##
**Cohomology of Drinfeld modular varieties. Part 1: Geometry, counting of points and local harmonic analysis.**
*(English)*
Zbl 0837.14018

Cambridge Studies in Advanced Mathematics. 41. Cambridge: Cambridge University Press. xiii, 344 p. £40.00; $ 64.95 (1996).

This volume is the first part of a book on the cohomology of Drinfeld modular varieties interpreted as parallels to Shimura varieties. Shimura varieties are quasi-projective varieties over number fields and their cohomology is related to automorphic forms over numbers fields. Drinfeld modular varieties are quasi-projective varieties over function fields and their cohomology is related to automorphic functions over function fields. This volume gives a very detailed and precise presentation of (mostly) preparatory material for the subject proper (to be treated in a second volume). It consists of:

(i) a preface sketching the main problem and the difficulties met to attack it;

(ii) eight chapters, the first five of which culminate in the sixth with a formula giving Lefschetz numbers in terms of elliptic orbital integrals, and the last two of which deal with fundamental results of local harmonic analysis to be used in the second volume of the book;

(iii) four appendices on central division algebras, Dieudonné theory, combinatorial formulas and the representation theory of locally compact, totally discontinuous, separated, topological groups, respectively;

(iv) four pages with 68 references, and

(v) a somewhat limited index.

Unfortunately, the author did not include a list of symbols. It should be stressed that everything is presented very explicitly and fully proved throughout the book, without making it easy reading. The subject matter is just too complicated.

Let \(p\) be a prime number and let \(X\) be a smooth, projective and connected curve over \(\mathbb{F}_p\) with generic point \(\eta\). Write \(F = \kappa (\eta)\) for its function field. For a place \(x\) of \(F\) let \(F_x\) be the completion of \(F\) at \(x\) and write \({\mathfrak O}_x\) for the corresponding valuation ring. After fixing a place \(\infty\) of \(F\) and identifying the set of closed points of \(X\) with the set of places of \(F\) one can define the ring of regular functions \(A\) of \(F : A = \{a \in F \mid x(a) \geq 0\), \(\forall x \in X \backslash \{\infty\}\}\). One defines a Drinfeld \(A\)-module of rank \(d > 0\) over the scheme \(S\) (of characteristic \(p)\) as a pair \((E, \varphi)\), where \(E\) is a commutative group scheme over \(S\) and \(\varphi : A \to \text{End} (E)\) is a homomorphism satisfying certain conditions involving \(d\). For a nonzero ideal \(I \subset A\) one can define a level-\(I\) structure \(\iota\) on \((E, \varphi)\). One defines the fibered category \({\mathcal M}^d_I\) over the category of characteristic \(p\) schemes by \({\mathcal M}^d_I (S) : =\) {the category of triples \((E,\varphi,\iota)\) where \((E,\varphi)\) is a Drinfeld \(A\)-module of rank \(d\) over \(S\) with level-\(I\) structure \(\iota\) on \((E, \varphi)\) (and some other condition on the characteristic)}. It is shown that for \(I \neq A\), \({\mathcal M}^d_I\) is representable by an affine scheme \(M^d_I\) of finite type over \(\mathbb{F}_p\). For \(I = A\), \({\mathcal M}^d_A\) is representable by a Deligne-Mumford algebraic stack \(M^d_A\) of finite type over \(\mathbb{F}_p\).

Let \(V(I) \subset X \backslash \{\infty\} = \text{Spec} (A)\) be the closed subscheme defined by \(I\). One has a morphism of algebraic stacks (schemes if \(I \neq A)\), \(\Theta : M^d_I \to X \backslash (\{\infty\} \cup V(I))\), smooth and of relative dimension \(d - 1\). For a place \(o\) of \(F\), \(o \neq \infty\), \(o \notin V(I)\), let \(\text{Frob}_o \in \text{Gal} (\overline{\kappa (o)}/ \kappa (o))\) be the geometric Frobenius. Define \(M^d_{I,o}(\overline{\kappa (o)})= \Theta^{-1} (o) (\overline{\kappa(o)})\). One can define the algebra of Hecke operators \({\mathcal H}_I^{\infty, o}\) acting on \(M^d_{I,o}\) by correspondences, and for \(f^{\infty,o} \in {\mathcal H}_I^{ \infty, o}\) one is interested in the action of \(\text{Frob}_o^r \times f^{\infty,o}\) on \(M^d_{I,o} (\overline {\kappa (o)})\), where \(r\) is a positive integer. In particular, the number of its fixed points, written \(\text{Lef}_r (f^{\infty,o})\), is of particular importance. Now, an element \(\gamma \in GL_d (F)\) is called elliptic if its minimal polynomial is irreducible over \(F\). Let \(GL_d (F)_\ell \) be a system of representatives in \(GL_d (F)\) of the elliptic conjugacy classes. An elliptic \(\gamma\) is said to be elliptic at the place \(\infty\) iff there is only one place \(\infty'\) of \(F' = F [\gamma]\) (this is a field for elliptic \(\gamma)\) dividing \(\infty\) of \(F\). \(\gamma\) is said to be \(r\)-admissible at the place \(o\) iff \(o (\text{det} (\gamma)) = r\) and there exists a place \(o'\) of \(F'\) dividing \(o\) such that \(x' (\gamma) = 0\) for all other places \(x'\) of \(F'\) which divide \(o\). Then, as a first result, one can deduce a formula for \(\text{Lef}_r (f^{\infty,o})\) as a (finite) sum over \(\gamma \in GL_d (F)_\ell\) which are elliptic at \(\infty\) and \(r\)-admissible at \(o\) of products of terms involving orbital and twisted orbital integrals. Then, by the so- called fundamental lemma one can replace the twisted orbital integrals by ordinary orbital integrals.

