on the compactification of the drinfeld modular curve of level γ

ON THE COMPACTIFICATION OF THE DRINFELD 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 toq´1. 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 dą0 and prime to n. We put

Γ₁pnq “

"

γ PGL₂pAq ˇ ˇ ˇ ˇ

γ ” ˆ˚ ˚

0 1

˙

modn

*

and Γ₁₁pnq “Γ₁pnq XSL₂pAq. LetK be the℘-adic completion ofFqptq, which is a complete discrete valuation field with uniformizer ℘.

For any k P Z and l P Z{pq ´1q, a Drinfeld modular form of level Γ1pnq, weight k and type l is a rigid analytic function f : Ω Ñ C8

satisfying f

ˆaz`b cz`d

˙

“ pad´bcq^´lpcz`dq^kfpzq for any zP Ω,

ˆa 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 and weight k are identified with global sections of the

Date: October 19, 2017.

1

k-th tensor power of a natural line bundle ¯ω_C8 on an algebraic curve X₁₁pnq_C8 over C8 called Drinfeld modular curve of level Γ₁₁pnq. The curve X₁₁pnq_C8 is the compactification of a moduli space Y₁₁pnq_C8 of Drinfeld modules of rank two endowed with some level structures such that Y₁₁pnq_C8pC8q is 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.

To investigate ℘-adic properties of Drinfeld modular forms, we need to define modelsX and ¯ωof X11pnq_C8 and ¯ω_C8 overAr1{ns in order to pass to OK. The problem is that, the study around cusps of Drinfeld modular curves in the literature [Dri, Gos1, Gek1, Gek2, Gek3, vdPT, vdH, B¨oc] is carried out by, first describing the formal completion for the case of the Drinfeld modular curve Xpnq of full level over Ar1{ns and then taking the quotient by an appropriate group acting onXpnq. Since this group action is not necessarily free at cusps (in fact, the element

ˆ1 1 0 1

˙

PΓ₁₁pnq{Γpnq stabilizes 8), it is unclear if the Hodge bundle on Xpnq descends to a model X over Ar1{ns and we need a more precise study of the formal completion along cusps.

In this paper, we resolve it by following the method of Katz-Mazur [KM] in the case of classical modular curves. For this, we need to assume that the level n has a prime factor of degree prime to q´1.

This ensures the existence of a subgroup ∆Ď pA{pnqq^ˆ which is a di- rect summand of F^ˆq. 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 F^ˆq-action. In particular, for any Ar1{ns-algebra R₀ which is an excellent regular ring, we have a fine moduli scheme Y₁^∆pnq_R0 classifying Drinfeld modules with Γ^∆₁pnq-structures and also its compactification X₁^∆pnq_R0. Then we can show that X₁^∆pnq_Ar1{ns is a model of X₁₁pnq_C8. It also enables us to control types of Drinfeld modular forms by a diamond operator [Hat].

Let Cusps{^∆_R

0 be the formal completion of X₁^∆pnqR0 along the cusps and Cusps^∆_R0 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^˚ : ˆOX₁^∆pnqR0,P₈^∆ ÑR₀rrxss.

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

¯

ω^∆_un on X₁^∆pnqR0 satisfying

px^∆₈q^˚pω¯_un^∆q “R₀rrxssdX,

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

(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 F^ˆq on X₁^∆pnq_R0 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 R_n the affine ring of a connected component of SpecpR₀rXs{pW_npXqqq. We also put

Γ¯¹₁ “

"

γ PSL₂pA{pnqq ˇ ˇ ˇ ˇ

γ ” ˆ1 0

˚ 1

˙

modn

* .

(1) We have a natural isomorphism

Cusps{^∆_R0 ˆR0 R_n » ž

pa,bq

SpecpR_nrrwssq,

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

F^ˆqztpa, bq P pA{pnqq² | pa, bq “ p1qu{Γ¯¹₁. (2) Cusps^∆_R

0 is finite etale over R₀. In particular, it defines an effective Cartier divisor of X₁^∆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{x“Φ^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_t of the usual Tate-Drinfeld module does not give (the inverse of) a uniformizer of the j-line at the infinity, contrary to the case of the Tate curve. For this, we use a descent TD^▽pΛqof the Tate-Drinfeld module by anF^ˆq-action on the coefficients to obtain a right j-invariant (see (5.1)). This enables us to study Drinfeld modular curves directly in §5, not via taking quotients of Xpnq. As a trade-off, we need to consider Γ^∆₁pnq-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 Drinfeld modular curves around cusps. This is bypassed by a direct computation of the formal completion 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 Fq, 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^pqq“T ˆS,FSS andL^pqq“F_S^˚pLq. For anyA-schemeS, the image of tP A by the structure mapA ÑOSpSq is denoted byθ.

