Here is another excellent review, this time from David Goss. In his review of a paper by Pink and Schieder, Goss gives not just a good description of what’s in the paper, but also tells us about where the results come from, what work others have done on related problems, and even a little about what might come next. In short, he gives the long-form answer to the question “Is this paper interesting?” The question Pink and Schieder address comes from number theory, but most of the hard work is algebraic geometry. To get started, Goss says why compactifications are useful, and gives examples of some general methods. In his discussion, Goss moves from some classical compactifications, such as the arithmetically interesting compactification of the upper half plane, to some very new ones related to Drinfeld modules. The family of ideas used here has its start in the work of Satake, which dates as far back as 1956. Goss sketches this history, insofar as it applies to the present paper. In this way, the review tells us not just what is in the article, but also where it fits into the rest of mathematics. And then, to finish with a flourish, Goss ends his review by pointing the reader in the direction of possible new results. All in all, David Goss provides us with an exceptional review!

Note: *Mathematical Reviews* is fortunate that Goss is also a frequent reviewer, having written 180 reviews (so far).

**MR3166390**

Pink, Richard(CH-ETHZ); Schieder, Simon(1-HRV)

Compactification of a Drinfeld period domain over a finite field. (English summary)

*J. Algebraic Geom.* 23 (2014), no. 2, 201–243.

11T60 (11F03 11G09 14G15 14G22 14M27)

Let $\Bbb F_q$ be the finite field with $q$-elements. Following the authors, for $d\geq 1$ we denote by $\Omega_d$ the affine subscheme of $\Bbb P^d$ obtained by removing the (finitely many) proper $\Bbb F_q$-linear subspaces. In this important and self-contained paper the authors present a normal “Satake” compactification of the “period domain”$\Omega_d$ as well as explicit computations of its invariants. It is important to note at the outset that this compactification is very different than simply the ambient space $\Bbb P^d$ itself.

Notice that the variety $\Omega_d$ is clearly an avatar of the rigid analytic space $\Omega^d$ of Drinfeld. Indeed, if $K$ is any local non-Archimedean field, then $\Omega^d$ is the admissible open subset of $\Bbb P^d$ obtained by removing all proper $K$-linear subspaces. When $K=k_\infty$, for $k$ a global function field over $\Bbb F_q$ and $\infty$ a fixed place of $k$, one sees that $\Omega^d$ is the space of $A$-lattices up to homothety (here $A\subset k$ is the subring of those functions regular outside $\infty$); it precisely gives an analytic uniformization of the moduli schemes of elliptic $A$-modules of rank $d$. Remarkably, the scheme $\Omega_d$ is, itself, isomorphic to such a modular scheme (when $A=\Bbb F_q[\theta]$; see below). In fact, its Satake compactification is crucial to the work of the first author in [Manuscripta Math. 140 (2013), no. 3-4, 333–361; MR3019130], where normal Satake compactifications were obtained in complete generality (for general $A$ and $d$).

A general principle in algebraic geometry is that “global sections of coherent sheaves on complete (`compact’) varieties are finite-dimensional over the base field”. (The necessity of having a compact variety is brought home simply by noticing that the ring of polynomials, the functions on the affine line, is obviously infinite-dimensional over the base field.) Thus, given a variety over a field $k$ a good deal of work has gone into establishing conditions guaranteeing that it is compactifiable (i.e., may be embedded in a complete variety) with the most general such criterion being due to M. Nagata.

For modular varieties, of one sort or another, one then wants to also have a good description of this compactification. An essential idea in this regard (due to I. Satake, for whom these spaces are named) is illustrated by the basic elliptic modular case: Let $\scr H$ be the upper half-plane and set $\scr H^\ast := \scr H\cup \Bbb P^1(\Bbb Q)$, which is given a natural topology extending the one on $\scr H$. The standard ${\rm SL}_2(\Bbb Z)$ action on $\scr H$ extends to $\scr H^\ast$, and if $\Gamma\subseteq {\rm SL}_2(\Bbb Z)$ is a subgroup of finite index, one finds that $\Gamma \backslash \scr H^\ast$ is compact. The finite set of points thus added to $\Gamma\backslash \scr H$ are called the “cusps”. An absolutely fundamental insight in the case of congruence subgroups of ${\rm SL}_2(\Bbb Z)$, due to J. Tate, is that these cusps may be described in a purely algebraic fashion using “Tate objects” arising from his non-Archimedean parametrization of elliptic curves. In general the Satake compactification (roughly!) involves attaching to the modular variety of objects of a given dimension a finite number of modular varieties of the same type of objects but of smaller dimension.

In the elliptic modular case the compactification is smooth, but in general it may be singular. However, the compactified varieties are usually normal. Modular forms in a given category then arise as sections of a line bundle. Very often (but certainly not always, e.g., for Siegel forms of higher genus) such compactifications need to be combined with “holomorphicity at infinity” in order to deduce the finite dimensionality of spaces of modular forms.