Another unpleasant quantity ocurring in the product terms of the above mentioned formula can be written as an ordinary orbital integral of the newly introduced concept of very cuspidal Euler-Poincaré function \(f_\infty \in {\mathcal C}^\infty_c (F^\times_\infty \backslash GL_d (F_\infty)//{\mathcal B}^0_\infty)\), where \({\mathcal B}^0_\infty \subset K_\infty = GL_d ({\mathfrak O}_\infty)\) denotes the Iwahori subgroup. Also, with \(K_o = GL_d ({\mathfrak O}_o)\), let \(f_o \in {\mathcal C}_c^\infty (GL_d (F_o)//K_o)\) be a suitable Hecke function (with prescribed Satake transform) and define \[ f_\mathbb{A} = f_\infty f^{\infty, o} f_o \in {\mathcal C}^\infty_c (F^{\times_\infty \backslash GL_d (\mathbb{A})//{\mathcal B}^0_\infty K_I^{\infty, o} K_o)}. \] Taking suitable Haar measures, we are now able to state the main result (chapter 6) on \(\text{Lef}_r (f^{\infty, o})\) in terms of the orbital integrals \[ O_\gamma (f_\mathbb{A}, dg_{\gamma, \mathbb{A}}) = \int_{GL_d (\mathbb{A})_\gamma \backslash GL_d (\mathbb{A})} f_\mathbb{A} (g^{-1}_\mathbb{A} \gamma g_\mathbb{A}) {dg_\mathbb{A} \over dg_{\gamma, \mathbb{A}}} \] and the volumes \(a(\gamma, dg_{\gamma, \mathbb{A}}) = \text{vol} (F^\times_\infty GL_d (F)_\gamma \backslash GL_d (\mathbb{A})_\gamma, {dg_{\gamma, \mathbb{A}} \over dz_\infty d \gamma'})\) as \[ \text{Lef}_r (f^{\infty, o}) = \sum_{\gamma \in GL_d (F)_{\ell}} a(\gamma, dg_{\gamma, \mathbb{A}}) O_\gamma (f_\mathbb{A}, dg_{\gamma, \mathbb{A}}). \] Here \(dg_{\gamma, \mathbb{A}}\) is a Haar measure on the centralizer \(GL_d (\mathbb{A})_\gamma\) of \(GL_d (\mathbb{A})\), and \(d \gamma'\) denotes the counting measure on \(GL_d (F)_\gamma\), the centralizer of \(\gamma\) in \(GL_d (F)\).

Chapter 7 deals with the construction and the main properties of the unramified principal series representations of \(G(F) = GL_d (F)\), \(F\) a non-archimedean local field. Subjects treated are parabolic induction and restriction, cuspidal representations, (unramified) principal series representations, and spherical representations. It covers work of Bernstein and Zelevinsky on the one hand, and of Casselman on the other hand.

In the last chapter it is proved that the Euler-Poincaré function introduced earlier is a pseudo-coefficient of the Steinberg representation. This result is due to Casselman and Kottwitz. It says that for an admissible irreducible representation \(({\mathcal V}, \pi)\) of \(F^\times G(F)\) and \(f \in {\mathcal C}^\infty_c (F^\times \backslash G(F)//{\mathcal B}^0)\) the very cuspidal Euler-Poincaré function, one has a description of \(\text{tr} (\pi (f))\), in particular, it will often vanish. It can be given a cohomological interpretation. The last two sections deal with unitarizable representations, i.e. those \(({\mathcal V}, \pi)\) such that there exists an \((F^\times \backslash G(F))\)-invariant, positive definite, Hermitian scalar product on \({\mathcal V}\). As an example one has the Steinberg representation. Howe and Moore gave a criterion of non-unitarizability. This is discussed in detail.

