_C8 on an algebraic curve X₁₁pnqC8 over C8 called Drinfeld modular curve of level Γ₁₁pnq. The curveX₁₁pnqC8 is the compactification of an affine algebraic curve Y₁₁pnqC8 such that Y₁₁pnqC8pC8qis identified with Γ₁₁pnqzΩ.

We also have a ℘-adic version of the notion of Drinfeld modular form—℘-adic Drinfeld modular form [Vin, Gos2]. The latter is defined as the ℘-adic limit in Krrxss of Fourier expansions at 8 of Drinfeld modular forms with expansion coefficients in Fqptq. It is expected that Drinfeld modular forms have deep ℘-adic properties which are compa- rable top-adic properties of elliptic modular forms. Note that, in order to develop a geometric theory of ℘-adic Drinfeld modular forms such that each form is determined by the Fourier expansion at 8, we need to consider it over a Drinfeld modular curve which is geometrically connected. Thus, to investigate ℘-adic properties of Drinfeld modular forms, we need to define models X and ¯ω of X₁₁pnqC8 and ¯ω_C8 over Ar1{ns so that we can pass to O_K.

The problem is that, in the literature [Dri, Gos1, Gek1, Gek2, Gek3, vdPT, vdH, B¨oc, Pin], arithmetic compactifications of Drinfeld modu- lar curves are constructed by taking the quotient of the Drinfeld modu- lar curveXpnqof full level overAr1{ns by an appropriate group acting on it. Since this group action is not necessarily free at cusps (in fact, the element

1 1 0 1

PΓ₁₁pnq{Γpnq stabilizes 8), it is unclear if the Hodge bundle onXpnqdescends to a line bundle over a model X overAr1{ns constructed thereby. Though Pink [Pin] proved, at least on the generic fiber, that the descent of the Hodge bundle works for the case where the level structure is “fine”, for our situation the fineness condition means that the Drinfeld modular curve is not geometrically connected, and thus it is not suitable for studying ℘-adic Drinfeld modular forms.

To construct a Hodge bundle over a geometrically connected Drinfeld modular curve overAr1{ns, it seems that we need a more subtle study of the formal completion along cusps.

In this paper, we carry it out by following the method of Katz-Mazur [KM, (8.11)] in the case of classical modular curves. For this, we need to assume that the level n has a prime factor of degree prime toq1.

This ensures the existence of a subgroup ∆„ pA{pnqq which is a di- rect summand of Fq. Under this mild assumption, a Γ^∆₁ pnq-structure is defined as a pair of a usual Γ1pnq-structure and an additional struc- ture admitting an Fq-action. In particular, for any Ar1{ns-algebra R₀ which is an excellent regular ring, we have a fine moduli scheme Y₁^∆pnqR0 classifying Drinfeld modules with Γ^∆₁pnq-structures and also

its compactification X₁^∆pnqR0. Then we can show that X₁^∆pnqAr1{ns is a model of X₁₁pnqC8. It also enables us to control types of Drinfeld modular forms by a diamond operator [Hat]. On the other hand, for the profinite completion ˆA of A, the Γ^∆₁ pnq-structure corresponds to a compact open subgroupK₁^∆pnqofGL₂pAˆqwhich is not fine in the sense of Pink.

Let Cusps{^∆_R0 be the formal completion of X₁^∆pnqR0 along the cusps and Cusps^∆_R

0 its reduction. Then we will prove the following theorems.

Theorem 1.1(Theorem 5.3). LetR₀ be a flat Ar1{ns-algebra which is an excellent regular domain.

(1) Let P₈^∆ be the 8-cusp of X₁^∆pnqR0. Then there exists a natural isomorphism of complete local rings

px^∆₈q : ˆO_X^∆

1 pnqR0,P₈^∆ ÑR₀rrxss.

(2) The Hodge bundle on Y₁^∆pnqR0 extends to an invertible sheaf

¯

ω^∆_un on X₁^∆pnqR0 satisfying

px^∆₈qpω¯_un^∆q R₀rrxssdX,

where dX denotes an invariant differential form of a Tate- Drinfeld module TD^OpΛq.

(3) The formation ofω¯_un^∆ is compatible with any base change R₀ Ñ R¹₀ of flat Ar1{ns-algebras which are excellent regular domains.

(4) There exist natural actions of Fq on X₁^∆pnqR0 and on ω¯_un^∆ cov- ering the former action.

Theorem 1.2(Theorem 6.3). LetR₀ be a flat Ar1{ns-algebra which is an excellent regular domain. Let W_npXq be the n-th Carlitz cyclotomic polynomial [Car] and Rn the affine ring of a connected component of SpecpR₀rXs{pW_npXqqq. We also put

Γ¯¹₁

"

γ PSL₂pA{pnqq γ

1 0 1

modn

* .

(1) We have a natural isomorphism Cusps{^∆_R0 R0 R_n º

pa,bq

SpecpR_nrrwssq,

where the direct sum is taken over a complete representative of the set

Fqztpa, bq P pA{pnqq² | pa, bq p1qu{Γ¯¹₁.

(2) Cusps^∆_R0 is finite etale over R₀. In particular, it defines an effective Cartier divisor ofX₁^∆pnqR0 overR₀.