The formalism of the normal Satake compactification applies to the category of Drinfeld modules which, in turn, contains an excellent theory of Tate objects [V. G. Drinfeld, Mat. Sb. (N.S.) 94(136) (1974), 594–627, 656; MR0384707 (52 #5580)]. Modular forms and their expansions at infinity (in the rank 2 case) were described in [D. Goss, Compositio Math.41 (1980), no. 1, 3–38; MR0578049 (82e:10053)]. The idea, very briefly, is as follows: Let $k$ be our global function field, as above, and let $\Bbb C_\infty$ be the completion of a fixed algebraic closure of $k_\infty$. A Drinfeld module of rank $d$ then corresponds to a lattice $M$ of rank $d$ lying in $\Bbb C_\infty$ and thus to points on $\Omega^d$ considered over the local field $k_\infty$. The group ${\rm GL}_d(A)$ acts on $\Omega^d$ in the natural fashion, and completely analogous to how ${\rm SL}_2(\Bbb Z)$ acts on $\scr H$, and the notion of modular form on $\Omega^d$ is then readily obtained.

In the original rank 2 case, Drinfeld compacted the corresponding curves by adding cusps in almost exact similarity with the classical elliptic curve case, as well as established that $\Omega^2$ is rigid analytically connected. Combining this with a finiteness statement at the cusps (using the exponential functions of Drinfeld modules of rank 1 as in the elliptic modular case) as well as rigid analytic GAGA, one also obtains the finite dimensionality of spaces of modular forms.

The moduli spaces of Drinfeld modules are affine, and thus spaces of forms, of all ranks, will be infinite-dimensional over$\Bbb C_\infty$ without cuspidal conditions of some sort. In the rank 2 case, one is able to obtain expansions at the cusps using the reciprocals of rank 1 Drinfeld modules (which give the associated Tate objects in this case); one finds then that the spaces of modular forms, which are regular at the cusps, are by rigid analytic GAGA then finite-dimensional over $\Bbb C_\infty$ (as they correspond to algebraic sections of the line bundle on a complete curve).

In the higher rank case, one expects something similar. The rigid analytic space $\Omega^d$ is, in fact, connected in general [see, e.g., M. van der Put, Nederl. Akad. Wetensch. Indag. Math. 49 (1987), no. 3, 313–318; MR0914089 (88m:32055)]. For general $d$, a Satake compactification was originally given by M. M. Kapranov [Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 568–583, 688; MR0903624 (89c:11095)] in the polynomial case. Using Tate objects, one can, in principle, give an expansion to modular forms of rank $d>2$ in a very similar fashion to that of the rank 2 case. Those that are “holomorphic at the cusps” should then also form finite-dimensional spaces via rigid analytic GAGA. This argument, also essentially due to Kapranov, was sketched briefly (too briefly!) in the final section of [D. Goss, in The arithmetic of function fields (Columbus, OH, 1991), 227–251, Ohio State Univ. Math. Res. Inst. Publ., 2, de Gruyter, Berlin, 1992;MR1196522 (94c:11036)].

Another approach to these issues was presented by the first author (R. Pink) of the present paper in [op. cit.;MR3019130] as mentioned above; there the moduli spaces for Drinfeld modules of arbitrary rank and base ring $A$ are given a normal Satake compactification. An invertible sheaf on this compactification is canonically given and modular forms are then defined as global sections of this sheaf (and so are obviously finite-dimensional by standard results in coherent cohomology). Moreover the Proj of the graded algebra of modular forms is isomorphic to the Satake compactification of Pink. Tate objects are not used in this exposition, but one fully expects both approaches to be equivalent.

In [op. cit.; MR3019130], Pink reduced to the case $A=\Bbb F_q[\theta]$ and level $(\theta)$ which is isomorphic to $\Omega_d$, thus giving the connection to the excellent paper being reviewed. The work in [R. Pink, op. cit.; MR3019130] is “axiomatic” and it remains to work out explicitly the intricate structures so obtained.

Using new results of A. Petrov, in [D. Goss, J. Number Theory 136 (2014), 330–338; MR3145337] a nontrivial construction of $\frak v$-adic modular forms à la J.-P. Serre was given for $\frak v\in {\rm Spec}({\Bbb F}_q[\theta])$. It would be very interesting, and potentially extremely useful, to merge the ideas of [R. Pink, op. cit.; MR3019130] and [D. Goss, op. cit.; MR0578049 (82e:10053)] in order to give a more functorial definition of such $\frak v$-adic forms à la N. M. Katz [in Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 69–190. Lecture Notes in Mathematics, 350, Springer, Berlin, 1973; MR0447119 (56 #5434)].

Reviewed by David Goss