Elsevier

Journal of Number Theory

Volume 232, March 2022, Pages 33-74
Journal of Number Theory

General Section
On Drinfeld modular forms of higher rank IV: Modular forms with level

Dedicated to the memory of David Goss
https://doi.org/10.1016/j.jnt.2019.04.019Get rights and content

Abstract

We construct and study a natural compactification Mr(N) of the moduli scheme Mr(N) for rank-r Drinfeld Fq[T]-modules with a structure of level NFq[T]. Namely, Mr(N)=ProjEis(N), the projective variety associated with the graded ring Eis(N) generated by the Eisenstein series of rank r and level N. We use this to define the ring Mod(N) of all modular forms of rank r and level N. It equals the integral closure of Eis(N) in their common quotient field F˜r(N). Modular forms are characterized as those holomorphic functions on the Drinfeld space Ωr with the right transformation behavior under the congruence subgroup Γ(N) of Γ=GL(r,Fq[T]) (“weak modular forms”) which, along with all their conjugates under Γ/Γ(N), are bounded on the natural fundamental domain F for Γ on Ωr.

Introduction

This is the fourth of a series of papers (see [18], [21], [19]) which aim to lay the foundations for a theory of Drinfeld modular forms of higher rank. These are modular forms for the modular group Γ=GL(r,Fq[T]) or its congruence subgroups, where “higher rank” refers to r larger or equal to 2. The case of r=2, remarkably similar in some aspects but rather different in others to the theory of classical elliptic modular forms for SL(2,Z) or its congruence subgroups, is meanwhile well-established and the subject of several hundred publications since about 1980.

We leave aside to deal with more general Drinfeld coefficient rings A than A=Fq[T], as the amount of technical and notational efforts required would obscure the overall picture. The interested reader may consult [15] to get an impression of the complications that - even for r=2 - result from class numbers h(A)>1 for general A.

While we developed some of the theory of modular forms “without level” in [18] and [19] and focussed on the connection with the geometry of the Bruhat-Tits building in [18] and [21], the current part IV is devoted to forms “with level”, i.e., forms for congruence subgroups of Γ. Again we restrict to the most simple case of full congruence subgroups Γ(N)={γΓ|γ1(modN)} for NA. Finer arithmetic/geometric properties of modular forms (or varieties) for other congruence subgroups ΓΓ(N) may be derived in the course of the further development of the theory from those for Γ(N), by taking invariants (or quotients) of the finite group Γ/Γ(N).

Let us introduce a bit of notation: F=Fq is the finite field with q elements, A=F[T] the polynomial ring in an indeterminate T, with quotient field K=F(T), and its completion K=F((T1)) at infinity, and C the completed algebraic closure of K. The Drinfeld symmetric space Ωr (where r2) is the complement in Pr1(C) of the K-rational hyperplanes. The modular group Γ=GL(r,A) acts in the usual fashion on Ωr, and we let Mr(N) be the quotient analytic space Γ(N)Ωr (which is also the set of C-points of an affine variety labelled by the same symbol, and which is smooth if NA is non-constant).

The modular forms dealt with will be holomorphic functions on Ωr with certain additional properties; so the theory is “over C”; we will only briefly touch on questions of rationality.

Our approach is based on

  • (i)

    the use of the natural fundamental domain F for Γ on Ωr introduced in [20]; it relies on the notion of successive minimum basis (SMB) of an A-lattice in C. On F, one may perform explicit calculations;

  • (ii)

    a natural compactification Mr(N) of Mr(N), the Eisenstein compactification, whose construction is influenced by but different from Kapranov's in [29].

The obvious examples of modular forms-to-be for Γ(N) are the Eisenstein series of level N. They generate a graded C-algebra Eis(N) (generated in dimension 1 if N is non-constant), and Mr(N) will be the associated projective variety Proj(Eis(N)), see Theorem 5.9. It is a closed subvariety of a certain projective space Pc1, where c is the number of cusps of Γ(N) (Corollary 4.7, Theorem 5.9), and is therefore supplied with a natural very ample line bundle M. We define strong modular forms of weight k for Γ(N) as sections of Mk, and thereby get the graded ring Modst(N) of strong modular forms, which encompasses Eis(N).