(3) For any pa, bq P pA{pnqq² satisfying pa, bq p1q, we denote by f_b the monic generator of the ideal Ann_ApbpA{pnqqq and by Φ^C_f

b

the f_b-multiplication map of the Carlitz module C. Then, at each point of Cusps^∆_R

0 in the component labeled by pa, bq, the invertible sheaf

Ω¹_X∆

1 pnqR0{R0p2Cusps^∆_R0q

is locally generated by the section dx{x², where x is defined by 1{xΦ^C_f

bp1{wq.

We also have similar results for the case of level Γ^∆₁ pnq XΓ0p℘q (§7).

For the proof of the above theorems, the main differences from [KM]

are twofold: First, thej-invariantj_tof the usual Tate-Drinfeld module, which is used to studyXpnq in the literature including [vdH], does not give (the inverse of) a uniformizer of thej-line at the infinity, contrary to the case of the Tate curve. For this, we use a descent TD^OpΛqof the Tate-Drinfeld module by an Fq-action on the coefficients to obtain a rightj-invariant (see (5.1)). This enables us to study Drinfeld modular curves directly by using a variant of [KM, Theorem 8.11.10], not via Xpnq, and thus to construct a Hodge bundle ¯ω_un^∆ on X₁^∆pnqR0 (§5).

As a trade-off, we need to consider Γ^∆₁ pnq-structures, not just Γ1pnq- structures, in order to kill an effect of the descent. The author learned the idea of the use of the descent from a work of Armana [Arm].

Second, since we are in the positive characteristic situation with wild ramification along cusps, we cannot use Abhyankar’s lemma to study the structure of X₁^∆pnqR0 and ¯ω^∆_un around cusps as in [KM, Theorem 8.6.8]. This is bypassed by a direct computation of the formal comple- tion along each cusp over R_n (§6).

In the paper [Hat], the above theorems are combined with a dual- ity theory of Taguchi [Tag] for Drinfeld modules of rank two, which compensates the lack of autoduality for Drinfeld modules, to develop a geometric theory of ℘-adic Drinfeld modular forms in a similar way to [Kat].

Acknowledgments. This work was supported by JSPS KAKENHI Grant Numbers JP26400016, JP17K05177.

2. Drinfeld modules

For any scheme S over F^q, we denote the q-th power Frobenius map on S by FS : S ÑS. For any S-scheme T and OS-module L, we put T^pqqT S,FSS andL^pqqF_SpLq. For anyA-schemeS, the image of tP A by the structure mapA ÑO_SpSq is denoted byθ.

For any schemeS overF^q and any invertibleOS-moduleL, we write the associated covariant line bundle to L as

VpLq Spec_SpSym_O

SpL^b1qq

with L^b1 : L^_ HomO_SpL,O_Sq. It represents the functor over S defined by T ÞÑL|TpTq, where L|T denotes the pull-back to T, and thus we identifyL withVpLq. We have theq-th power Frobenius map

τ :LÑL^bq, l ÞÑl^bq,

by which we identifyL^pqqwithL^bq. This map induces a homomorphism of group schemes over S

τ :VpLq ÑVpL^bqq.

Definition 2.1([Lau], Remark (1.2.2)). LetSbe a scheme overAand r a positive integer. A (standard) Drinfeld (A-)module of rank r over S is a pairE pL,Φ^Eqof an invertible sheafLonS and anFq-algebra homomorphism

Φ^E :AÑEnd_SpVpLqq satisfying the following conditions for any aPAzt0u:

the image Φ^E_a of a by Φ^E is written as Φ^E_a

rdeg¸paq i0

α_ipaqτⁱ, α_ipaq PL^b1qipSq with α_rdegpaqpaqnowhere vanishing.

α0paq is equal to the image of a by the structure map A Ñ O_SpSq.

We often refer to the underlying A-module scheme VpLq as E. A morphism pL,Φq Ñ pL¹,Φ¹q of Drinfeld modules over S is defined to be a morphism of A-module schemes VpLq ÑVpL¹q over S.

We denote the Carlitz module overS byC: it is the Drinfeld module pO_S,Φ^Cq of rank one over S defined by Φ^C_t θ τ. We identify the underlying group scheme of C with Ga Spec_SpO_SrZsq using 1POSpSq.

Lemma 2.2. (1) Let E be a line bundle over S. Let H be a finite locally free closed Fq-submodule scheme of E over S. Suppose that the rank of H is a constant q-power. Then E{H is a line bundle overS.

(2) Let E be a Drinfeld module of rank r. Let H be a finite locally free closed A-submodule scheme of E of constant q-power rank overS. Suppose either

H is etale over S, or

S is reduced and for any maximal point η of S, the fiber H_η of H overη is etale.

Then E{H is a Drinfeld module of rank r with the induced A- action.

Proof. The assertion (1) follows in the same way as [Leh, Ch. 1, Propo- sition 3.2]. For (2), we may assume that S SpecpBq is affine, the underlying invertible sheaves of E andE{H are trivial andH is free of rankqⁿ over S. We write the t-multiplication maps ofE and E{H as Φ^E_tpXq θX a₁X^q a_rX^qr, Φ^E_t^{HpXq b₀X b₁X^q b_sX^qs with b_s 0. From the proof of [Leh, Ch. 1, Proposition 3.2], we may also assume that the map E ÑE{H is defined by an Fq-linear monic additive polynomial

X ÞÑPpXq p₁X p_n1X^qn1 X^qn.