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

V˚pLq “Spec_SpSym_OSpL^b´1qq

with L^b´1 :“ L^_ “ Hom_OSpL,OSq. It represents the functor over S defined by T ÞÑL|TpTq, where L|T denotes the pull-back to T, and thus we identifyL withV˚pLq. 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 :V˚pLq ÑV˚pL^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_SpV˚pLqq satisfying the following conditions for any aPAzt0u:

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

rdegpaq

ÿ

i“0

αipaqτⁱ, αipaq PL^b1´qipSq with α_rdegpaqpaqnowhere vanishing.

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

We often refer to the underlying A-module scheme V˚pLq as E. A morphism pL,Φq Ñ pL¹,Φ¹q of Drinfeld modules over S is defined to be a morphism of A-module schemes V˚pLq ÑV˚pL¹q over S.

We denote the Carlitz module overS byC: it is the Drinfeld module pOS,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_SpOSrZsq 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 Hη 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`a<sub>1</sub>X<sup>q</sup>`¨ ¨ ¨`a<sub>r</sub>X<sup>qr</sup>, Φ<sup>E{H</sup><sub>t</sub> pXq “b<sub>0</sub>X`b₁X^q`¨ ¨ ¨`b_sX^qs

with bs ‰ 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` ¨ ¨ ¨ `pn´1X^qn´1 `X^qn.

From the equality Φ^E{H_t pPpXqq “PpΦ^E_t pXqq, we obtainr“s,b_r “a^q_rⁿ andp₁pb₀´θq “0. IfHis etale over B, then we havep₁ PB^ˆand 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 ofś

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 ą 0 in A“Fqrts, 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¯^b1´qipS¯q. We put

F_d,E¯ “τ^d: ¯E ÑE¯^pqdq, V_d,E¯ “α_dpEq ` ¨ ¨ ¨ `¯ α_2dpEqτ¯ ^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_d˝F_d“Φ^E_℘^¯ andF_d˝V_d“ Φ^E_℘^¯pqdq [Sha,§2.8].

Definition 2.3. We say ¯E is ordinary ifα_dpEq P¯ L¯^b1´qdpSq¯ is nowhere vanishing, and supersingular if αdpEq “¯ 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

ÿ

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^ˆ-torsor IpE,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 overA_n of relative dimension one.

Suppose that there exists a prime factorq of n such that its residue extensionkpqq{Fq is of degree prime toq´1. In this case, the inclusion F^ˆq Ñkpqq^ˆ splits and we can choose a subgroup ∆ Ď pA{pnqq^ˆ such that the natural map ∆ Ñ pA{pnqq^ˆ{F^ˆq 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^∆ “V˚pL^∆unq and put

ω_un^∆ :“ω_E∆

un “ pL^∆unq^_, where ω_E∆

un 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 anA_n-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^∆pnqR“Y₁^∆pnq ˆAnSpecpRqand similarly for other Drinfeld modular curves.

Lemma 3.1. 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 pairpE,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`a<sub>1</sub>τ `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`ˆa<sub>1</sub>τ `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^pqdq _„ pE{Gq{KerpF_d,Eq _„^{℘ //}

F_d,E{G

oo 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. □ Put K₈ “ Fqpp1{tqq and let C8 be the p1{tq-adic completion of an algebraic closure of K₈. Let Af be the ring of finite adeles (namely, the restricted direct product over the set of places of Fqptq 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 GL₂pAqˆ ˇ ˇ ˇ ˇ

g modnAˆP

ˆ∆ A{pnq

0 1

˙*

,

Γpnq “

"

g P GL₂pAq ˇ ˇ ˇ ˇ

g modpnq “ ˆ1 0

0 1

˙*

and Γ^∆₁pnq “ GL₂pAq X K₁^∆pnq. Since A^ˆ “ F^ˆq, we have Γ^∆₁pnq Ď SL₂pAq. This yields

Γ^∆₁pnq “

"

g PSL₂pAq ˇ ˇ ˇ ˇ

g modpnq P

ˆ1 A{pnq

0 1

˙*

.