Each chapter (except the sixth) ends with comments and references. These are often very useful and motivating. One may eagerly look forward to the second volume of this beautiful book!

(i) a preface sketching the main problem and the difficulties met to attack it;

(ii) eight chapters, the first five of which culminate in the sixth with a formula giving Lefschetz numbers in terms of elliptic orbital integrals, and the last two of which deal with fundamental results of local harmonic analysis to be used in the second volume of the book;

(iii) four appendices on central division algebras, Dieudonné theory, combinatorial formulas and the representation theory of locally compact, totally discontinuous, separated, topological groups, respectively;

(iv) four pages with 68 references, and

(v) a somewhat limited index.

Unfortunately, the author did not include a list of symbols. It should be stressed that everything is presented very explicitly and fully proved throughout the book, without making it easy reading. The subject matter is just too complicated.

Let \(p\) be a prime number and let \(X\) be a smooth, projective and connected curve over \(\mathbb{F}_p\) with generic point \(\eta\). Write \(F = \kappa (\eta)\) for its function field. For a place \(x\) of \(F\) let \(F_x\) be the completion of \(F\) at \(x\) and write \({\mathfrak O}_x\) for the corresponding valuation ring. After fixing a place \(\infty\) of \(F\) and identifying the set of closed points of \(X\) with the set of places of \(F\) one can define the ring of regular functions \(A\) of \(F : A = \{a \in F \mid x(a) \geq 0\), \(\forall x \in X \backslash \{\infty\}\}\). One defines a Drinfeld \(A\)-module of rank \(d > 0\) over the scheme \(S\) (of characteristic \(p)\) as a pair \((E, \varphi)\), where \(E\) is a commutative group scheme over \(S\) and \(\varphi : A \to \text{End} (E)\) is a homomorphism satisfying certain conditions involving \(d\). For a nonzero ideal \(I \subset A\) one can define a level-\(I\) structure \(\iota\) on \((E, \varphi)\). One defines the fibered category \({\mathcal M}^d_I\) over the category of characteristic \(p\) schemes by \({\mathcal M}^d_I (S) : =\) {the category of triples \((E,\varphi,\iota)\) where \((E,\varphi)\) is a Drinfeld \(A\)-module of rank \(d\) over \(S\) with level-\(I\) structure \(\iota\) on \((E, \varphi)\) (and some other condition on the characteristic)}. It is shown that for \(I \neq A\), \({\mathcal M}^d_I\) is representable by an affine scheme \(M^d_I\) of finite type over \(\mathbb{F}_p\). For \(I = A\), \({\mathcal M}^d_A\) is representable by a Deligne-Mumford algebraic stack \(M^d_A\) of finite type over \(\mathbb{F}_p\).

Let \(V(I) \subset X \backslash \{\infty\} = \text{Spec} (A)\) be the closed subscheme defined by \(I\). One has a morphism of algebraic stacks (schemes if \(I \neq A)\), \(\Theta : M^d_I \to X \backslash (\{\infty\} \cup V(I))\), smooth and of relative dimension \(d - 1\). For a place \(o\) of \(F\), \(o \neq \infty\), \(o \notin V(I)\), let \(\text{Frob}_o \in \text{Gal} (\overline{\kappa (o)}/ \kappa (o))\) be the geometric Frobenius. Define \(M^d_{I,o}(\overline{\kappa (o)})= \Theta^{-1} (o) (\overline{\kappa(o)})\). One can define the algebra of Hecke operators \({\mathcal H}_I^{\infty, o}\) acting on \(M^d_{I,o}\) by correspondences, and for \(f^{\infty,o} \in {\mathcal H}_I^{ \infty, o}\) one is interested in the action of \(\text{Frob}_o^r \times f^{\infty,o}\) on \(M^d_{I,o} (\overline {\kappa (o)})\), where \(r\) is a positive integer. In particular, the number of its fixed points, written \(\text{Lef}_r (f^{\infty,o})\), is of particular importance. Now, an element \(\gamma \in GL_d (F)\) is called elliptic if its minimal polynomial is irreducible over \(F\). Let \(GL_d (F)_\ell \) be a system of representatives in \(GL_d (F)\) of the elliptic conjugacy classes. An elliptic \(\gamma\) is said to be elliptic at the place \(\infty\) iff there is only one place \(\infty'\) of \(F' = F [\gamma]\) (this is a field for elliptic \(\gamma)\) dividing \(\infty\) of \(F\). \(\gamma\) is said to be \(r\)-admissible at the place \(o\) iff \(o (\text{det} (\gamma)) = r\) and there exists a place \(o'\) of \(F'\) dividing \(o\) such that \(x' (\gamma) = 0\) for all other places \(x'\) of \(F'\) which divide \(o\). Then, as a first result, one can deduce a formula for \(\text{Lef}_r (f^{\infty,o})\) as a (finite) sum over \(\gamma \in GL_d (F)_\ell\) which are elliptic at \(\infty\) and \(r\)-admissible at \(o\) of products of terms involving orbital and twisted orbital integrals. Then, by the so- called fundamental lemma one can replace the twisted orbital integrals by ordinary orbital integrals.