From the equality Φ^E_t^{HpPpXqq PpΦ^E_t pXqq, we obtainrs,br a^q_rⁿ andp1pb0θq 0. IfHis etale overB, then we havep1 PBand thus b₀ θ. If the latter assumption in the lemma holds, then p₁ P B is a non-zero divisor in the ring B{p for any minimal prime ideal p. Since B is reduced, it is a subring of±

B{p, where the product is taken over the set of minimal prime ideals p of B. This also yields b₀ θ, and thus E{H is a Drinfeld module of rank r in both cases.

Next let ℘ be a monic irreducible polynomial of degree d ¡ 0 in AFqrts, as before. Let ¯Sbe anA-scheme of characteristic℘and ¯E pL,¯ Φ^E¯q a Drinfeld module of rank two over ¯S. By [Sha, Proposition 2.7], we can write as

Φ^E_℘^¯ pα_dpE¯q α_2dpE¯qτ^dqτ^d, α_ipE¯q PL¯^b1qipS¯q. We put

F_d,E¯ τ^d: ¯E ÑE¯^pqdq, V_d,E¯ α_dpE¯q α_2dpE¯qτ^d : ¯E^pqdq ÑE.¯ We also denote them by F_d and V_d if no confusion may occur. They are isogenies of Drinfeld modules satisfyingV_dF_dΦ^E_℘^¯ andF_dV_d Φ^E_℘^¯pqdq [Sha,§2.8].

Definition 2.3. We say ¯E is ordinary ifα_dpE¯q PL¯^b1qdpS¯qis nowhere vanishing, and supersingular if α_dpE¯q 0.

By [Sha, Proposition 2.14], ¯E is ordinary if and only if KerpV_dq is etale.

3. Drinfeld modular curves

Let n be a non-constant monic polynomial in A Fqrts which is prime to ℘. Put A_n Ar1{ns. For any Drinfeld module E of rank two over an A-scheme S and a non-constant monic polynomial mPA, a Γpmq-structure on E is an A-linear homomorphism α :pA{pmqq² Ñ EpSqinducing the equality of effective Cartier divisors of E

¸

aPpA{pmqq²

rαpaqs Erms.

If m is invertible in S, then it is the same as an isomorphism of A- module schemes α : pA{pmqq² Ñ Erms over S, where the underline means the constant A-module scheme. If m has at least two different prime factors, then the functor over A sending S to the set of isomor- phism classes of such pairs pE, αq over S is represented by a regular affine schemeYpmqof dimension two which is flat and of finite type over A. Over Ar1{ms, for any non-constant m this functor is representable by an affine scheme Ypmq which is smooth of relative dimension one over Ar1{ms. The natural left action of GL₂pA{pmqq on pA{pmqq² in- duces a right action of this group on Ypmq.

For any Drinfeld moduleE of rank two over anA_n-schemeS, we de- fine a Γ₁pnq-structure onE as a closed immersion ofA-module schemes λ : Crns Ñ E over S. Since Crns is etale over S, we see that over a finite etale cover ofA_n a Γ₁pnq-structure onEis identified with a closed immersion of A-module schemes A{pnq Ñ E. Then [Fli, Proposition 4.2 (2)] implies thatE has no non-trivial automorphism fixingλ. Note that the quotient Erns{Impλqis a finite etale A-module scheme overS which is etale locally isomorphic to A{pnq, and thus the functor

Isom_A,SpA{pnq, Erns{Impλqq

is represented by a finite etale pA{pnqq-torsor I_pE,λq overS.

Consider the functor over A_n sending an A_n-scheme S to the set of isomorphism classes rpE, λqs of pairs pE, λq consisting of a Drinfeld module E of rank two over S and a Γ₁pnq-structure λ on E. Then we can show that this functor is representable by an affine scheme Y₁pnq which is smooth over A_n of relative dimension one.

Suppose that there exists a prime factorq of n such that its residue extensionkpqq{Fq is of degree prime toq1. In this case, the inclusion Fq Ñkpqq splits and we can choose a subgroup ∆ „ pA{pnqq such that the natural map ∆ Ñ pA{pnqq{Fq is an isomorphism. For such

∆, we define a Γ^∆₁ pnq-structure on E as a pair pλ,rµsq of a Γ₁pnq- structure λ on E and an element rµs P pI_pE,λq{∆qpSq. We have a fine

moduli scheme Y₁^∆pnq of the isomorphism classes of triples pE, λ,rµsq, which is finite etale over Y₁pnq. The universal Drinfeld module over Y₁^∆pnqis denoted by E_un^∆ VpL^∆_unq and put

ω_un^∆ :ω_Eun^∆ pL^∆_unq^_,

where ω_Eun^∆ denotes the sheaf of invariant differential forms on E_un^∆. For any Drinfeld module E over an A_n-scheme S, a Γ₀p℘q-structure on E is a finite locally free closed A-submodule scheme G of Er℘s of rankq^doverS. Then we have a fine moduli schemeY₁^∆pn, ℘qclassifying tuplespE, λ,rµs,Gqconsisting of a Drinfeld moduleE of rank two over anAn-schemeS, a Γ^∆₁pnq-structure pλ,rµsqand a Γ0p℘q-structureG on E. From the theory of Hilbert schemes, we see that the natural map Y₁^∆pn, ℘q ÑY₁^∆pnq is finite, and it is also etale over A_nr1{℘s. For any A_n-algebra R, we write asY₁^∆pnqRY₁^∆pnq AnSpecpRqand similarly for other Drinfeld modular curves.