In particular, the group Γ^∆₁ pnqis independent of the choice of ∆. Note that the natural right action ofg PGL₂pA{pnqqonYpnq_C8 corresponds to the left action of^tg on ΓpnqzΩ via the M¨obius transformation. Since F^ˆq detpK₁^∆pnqq “Aˆ^ˆ, [Dri, Proposition 6.6] implies that the analytifi- cation of Y₁^∆pnq_C8 is identified with

GL₂pFqptqqzΩˆGL₂pAfq{K₁^∆pnq “Γ^∆₁ pnqzΩ,

and thus the fiber Y₁^∆pnq_K8 is geometrically connected. Similarly, we see thatY₁^∆pn, ℘q_K8 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^bq`1₁ ba^b´1₂ P OSpSq.

Consider the finite flat map

jt :Y₁^∆pnq Ñ A¹An “SpecpA_nrjsq, j ÞÑjtpE_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 P¹_An inY₁^∆pnqandY₁^∆pn, ℘qvia this map, respectively. As in [Sha,§7.2], we see that X₁^∆pnq is smooth over A_n 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 SpecpA_nq 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^∆pnq_RandX₁^∆pn, ℘q_R of Y₁^∆pnq_R and Y₁^∆pn, ℘q_R. From the smoothness of X₁^∆pnq, we have X₁^∆pnqR “ X₁^∆pnq ˆAn 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{pnqq^ˆandpA{pnqq^ˆ{∆“F^ˆ_q onX₁^∆pnq_R. We denote them by xay_n and xcy_∆, respectively.

Lemma 3.2. Let S be a scheme over A and E a Drinfeld module of rank two over S. If jtpEq P OSpSq is invertible, then for the big fppf sheaf Aut_A,SpEq defined by

T ÞÑAut_A,TpE|_Tq,

the natural map F^ˆ_q 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^ˆ. Write as Φ^E_t “θ`a<sub>1</sub>τ`a₂τ². From the assumption, we have a₁ P B^ˆ and the equality Φ^E_t pbXq “ bΦ^E_tpXq yields b^q´1 “1.

Since the group scheme µ_q´1 over Fq is isomorphic to the constant group scheme F^ˆq, so is µ_q´1|B over the Fq-algebra B. This concludes

the proof. □

Lemma 3.3. 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 OSpSq^ˆ. Then the big fppf sheaf Isom_A,SpE, E¹q over S defined by

T ÞÑIsom_A,TpE|_T, E¹|Tq

is represented by a Galois covering of S with Galois group F^ˆq.

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`a<sub>1</sub>τ `a₂τ², Φ^E_t¹ “θ`a<sup>1</sup><sub>1</sub>τ `a¹₂τ²

with some a₁, a¹₁ P B and a₂, a¹₂ P B^ˆ. By assumption, we have a^{q`1</sup><sub>1</sub> {a<sub>2</sub> “ pa<sup>1</sup><sub>1</sub>q<sup>q`1}{a¹₂ PB^ˆ and thus a₁, a¹₁ PB^ˆ. Hence the scheme

J “SpecpBrYs{pY^q´1´a₁{a¹₁qq

is a finite etaleF^ˆ_q-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 3.2. □

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 ř

iě0aiXⁱ satisfying

iÑ8limpv_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} ˆ1

x

˙ ˇ ˇ ˇ ˇ

aPA

*

ĎR₀ppxqq, (4.1)

efΛpXq “X ź

α‰0PfΛ

ˆ 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 `₁

x

˘ Px^qdegpfqF^ˆ_qp1`xR₀rrxssq.

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

iaiXⁱ P R₀ppxqqrrXss, we put ν_f^˚phqpXq “ř

iν_f⁷pa_iqXⁱ. Then we have ν_f⁷pΛq “ fΛ and ν_f^˚pe_ΛqpXq “e_fΛpXq.

For any elementa PA, consider the power series (4.3) Φ^fΛ_a pXq “ e_fΛpΦ^C_ape^´1_fΛpXqqq 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_aq^´1pfΛq “ tyPK₀ppxqq^alg |Φ^C_apyq PfΛu,

which is an A-module, and let Σ_a Ď pΦ^C_aq^´1pfΛqbe a representative of the set

ppΦ^C_aq^´1pfΛq{fΛqzt0u.