Another unpleasant quantity ocurring in the product terms of the above mentioned formula can be written as an ordinary orbital integral of the newly introduced concept of very cuspidal Euler-Poincaré function \(f_\infty \in {\mathcal C}^\infty_c (F^\times_\infty \backslash GL_d (F_\infty)//{\mathcal B}^0_\infty)\), where \({\mathcal B}^0_\infty \subset K_\infty = GL_d ({\mathfrak O}_\infty)\) denotes the Iwahori subgroup. Also, with \(K_o = GL_d ({\mathfrak O}_o)\), let \(f_o \in {\mathcal C}_c^\infty (GL_d (F_o)//K_o)\) be a suitable Hecke function (with prescribed Satake transform) and define \[ f_\mathbb{A} = f_\infty f^{\infty, o} f_o \in {\mathcal C}^\infty_c (F^{\times_\infty \backslash GL_d (\mathbb{A})//{\mathcal B}^0_\infty K_I^{\infty, o} K_o)}. \] Taking suitable Haar measures, we are now able to state the main result (chapter 6) on \(\text{Lef}_r (f^{\infty, o})\) in terms of the orbital integrals \[ O_\gamma (f_\mathbb{A}, dg_{\gamma, \mathbb{A}}) = \int_{GL_d (\mathbb{A})_\gamma \backslash GL_d (\mathbb{A})} f_\mathbb{A} (g^{-1}_\mathbb{A} \gamma g_\mathbb{A}) {dg_\mathbb{A} \over dg_{\gamma, \mathbb{A}}} \] and the volumes \(a(\gamma, dg_{\gamma, \mathbb{A}}) = \text{vol} (F^\times_\infty GL_d (F)_\gamma \backslash GL_d (\mathbb{A})_\gamma, {dg_{\gamma, \mathbb{A}} \over dz_\infty d \gamma'})\) as \[ \text{Lef}_r (f^{\infty, o}) = \sum_{\gamma \in GL_d (F)_{\ell}} a(\gamma, dg_{\gamma, \mathbb{A}}) O_\gamma (f_\mathbb{A}, dg_{\gamma, \mathbb{A}}). \] Here \(dg_{\gamma, \mathbb{A}}\) is a Haar measure on the centralizer \(GL_d (\mathbb{A})_\gamma\) of \(GL_d (\mathbb{A})\), and \(d \gamma'\) denotes the counting measure on \(GL_d (F)_\gamma\), the centralizer of \(\gamma\) in \(GL_d (F)\).

Chapter 7 deals with the construction and the main properties of the unramified principal series representations of \(G(F) = GL_d (F)\), \(F\) a non-archimedean local field. Subjects treated are parabolic induction and restriction, cuspidal representations, (unramified) principal series representations, and spherical representations. It covers work of Bernstein and Zelevinsky on the one hand, and of Casselman on the other hand.

In the last chapter it is proved that the Euler-Poincaré function introduced earlier is a pseudo-coefficient of the Steinberg representation. This result is due to Casselman and Kottwitz. It says that for an admissible irreducible representation \(({\mathcal V}, \pi)\) of \(F^\times G(F)\) and \(f \in {\mathcal C}^\infty_c (F^\times \backslash G(F)//{\mathcal B}^0)\) the very cuspidal Euler-Poincaré function, one has a description of \(\text{tr} (\pi (f))\), in particular, it will often vanish. It can be given a cohomological interpretation. The last two sections deal with unitarizable representations, i.e. those \(({\mathcal V}, \pi)\) such that there exists an \((F^\times \backslash G(F))\)-invariant, positive definite, Hermitian scalar product on \({\mathcal V}\). As an example one has the Steinberg representation. Howe and Moore gave a criterion of non-unitarizability. This is discussed in detail.

Each chapter (except the sixth) ends with comments and references. These are often very useful and motivating. One may eagerly look forward to the second volume of this beautiful book!

Reviewer: W.W.J.Hulsbergen (Haarlem)