A Drinfeld modular curve similar to Y₁^∆pn, ℘q is also studied in [Gek3]. In particular, in [Gek3, p. 232], it is claimed that we can argue as in [DR, VI, Th´eor`eme 6.9] to obtain its Drinfeld analogue. However, the proof of the key lemma [DR, V, Lemme 1.12] seems incorrect as pointed out by [Buz]. Here we give a proof in our case.

Lemma 3.1. The natural map π :Y₁^∆pn, ℘q Ñ Y₁^∆pnq is flat.

Proof. Note that Y₁^∆pnq is reduced. By [DR, V, Lemme 1.13], it is enough to show that the rank of geometric fibers of π is constant. At any geometric point of characteristic different from ℘, the map π is finite etale of rank q^d 1.

Consider fibers over p℘q. Let k be an algebraically closed field con- taining kp℘q A{p℘q and E a Drinfeld module over k with some Γ^∆₁pnq-structure. Then E defines a geometric point y P Y₁^∆pnqkp℘qpkq. We denote byZ_y the fiber ofπ overy. Note that, ifE is ordinary, then the argument of [DR, DeRa-97, b1)] works verbatim to show that Z_y is of rank q^d 1.

Suppose that E is supersingular. By [Sha, Definition 2.11], the Fq- module scheme Er℘s is isomorphic to the additive Fq-module scheme (3.1) SpecpkrXs{pX^q2dqq.

Note thatEr℘sis a finiteϕ-module [Tag, Definition (1.3)] (for this, see for example [Hat, Lemma 2.6]). Let pM, ϕMqbe the finiteϕ-sheaf over k corresponding to Er℘s. Then the k-vector space M is of dimension 2dand endowed with ak-linearkp℘q-action which commutes withϕ_M. For any iP Z{dZ, we denote by M_i the maximal k-subspace of M on

which kp℘q acts via

kp℘q QaÞÑa^qi Pk.

Then we have ϕ_MpM_iq „M_{i 1}.

Since (3.1) implies ϕ^2d_M 0 andϕ^2d_M¹ 0, there exist hPZ{dZ and xPM_hsatisfyingϕ^2d_M¹pxq 0. We claim that for anyiP t0, . . . , d1u, the non-zero elementsϕⁱ_Mpxqandϕ^{i d}_M pxqare linearly independent over k. Indeed, if we have ϕ^{i d}_M pxq bϕⁱ_Mpxq with some b P k, then the k-subspace generated by ϕⁱ_Mpxq, ϕⁱ_M¹pxq, . . . , ϕ^{i d}_M pxq gives a finite ϕ- subsheaf of M. It corresponds to the finite etale group scheme over k defined by the equation T^qd bT appearing as a quotient of Er℘s, which contradicts (3.1). Thus the set

tX_i :ϕⁱ_Mpxq |i0, . . . ,2d1u

forms a basis of M satisfying ϕ_MpX_iq X_{i 1} and ϕ_MpX_2d1q 0.

Moreover, we have M_{i h} kX_i `kX_{i d} for any i0, . . . , d1.

Note thatN À_2d1

id kX_i is ak-subspace ofM of dimensiondwhich is stable under ϕ_M and the kp℘q-action. Since the only factor of X^q2d of degreeq^dinkrXsisX^qd, the finiteϕ-module overkcorresponding to M{N gives the unique Γ₀p℘q-structure onE. Thus we have 7Z_ypkq 1 and, sinceZ_y is finite overk, it is local. HenceZ_y is determined by the valued points over local schemes which are finite over k.

LetpB, mBqbe any local ring which is finite overk. SinceB is finite, we may identify its residue field withk. We putM_B MbkB, which has a natural structure of a finite ϕ-sheaf over B with kp℘q-action induced by that of M. Here, for the q-th power Frobenius map σ_B on B, the Frobenius structure onM_Bis given byϕ_MB ϕ_Mbσ_B. By [Hat, Lemma 2.6], to give a Γ₀p℘q-structure on E_B E k SpecpBq is the same as to give a finite ϕ-subsheaf ˆN of M_B stable under kp℘q-action which is a direct summand of rank d as a B-module. By 7Z_ypkq 1, the finite ϕ-sheaf ˆN bBk agrees with N.

Since kp℘qacts k-linearly on ˆN, it is decomposed as Nˆ à

iPZ{dZ

Nˆ_i,

where ˆN_i is the maximal B-submodule on which kp℘q acts via a ÞÑ a^{qi h}. For any id, . . . ,2d1, we have ˆN_ibBk kX_i.

Take any lift ˆX_dP Nˆ₀ of the element X_d PN. By ˆN₀ „M_hbkB, it is written uniquely as

Xˆ_db₀pX₀b1q b_dpX_db1q, b₀, b_dPB,

where b0 P mB and bd 1 modmB. Since bd is a unit in B, replacing Xˆd with a scalar multiple we may assume bd1.