Since R0 is flat over A, we have (4.5) Φ^f_a^ΛpXq “ aX ź

βPΣa

ˆ

1´ X

e_fΛpβq

˙

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

Lemma 4.1. If we write asΦ^Λ_t “θ`a<sub>1</sub>τ`a₂τ² for somea_i PR₀rrxss, then we have

a₁ P1`xR₀rrxss, a₂ Px^q´1R₀rrxss^ˆ.

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 F^ˆqu, Σ₀ “ tζ P K₀ppxqq^alg | Φ^C_t pζq “ 0u and Σ_t “ pΣ˜ `Σ₀q Y pΣ₀zt0uq. By (4.5), we have a₂ “ θ{pś

βPΣte_Λpβqq.

The denominator ś

βPΣteΛpβq is equal to ź

βPΣ˜

ź

ζPΣ0

pβ`ζq ź

α‰0PΛ

ˆα´ pβ`ζq α

˙

¨ ź

ζPΣ0zt0u

ζ ź

α‰0PΛ

ˆα´ζ α

˙ .

The first term is equal to ź

βPΣ˜

Φ^C_tpβq ź

α‰0PΛ

ˆΦ^C_t pα´βq α^q

˙

“

´ś

cPF^ˆq c¯ x^q´1 ¨ ź

cPF^ˆq

ź

α‰0PΛ

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 α“x^´qrhwith r“degpaq and h P R₀rrxss^ˆ, which yields pθα`α<sup>q</sup>´c{xq{α<sup>q</sup> P 1`xR₀rrxss. By a similar computation, the second term is equal to

θ ź

α‰0PΛ

θ`α^q´1

α^q´1 Pθp1`xR₀rrxssq.

Hence we obtain the assertion on a₂. □

Using Lemma 4.1 and the map ν_f, 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_f^˚pλ^Λ_8,mq “ λ^fΛ_8,m. In particular, the Tate-Drinfeld module TDpfΛq is endowed with a natural Γ₁pnq-structure λ^f8,n^Λ 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 ĎR₀rrxssxZy, the map

R₀rrxssrXs ÑR₀rrxssttZuu, X ÞÑe_fΛpZq induces a homomorphism of Hopf algebras

R0ppxqqrXs ÑR0rrxssxZyr1{xs{pΦ^C_mpZqqⁱÑ^´1 R0ppxqqrZs{pΦ^C_mpZqq, which we denote bypλ^f_8,m^Λ q^˚. 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^f_8,m^Λ q^˚ 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 ν_f.

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. Let Dbe the integral closure of R₀rrxss in D. We consider D as a topological ring by taking tx^lDulPZě0 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,m^fΛ “TDpfΛqrms{Impλ^fΛ_8,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,m^fΛ which is compatible with the map νf such that the image of µ^f₈^Λp1q P H8,m^fΛ pT0q in H^fΛ8,mpB_0,m^fΛq is

equal to the image e_fΛpηq of the element e_fΛpηq P TDpfΛqrmspB_0,m^fΛq.

In particular, we have an exact sequence of A-module schemes over T0

(4.7) 0 ^//Crms ^λ

fΛ

8,m//TDpfΛqrms ^π

fΛ

8,m//A{pmq ^//0.

Proof. By Lemma 4.3, we have an element e_fΛpηq PTDpfΛqrmspB_0,m^fΛq.

Since its image e_fΛpηq in H^f8,m^Λ pB_0,m^fΛq is invariant under the action of Crms on B_0,m^fΛ, we obtain efΛpηq P H^fΛ8,mpT0q. This yields an A-linear homomorphism A{pmq Ñ H^fΛ8,m over T₀ which is compatible with the map νf.

To see that it is an isomorphism, using the map νf 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^Λ8,m is etale.

Now it is enough to show aeΛpηq ‰ 0 in H^Λ8,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 ă degpmq.

The condition ae_Λpηq “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^Λ8,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_q´1 :T₀ ÑS₀

defined by y ÞÑ x^q´1. The S₀-scheme T₀ is a finite etale F^ˆq-torsor, where cP F^ˆq acts on it by the R₀-linear map

g_c:R₀ppxqq ÑR₀ppxqq, xÞÑc^´1x.