General SectionOn Drinfeld modular forms of higher rank IV: Modular forms with level
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 or its congruence subgroups, where “higher rank” refers to r larger or equal to 2. The case of , remarkably similar in some aspects but rather different in others to the theory of classical elliptic modular forms for 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 , 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 - result from class numbers 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 for . Finer arithmetic/geometric properties of modular forms (or varieties) for other congruence subgroups may be derived in the course of the further development of the theory from those for , by taking invariants (or quotients) of the finite group .
Let us introduce a bit of notation: is the finite field with q elements, the polynomial ring in an indeterminate T, with quotient field , and its completion at infinity, and the completed algebraic closure of . The Drinfeld symmetric space (where ) is the complement in of the -rational hyperplanes. The modular group acts in the usual fashion on , and we let be the quotient analytic space (which is also the set of -points of an affine variety labelled by the same symbol, and which is smooth if is non-constant).
The modular forms dealt with will be holomorphic functions on with certain additional properties; so the theory is “over ”; 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 introduced in [20]; it relies on the notion of successive minimum basis (SMB) of an A-lattice in . On F, one may perform explicit calculations;
- (ii)
a natural compactification of , the Eisenstein compactification, whose construction is influenced by but different from Kapranov's in [29].
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 (and in fact over , the ring of modular forms of type 0 for ), but we don't know whether is integrally closed. We define the Satake compactification of as the normalization of (as Kapranov does) and a modular form of weight k for as a section of the pull-back of to . This yields the graded ring of all modular forms. Hence we have inclusions of finitely generated graded integral -algebras, where is the integral closure of in their common quotient field . Elements of 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 (), is bounded on the fundamental domain F. Further has always finite codimension in (Corollary 7.11), while is either zero or infinite, according to whether agrees with 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 is not very far from : the normalization map is bijective on -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 with its strong topology, which upon dividing out the action of will yield the underlying topological space for the Eisenstein compactification . Its points correspond to homothety classes of pairs , where is a K-subspace of and i is a discrete embedding of into . For technical purposes we also consider the -torsor over whose points correspond to pairs (not homothety classes) as above. Further, the fundamental domains on and F on for Γ are introduced. Although and come with the same information, it will sometimes be more convenient to work with and instead of and F. We take particular care to give a consistent description of the group actions on and related objects.
In Section 2 the (well-known) relationship of 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 bijection is a homeomorphism for the strong topologies on both sides. We further introduce and describe the function fields of the analytic spaces and .
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 of Eisenstein series of level N and weight k has dimension , the number of cuspidal divisors of , independently of k. Further (Proposition 4.8), separates points of , which will give rise to its projective embedding. This latter is defined and investigated in Section 5; we thereby interpret as the Eisenstein compactification of .
Section 6 is of a more technical nature. There we construct tubular neighborhoods along the cuspidal divisors of , see Theorem 6.9.
In Section 7, the rings and of modular forms are introduced and their relation with the Eisenstein ring and the compactifications and 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 of -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 ; ditto, a -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 , 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 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 the finite field with q elements; the polynomial ring in an indeterminate T, with quotient field and its completion at infinity; completed algebraic closure of , with absolute value and valuation normalized by , ; with center of scalar matrices; , ; ; , attached to , where ; , .
If the group G acts on the space X then , Gx and denote the stabilizer of , its orbit, and the space of all orbits, respectively. Also, for , is the image of Y in . The multiplicative group of the ring R is ; the R-module generated by is written either as or as . We use the convention , .
Section snippets
We let V be the K-vector space , where , and the set of K-subspaces of V. An A-lattice in is a free A-submodule L of U of full rank , that is . A subset of is discrete if the intersection with each ball of finite radius in is finite. A discrete embedding of (“embedding” for short) is some K-linear injective map such that is discrete in for one fixed (or equivalently, for each) A-lattice L in U. We put
Given an A-lattice Λ in of rank , we dispose of
- •
the exponential function
- •
the Drinfeld A-module of rank r, defined by the operator polynomial
- •
the Eisenstein series
If with
The boundary components
From now on, we assume that .
The set of s-dimensional subspaces U of is in canonical bijection with through
As the action of Γ on is transitive, we may replace the left hand side with . Let be the set of primitive elements of , that is, of elements that belong to a basis of the free -module . Then, as is easily verified, the map that associates with the double class of
Behavior of Eisenstein series at the boundary
In the whole section, N is a fixed monic element of A of degree .
The projective embedding
In this section, we show that is the set of -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 , the Eisenstein compactification of .
Throughout, of degree 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 of of codimension 1, possesses a neighborhood Z isomorphic with , where W is an open admissible affinoid neighborhood of x on and B a ball, and such that the map derived from the canonical projection is the projection to the second factor.
Modular forms
In this section, we define the ring of modular forms for and relate it with the ring of sections of the very ample line bundle of given by the embedding .
Theorem 7.9 gives several different descriptions of modular forms. The assumptions of the preceding sections remain in force. Thus and is monic of degree .
Examples and concluding remarks
The preceding immediately raises a number of important questions and desiderata.
8.1 Question Do the Eisenstein and Satake compactifications and always coincide, i.e., is always normal?
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