By Nakayama’s lemma, the elements (3.2) ϕⁱ_M

BpXˆ_dq b^q₀ⁱpX_ib1q X_{d i}b1PN ,ˆ i0, . . . , d1 form a basis of theB-module ˆN. For i0, . . . , d1, we have

ϕ^{d i}_M

BpXˆdq b^q₀^{d i}pXd ib1q

and they are linear combinations of the elements (3.2) if and only if b^q₀^{d 1} 0. Thus, to give ˆN is the same as to give an element b P B satisfying b^{qd 1} 0, via

bÞÑNˆ ^dà¹

i0

Bϕⁱ_M

BpbpX₀ b1q X_db1q. Hence Z_y is identified with

SpecpkrTs{pT^{qd 1}qq,

which is of rankq^d 1. This concludes the proof.

Lemma 3.2. Y₁^∆pn, ℘qis smooth over A_n outside finitely many super- singular points on the fiber over p℘q.

Proof. LetB be an Artinian local A_n-algebra of characteristic℘ and J an ideal ofB satisfyingJ² 0. Let E be an ordinary Drinfeld module of rank two over B{J and G a Γ₀p℘q-structure onE. Since B is local, the underlying invertible sheaf ofEis trivial. It is enough to show that the isomorphism class of the pair pE,Gq lifts to B.

Since E is ordinary and B{J is Artinian local, we have either G KerpF_d,Eq or the compositeGÑEr℘s ÑKerpV_d,Eqis an isomorphism.

In the former case, write as Φ^E_t θ a₁τ a₂τ². For any lift ˆa_i P B of a_i, we can define a structure of a Drinfeld module of rank two over B on ˆE SpecpBrXsq by putting Φ^E_t^ˆ θ ˆa₁τ ˆa₂τ², which is also ordinary. Then G lifts to KerpF_d,Eˆq. In the latter case G is etale and, by Lemma 2.2 (2), E{G has a structure of a Drinfeld module of rank two. Moreover, it is also ordinary sincepE{Gqr℘shas the etale quotient G. Thus we have isomorphisms

pE{Gq^{pqdqFoo d,E{ G}pE{Gq{KerpF_d,Eq ^{℘ //}E

sending KerpV_d,E{Gq to G. Since the above argument shows that E{G also lifts to an ordinary Drinfeld module ˆF of rank two over B, the pair pE,Gqlifts to the pair pFˆ^pqdq,KerpV_d,Fˆqq over B.

Remark 3.3. As in [DR, DeRa-99], Lemma 3.1 implies thatY₁^∆pn, ℘q is Cohen-Macaulay. Moreover, combined with Lemma 3.2, it also im- plies that Y₁^∆pn, ℘q is normal, and thus it agrees with the quotient of Ypn℘qby a finite group, as considered in [Gek3].

Put K₈ Fqpp1{tqq and let C8 be the p1{tq-adic completion of an algebraic closure of K₈. Let A^f be the ring of finite adeles (namely, the restricted direct product over the set of places of F^qptq other than the p1{tq-adic one) and ˆA its subring of elements which are integral at all finite places. Let Ω be the Drinfeld upper half plane over C8. Put

K₁^∆pnq

"

g P GL₂pAˆq

g modnAˆP

∆ A{pnq

0 1

* ,

Γpnq

"

g P GL₂pAq

g modpnq 1 0

0 1

*

and Γ^∆₁pnq GL₂pAq X K₁^∆pnq. Since A Fq, we have Γ^∆₁pnq „ SL₂pAq. This yields

Γ^∆₁pnq

"

g PSL₂pAq

g modpnq P

1 A{pnq

0 1

* .

In particular, the group Γ^∆₁ pnqis independent of the choice of ∆. Note that the natural right action ofg PGL₂pA{pnqqonYpnqC8 corresponds to the left action of^tg on ΓpnqzΩ via the M¨obius transformation. Since Fq detpK₁^∆pnqq Aˆ, [Dri, Proposition 6.6] implies that the analytifi- cation of Y₁^∆pnqC8 is identified with

GL₂pFqptqqzΩGL₂pAfq{K₁^∆pnq Γ^∆₁ pnqzΩ,

and thus the fiber Y₁^∆pnqK₈ is geometrically connected. Similarly, we see thatY₁^∆pn, ℘qK₈ is also geometrically connected.

For any Drinfeld module E of rank two over S, we write the t- multiplication map of E as Φ^E_t θ a₁τ a₂τ² and put

j_tpEq a^b₁^{q 1}ba^b₂ ¹ P O_SpSq. Consider the finite flat map

j_t :Y₁^∆pnq Ñ A¹An SpecpA_nrjsq, j ÞÑj_tpE_un^∆q

and a similar finite map for Y₁^∆pn, ℘q. We define the compactifications X₁^∆pnq and X₁^∆pn, ℘q of Y₁^∆pnq and Y₁^∆pn, ℘q as the normalizations of P¹An inY₁^∆pnqandY₁^∆pn, ℘qvia this map, respectively. As in [Sha,§7.2], we see that X₁^∆pnq is smooth over An and X₁^∆pn, ℘q is smooth over A_nr1{℘s. By a similar argument to the proof of [KM, Corollary 10.9.2], Zariski’s connectedness theorem implies that each fiber of the map

X₁^∆pnq Ñ SpecpAnq is geometrically connected, and so is X₁^∆pn, ℘q Ñ SpecpA_nr1{℘sq. For any A_n-algebra R which is Noetherian, excellent and regular, we also have the compactificationsX₁^∆pnqRandX₁^∆pn, ℘qR

of Y₁^∆pnqR and Y₁^∆pn, ℘qR. From the smoothness of X₁^∆pnq, we have X₁^∆pnqR X₁^∆pnq An SpecpRq. The base change compatibility also holds for X₁^∆pn, ℘qR if ℘ is invertible inR.