The Eisenstein compactification is natural and explicit, and has good functorial properties (see Remark 5.10; it is, e.g., compatible with level change), but unfortunately we presently cannot assure that it is normal. Correspondingly, strong modular forms are integral over Eis(N) (and in fact over Mod=Mod(1), the ring of modular forms of type 0 for Γ(1)=Γ), but we don't know whether Modst(N) is integrally closed. We define the Satake compactification Mr(N)Sat of Mr(N) as the normalization of Mr(N) (as Kapranov does) and a modular form of weight k for Γ(N) as a section of the pull-back of Mk to Mr(N)Sat. This yields the graded ring Mod(N) of all modular forms. Hence we have inclusionsEis(N)Modst(N)Mod(N) of finitely generated graded integral C-algebras, where Mod(N) is the integral closure of Eis(N) in their common quotient field F˜r(N). Elements of Mod(N) have a nice characterization given by Theorem 7.9: A weak modular form f of weight k is modular if and only if f, together with all its conjugates f[γ]k (γΓ/Γ(N)), is bounded on the fundamental domain F. Further Eis(N) has always finite codimension in Modst(N) (Corollary 7.11), while dim(Mod(N)/Modst(N)) is either zero or infinite, according to whether Mr(N)Sat agrees with Mr(N) or not (Corollary 7.14). Except for some examples presented in Section 8, where the two compactifications and also the three rings in (0.6) agree (these examples depend crucially on work of Cornelissen [8] and Pink-Schieder [33]), we don't know what happens in general: more research is needed! At least Mr(N)Sat is not very far from Mr(N): the normalization mapν:Mr(N)SatMr(N) is bijective on C-points (Corollary 7.6, see also Proposition 1.18 in [29]), and is an isomorphism on the complement of a closed subvariety of codimension ≥2 (Corollary 6.10).

We now describe the plan of the paper. In the first section, we introduce the space Ωr with its strong topology, which upon dividing out the action of Γ(N) will yield the underlying topological space for the Eisenstein compactification Mr(N). Its points correspond to homothety classes of pairs (U,i), where U0 is a K-subspace of Kr and i is a discrete embedding of UAr into C. For technical purposes we also consider the Gm-torsor Ψr over Ωr whose points correspond to pairs (not homothety classes) (U,i) as above. Further, the fundamental domains F˜ on Ψr and F on Ωr for Γ are introduced. Although Ωr and Ψr come with the same information, it will sometimes be more convenient to work with Ψr and F˜ instead of Ωr and F. We take particular care to give a consistent description of the group actions on Ωr and related objects.

In Section 2 the (well-known) relationship of Ωr with the moduli of Drinfeld modules of rank r is presented. We further show the crucial technical result Theorem 2.3, which asserts that the bijectionj:ΓΩrProj(Mod) is a homeomorphism for the strong topologies on both sides. We further introduce and describe the function fields of the analytic spaces Mr(N)=Γ(N)Ωr and M˜r(N)=Γ(N)Ψr.

In Sections 3 and 4, the boundary components and the (non-)vanishing of Eisenstein series on them are studied. We find in Corollary 4.7 that the space Eisk(N) of Eisenstein series of level N and weight k has dimension cr(N), the number of cuspidal divisors of Γ(N)Ωr, independently of k. Further (Proposition 4.8), Eis1(N) separates points of Γ(N)Ωr, which will give rise to its projective embedding. This latter is defined and investigated in Section 5; we thereby interpret Γ(N)Ωr as the Eisenstein compactification Mr(N) of Mr(N).

Section 6 is of a more technical nature. There we construct tubular neighborhoods along the cuspidal divisors of Mr(N), see Theorem 6.9.

In Section 7, the rings Modst(N) and Mod(N) of modular forms are introduced and their relation with the Eisenstein ring Eis(N) and the compactifications Mr(N) and Mr(N)Sat is discussed.

We conclude in Section 8 with the two classes of examples where our knowledge is more satisfactory than in the general situation, namely the special cases where either the rank r equals 2 or where the conductor N has degree 1.

The point of view (and the notation, see below) of this paper widely agrees with that of the preceding [18], [21], [19], to which we often refer. As in these, our basic references for rigid analytic geometry are the books [12] by Fresnel-van der Put and [7] of Bosch-Güntzer-Remmert. The canonical topology on the set X(C) of C-points of an analytic space X ([7] Section 7.2) is labelled as the strong topology, so functions continuous with respect to it are strongly continuous, etc. In general, we don't distinguish in notation between X and X(C); ditto, a C-variety and its analytification are usually described by the same symbol. It will (hopefully) always be clear from the context whether e.g. the “algebraic” or the “analytic” local ring is intended.

After this paper was largely completed, I got access to the recent preprints [4], [5], [6] of Dirk Basson, Florian Breuer, and Richard Pink, which go about the same topic: providing a foundation for the theory of higher rank Drinfeld modular forms. As it turns out, the relative perspectives of Basson-Breuer-Pink's work and of the current paper are rather different. While BBP deal with the most general Drinfeld coefficient rings A and arithmetic subgroups of GL(r,A), for which they establish basic but sophisticated facts like e.g. the existence of expansions around infinity of weak modular forms, we restricted to the coefficient ring A=Fq[T] and full congruence subgroups and focus on the role of Eisenstein series, their arithmetic properties, and their impact on compactifications of the moduli schemes. Apart from examples, there is little overlap between the two works; so the reader who wants to enter into the field might profit from studying the two of them.