On the other hand, the maps

rpE, λ,rµsqs ÞÑ rpE, aλ,rµsqs, rpE, λ,rµsqs ÞÑ rpE, λ, crµsqs induce actions of the groupspA{pnqqandpA{pnqq{∆Fq onX₁^∆pnqR. We denote them by xayn and xcy∆, respectively.

Lemma 3.4. Let S be a scheme over A and E a Drinfeld module of rank two over S. If j_tpEq P O_SpSq is invertible, then for the big fppf sheaf Aut_A,SpEq defined by

T ÞÑAut_A,TpE|Tq,

the natural map Fq ÑAut_A,SpEq is an isomorphism.

Proof. We may assume thatS SpecpBq is affine and the underlying invertible sheaf of E is trivial. By [Fli, Proposition 4.2 (2)], any auto- morphism ofE SpecpBrXsqis linear, namely it is given byX ÞÑbX for somebP B. Write as Φ^E_t θ a₁τ a₂τ². From the assumption, we have a₁ P B and the equality Φ^E_t pbXq bΦ^E_tpXq yields b^q1 1.

Since the group scheme µ_q1 over Fq is isomorphic to the constant group scheme Fq, so is µ_q1|B over the Fq-algebra B. This concludes

the proof.

Lemma 3.5. Let S be a scheme over A. Let E and E¹ be Drinfeld modules of rank two over S satisfying j_tpEq j_tpE¹q P O_SpSq. Then the big fppf sheaf IsomA,SpE, E¹q over S defined by

T ÞÑIsom_A,TpE|T, E¹|Tq

is represented by a Galois covering of S with Galois group Fq.

Proof. By gluing, we reduce ourselves to the case where S SpecpBq is affine and the underlying line bundles of E and E¹ are trivial. We write the t-multiplication maps of E and E¹ as

Φ^E_t θ a₁τ a₂τ², Φ^E_t¹ θ a¹₁τ a¹₂τ²

with some a₁, a¹₁ P B and a₂, a¹₂ P B. By assumption, we have a^q₁ ¹{a2 pa¹₁q^{q 1}{a¹₂ PB and thus a1, a¹₁ PB. Hence the scheme

J SpecpBrYs{pY^q1a₁{a¹₁qq

is a finite etaleFq-torsor overB. ByY ÞÑ pX ÞÑY Xq, we obtain a map of functors J Ñ Isom_A,SpE, E¹q. To show that it is an isomorphism, we may prove it over J. In this case, it follows from Lemma 3.4.

4. Tate-Drinfeld modules

To investigate the structure around cusps of Drinfeld modular curves and extend the sheafω^∆_un, we need to introduce Tate-Drinfeld modules.

Let R₀ be a flat A_n-algebra which is an excellent Noetherian domain with fraction field K₀. Let R₀ppxqq and K₀ppxqqbe the Laurent power series rings over R₀ and K₀, respectively. Put T₀ SpecpR₀ppxqqq. We denote the normalizedx-adic valuation on K₀ppxqqbyv_x. We also denote the ring of entire series over K₀ppxqqbyK₀ppxqqttXuu; it is the subring of K₀ppxqqrrXss consisting of elements °

i¥0a_iXⁱ satisfying

ilimÑ8pv_xpa_iq iρq 8 for any ρPR. We putR₀rrxssttXuu K₀ppxqqttXuu XR₀rrxssrrXss.

LetpC,Φ^Cqbe the Carlitz module overR₀. For any non-zero element f PA, put

fΛ

"

Φ^C_{f a} 1

x aPA

*

„R₀ppxqq, (4.1)

e_fΛpXq X ¹

α0PfΛ

1 X

α

PX xX²R₀rrxssrrXss (4.2)

as in [Leh, Ch. 5,§2]. Note that any non-zero element offΛ is invertible in R₀ppxqq. We consider fΛ as anA-module via Φ^C. Then it is a free A-module of rank one, and it is also discrete insideK₀ppxqq. Hence the power series e_fΛpXq is entire, and it is an element ofR₀rrxssttXuu.

Put

F_fpxq 1

Φ^C_{f x}¹ Px^qdegpfqFqp1 xR₀rrxssq.

Then xÞÑF_fpxqdefines an R₀-algebra homomorphism ν_f⁷ :R₀ppxqq Ñ R₀ppxqq and a map ν_f : T₀ Ñ T₀. For any element hpXq °

ia_iXⁱ P R₀ppxqqrrXss, we put ν_fphqpXq °

iν_f⁷pa_iqXⁱ. Then we have ν_f⁷pΛq fΛ and ν_fpe_ΛqpXq e_fΛpXq.

For any elementa PA, consider the power series (4.3) Φ^fΛ_a pXq e_fΛpΦ^C_ape_fΛ¹pXqqq PR₀rrxssrrXss. Note that (4.2) yields

(4.4) Φ^f_a^ΛpXq Φ^C_apXqmodxR₀rrxss for any aPA.

LetK0ppxqq^alg be an algebraic closure of K0ppxqq. For any a PA, put pΦ^C_aq¹pfΛq tyPK₀ppxqq^alg |Φ^C_apyq PfΛu,

which is an A-module, and let Σ_a „ pΦ^C_aq¹pfΛqbe a representative of the set

ppΦ^C_aq¹pfΛq{fΛqzt0u. Since R₀ is flat over A, we have

(4.5) Φ^f_a^ΛpXq aX ¹

βPΣa

1 X

e_fΛpβq

(see for example the proof of [B¨oc, Proposition 2.9]). In particular, it is an F^q-linear additive polynomial of degree q^{2 degpaq}.

Lemma 4.1. If we write asΦ^Λ_t θ a1τ a2τ² for someai PR0rrxss, then we have

a1 P1 xR0rrxss, a2 Px^q1R0rrxss.

Proof. The assertion on a₁ follows from (4.4). That ona₂ is proved by the computation in the proof of [B¨oc, Lemma 2.10]. Indeed, we choose a root ηP K₀ppxqq^alg of the equation

Φ^C_tpXq θX X^q 1 x.

Put ˜Σ tcη | c P Fqu, Σ₀ tζ P K₀ppxqq^alg | Φ^C_t pζq 0u and Σ_t pΣ˜ Σ₀q Y pΣ₀zt0uq. By (4.5), we have a₂ θ{p±

βPΣte_Λpβqq. The denominator ±

βPΣteΛpβq is equal to

¹

βPΣ˜

¹

ζPΣ0

pβ ζq ¹

α0PΛ

α pβ ζq α

¹

ζPΣ0zt0u

ζ ¹

α0PΛ

αζ α

.

The first term is equal to

¹

βPΣ˜

Φ^C_tpβq ¹

α0PΛ

Φ^C_t pαβq α^q

±

cPFq c

x^q1 ¹

cPFq

¹

α0PΛ

θα α^q_x^c α^q .

By the definition (4.1) of Λ, any α 0 P Λ can be written as α Φ^C_ap1{xqfor somea 0PA. Thus we have αx^qrhwith rdegpaq and h P R₀rrxss, which yields pθα α^qc{xq{α^q P 1 xR₀rrxss. By a similar computation, the second term is equal to

θ ¹

α0PΛ

θ α^q1

α^q1 Pθp1 xR₀rrxssq.

Hence we obtain the assertion on a₂.

Using Lemma 4.1 and the map νf, we see that the polynomials Φ^fΛ_a define a structure of a Drinfeld module of rank two over T₀. We refer to it as the Tate-Drinfeld module TDpfΛq over T₀.

Lemma 4.2. For any monic polynomialmP A, there exists a natural A-linear closed immersion λ^fΛ_8,m : Crms Ñ TDpfΛq over T₀ satisfying ν_fpλ^Λ_8,mq λ^f₈^Λ_,m. In particular, the Tate-Drinfeld module TDpfΛq is endowed with a natural Γ₁pnq-structure λ^f₈^Λ_,n over T₀.

Proof. LetR₀rrxssxZy be thex-adic completion of the ringR₀rrxssrZs. We have a natural map

i:R₀rrxssrZs{pΦ^C_mpZqq ÑR₀rrxssxZy{pΦ^C_mpZqq.

Since Φ^C_mpZq P R₀rZs is monic, the ring on the left-hand side is finite over the x-adically complete Noetherian ring R₀rrxss. Hence this ring is also x-adically complete and the map i is an isomorphism. Since R₀rrxssttZuu „R₀rrxssxZy, the map

R₀rrxssrXs ÑR₀rrxssttZuu, X ÞÑe_fΛpZq induces a homomorphism of Hopf algebras

R0ppxqqrXs ÑR0rrxssxZyr1{xs{pΦ^C_mpZqqⁱѹ R0ppxqqrZs{pΦ^C_mpZqq, which we denote bypλ^f₈^Λ_,mq. In the ringRrrxssxZy, we have Φ^fΛ_a pe_fΛpZqq e_fΛpΦ^C_apZqq for any a P A and this implies that the map pλ^f₈^Λ,mq is compatible with A-actions. Thus we obtain a homomorphism of finite locally free A-module schemes over T0

λ^fΛ_8,m :Crms ÑTDpfΛqrms which is compatible with the map νf.

To prove that it is a closed immersion, it is enough to show that the map R₀rrxssrXs ÑR₀rrxssxZy{pΦ^C_mpZqq defined by X ÞÑe_fΛpZq is surjective. Since the right-hand side is x-adically complete, it suffices to show the surjectivity modulo x, which follows from (4.2).

Lemma 4.3. LetDbe any finite flat R₀ppxqq-algebra whose restriction toFracpR₀ppxqqq is etale, andδany element ofD. LetDbe the integral closure of R₀rrxss in D. We consider D as a topological ring by taking tx^lDulPZ¥0 as a fundamental system of neighborhoods of 0P D. Then, for any FpXq P R₀ppxqqttXuu, the evaluation Fpδq converges for any δ PD. In particular, we have an Fq-linear map e_fΛ :D ÑD which is functorial on D.

Proof. We haveDr1{xs D. SinceR₀is excellent, so is the power series ring R₀rrxss. Thus D is finite over R₀rrxss and x-adically complete.

(Here the fact that R0rrxss is excellent follows from an unpublished work of Gabber [KS, Main Theorem 2]. If we assume thatR₀is regular, then the finiteness ofD follows from [Mat, Proposition (31.B)]. This is the only case we need.) This implies that the evaluationFpδqconverges

and defines an element of D.