Finally, I wish to point to the recent thesis [25] of Simon Häberli, whose purpose is similar. In contrast with the present article, Häberli gives a direct construction of the Satake compactification, which he uses for the description of modular forms.

Notation

  • F=Fq the finite field with q elements;

  • A=F[T] the polynomial ring in an indeterminate T, with quotient field K=F(T) and its completion K=F((T1)) at infinity;

  • C= completed algebraic closure of K, with absolute value |.| and valuation v:CQ normalized by v(T)=1, |T|=q;

  • Ψr={ω=(ω1,,ωr)Cr|the ωi are K-linearly independent};

  • Ωr={ω=(ω1::ωr)Pr1(C)|ω represented by (ω1,,ωr)Ψr};

  • Γ=Γr=GL(r,A) with center ZF of scalar matrices;

  • Γ(N)={γΓ|γ1(modN)}, NA;

  • U=set of K-subspaces U0 of V=Kn;

  • ΨUΨs, ΩUΩs attached to UU, where dimU=s;

  • Ψr=UUΨU, Ωr=UUΩU.

If the group G acts on the space X then Gx, Gx and GX denote the stabilizer of xX, its orbit, and the space of all orbits, respectively. Also, for YX, GY is the image of Y in GX. The multiplicative group of the ring R is R; the R-module generated by x1,,xr is written either as Rxi or as x1,,xrR. We use the convention N={1,2,3,}, N0={0,1,2,}.

Section snippets

We let V be the K-vector space Kr, where r2, and U the set of K-subspaces U0 of V. An A-lattice in UU is a free A-submodule L of U of full rank rkA(L)=dimK(U), that is KL=KL=U. A subset of C is discrete if the intersection with each ball of finite radius in C is finite. A discrete embedding of UU (“embedding” for short) is some K-linear injective map i:UC such that i(L) is discrete in C for one fixed (or equivalently, for each) A-lattice L in U. We putΨU:=set of discrete embeddings

Given an A-lattice Λ in C of rank rN, we dispose of

  • the exponential function eΛ:CCeΛ(z)=zλΛ(1z/λ)=i0αi(Λ)zqi;

  • the Drinfeld A-module ϕΛ of rank r, defined by the operator polynomialϕTΛ(X)=TX+g1(Λ)Xq++gr(Λ)Xqr, and

  • the Eisenstein seriesEk(Λ)=λΛλk(kN).

We further put g0(Λ)=T, E0(Λ)=1. These are connected byeΛ(Tz)=ϕTΛ(eΛ(z));i,j0i+j=kαiEqj1qi=i+j=kαiqjEqj1=1 if k=0 and 0 otherwise, which determines a number of further relations, see e.g. [16] Sect. 2.

If Λ=Λω=1irAωi with ω=(ω1,

The boundary components

From now on, we assume that r2.

The set Us of s-dimensional subspaces U of V=Kr is in canonical bijection with GL(r,K)/Ps(K) throughGL(r,K)/Ps(K)Us.γVsγ1

As the action of Γ on Us is transitive, we may replace the left hand side with Γ/ΓPs(K). Let (A/N)primr be the set of primitive elements of (A/N)r, that is, of elements that belong to a basis of the free (A/N)-module (A/N)r. Then, as is easily verified, the mapΓ(N)Γ/ΓPr1(K)(A/N)rprim/F=:Cr(N) that associates with the double class of γ

Behavior of Eisenstein series at the boundary

In the whole section, N is a fixed monic element of A of degree d1.

The projective embedding

In this section, we show that Γ(N)Ωr is the set of C-points of a closed subvariety of some projective space. This allows us to endow it with the structure of projective variety, which then will be labelled with the symbol Mr(N), the Eisenstein compactification of Mr(N).

Throughout, NA of degree d1 is fixed.

Tubular neighborhood of cuspidal divisors

In this section we show that each point x on a cuspidal divisor, i.e., on a boundary component MU(N) of Mr(N) of codimension 1, possesses a neighborhood Z isomorphic with B×W, where W is an open admissible affinoid neighborhood of x on MU(N) and B a ball, and such that the map π:ZW derived from the canonical projection πU:Mr(N)MU(N) is the projection to the second factor.

Modular forms

In this section, we define the ring of modular forms for Γ(N) and relate it with the ring of sections of the very ample line bundle of Mr(N) given by the embedding jN.

Theorem 7.9 gives several different descriptions of modular forms. The assumptions of the preceding sections remain in force. Thus r2 and NA is monic of degree d1.

Examples and concluding remarks

The preceding immediately raises a number of important questions and desiderata.

8.1 Question

Do the Eisenstein and Satake compactifications Mr(N) and Mr(N)Sat always coincide, i.e., is Mr(N) always normal?

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