PutH₈^fΛ,m TDpfΛqrms{Impλ^fΛ₈,mq and

(4.6) B_0,m^fΛ R0ppxqqrηs{pΦ^C_mpηq Φ^C_fp1{xqq.

Then SpecpB_0,m^fΛq is a finite flat Crms-torsor over T₀. Sincemis invert- ible in K₀, it is etale over FracpR₀ppxqqq.

Lemma 4.4. For any monic polynomial m P A, there exists an A- linear isomorphism µ^fΛ_8,m :A{pmq ÑH₈^fΛ_,m which is compatible with the map ν_f such that the image of µ^f₈^Λp1q P H₈^fΛ_,mpT₀q in H^fΛ_8,mpB_0,m^fΛq is equal to the image efΛpηq of the element efΛpηq P TDpfΛqrmspB_0,m^fΛq. In particular, we have an exact sequence of A-module schemes over T0

(4.7) 0 ^//Crms ^λ

fΛ8,m//TDpfΛqrms ^π

fΛ8,m//A{pmq ^//0.

Proof. By Lemma 4.3, we have an element e_fΛpηq PTDpfΛqrmspB_0,m^fΛq. Since its image efΛpηq in H^f₈^Λ,mpB_0,m^fΛq is invariant under the action of Crms on B_0,m^fΛ, we obtain e_fΛpηq P H^fΛ₈,mpT₀q. This yields an A-linear homomorphism A{pmq Ñ H^fΛ₈,m over T₀ which is compatible with the map ν_f.

To see that it is an isomorphism, using the map ν_f we reduce our- selves to the case of f 1. Since the element m is invertible in K0, using co-Lie complexes we obtain the exact sequence

0 ^//ω_H^Λ

8,m //ω_TDpΛqrms^pλ

Λ8,mq//ω_Crms ^// 0.

We also see that the natural sequence

0 ^//ω_TDpΛq ^{m //}ω_TDpΛq ^//ω_TDpΛqrms ^//0

is exact and similarly forCrms. Since we havedpe_ΛpZqq dZ, the map pλ^Λ_8,mq is an isomorphism. Hence ω_H^Λ

8,m 0 andH^Λ_8,m is etale.

Now it is enough to show ae_Λpηq 0 in H^Λ_8,mpB_0,m^fΛq for any non- zero element a P A{pmq. For this, we may assume R₀ K₀. In this case, note that the polynomial Φ^C_mpXq 1{x is irreducible over K0ppxqq, since the equation Φ^C_mp1{Xq 1{x gives an Eisenstein exten- sion over K0rrxss. Hence we may consider the ring B_0,m^Λ as a subfield

of K0ppxqq^alg. Let ˆa P A be a lift of a satisfying degpˆaq   degpmq. The condition aeΛpηq 0 implies Φ^C_ˆapηq ζ mod Λ for some root ζ of Φ^C_mpXq in K0ppxqq^alg. By inspecting x-adic valuations it forces ζ 0, and the irreducibility of Φ^C_mpXq 1{x implies ˆa 0. This concludes

the proof.

We often write λ^Λ_8,n, H^Λ_8,n, B_0,n^Λ , µ^Λ_8,n and π_8,n^Λ as λ₈, H₈, B₀, µ₈ and π₈, respectively.

PutS₀ SpecpR₀ppyqqq and consider the morphism σ_q1 :T₀ ÑS₀

defined by y ÞÑ x^q1. The S₀-scheme T₀ is a finite etale Fq-torsor, where cP Fq acts on it by the R₀-linear map

g_c:R₀ppxqq ÑR₀ppxqq, xÞÑc¹x.

Since Λ is stable under this Fq-action, we see that the coefficients of e_ΛpXq and Φ^Λ_apXq are in R₀rrx^q1ss for any a P A [Arm, §5C1]. This means that there exists a unique pair of a Drinfeld module and its Γ₁pnq-structure over S₀

pTD^OpΛq, λ^O₈q

satisfyingσ_q1pTD^OpΛq, λ^O₈q pTDpΛq, λ₈q. OverT₀, the Tate-Drinfeld module TD^OpΛq|T0 TDpΛq has a Γ^∆₁pnq-structure

pTDpΛq, λ₈,rµ₈sq

with the element rµ₈s P pI_pTDpΛq,λ8q{∆qpT0q defined by µ₈. We also put

H₈^O TD^OpΛqrns{Impλ^O₈q, I₈^O Isom_A,S0pA{pnq,H^O₈q.

Lemma 4.5. There exists an isomorphism of finite etale Fq-torsors over S0

T0 ÑI₈^O{∆.

Proof. It is enough to give an Fq-equivariant morphism T₀ ÑI₈^O over S0, which amounts to giving anA-linear isomorphismµ:A{pnq ÑH₈ overT₀ satisfying cµg_cpµq for anycPFq. The mapg_c extends to a similarR0ppxqq-linear isomorphism onB0 viaη ÞÑcη, which we denote by ˜gc. Then the inclusion H₈pR0ppxqqq Ñ H₈pB0q is compatible with g_c and ˜g_c. Consider the isomorphism µ₈ of Lemma 4.4. We have

˜

g_cpe_Λpηqq e_Λpcηq inB₀ and this yields cµ₈ g_cpµ₈q.