# Reductions of Hecke correspondences on Anderson modular objects

A. Grishkov, D. Logachev111E-mails: shuragrigmail.com; logachev94gmail.com
(corresponding author)*************** First author: Departamento de Matemática
e estatistica Universidade de São Paulo. Rua de Matão 1010, CEP 05508-090, São
Paulo, Brasil, and Omsk State University n.a. F.M.Dostoevskii. Pr. Mira 55-A,
Omsk 644077, Russia. Second author: Departamento de Matemática, Universidade
Federal do Amazonas, Manaus, Brasil

We formulate some properties of a conjectural object $X_{fun}(r,n)$
parametrizing Anderson t-motives of dimension $n$ and rank $r$. Namely, we
give formulas for $\goth p$-Hecke correspondences of $X_{fun}(r,n)$ and its
reductions at $\goth p$ (where $\goth p$ is a prime of
$\mathbb{F}_{q}[\theta]$). Also, we describe their geometric interpretation.
These results are analogs of the corresponding results of reductions of
Shimura varieties. Finally, we give conjectural formulas for Hodge numbers
(over the fields generated by Hecke correspondences) of middle cohomology
submotives of $X_{fun}(r,n)$.

Moduli objects of Anderson t-motives, Hecke correspondences, reduction

###### :

11G09

††support: Thanks: The authors are grateful to FAPESP, São Paulo, Brazil for a
financial support (process No. 2017/19777-6). The first author is grateful to
SNPq, Brazil, to RFBR, Russia, grant 16-01-00577a (Secs. 1-4), and to Russian
Science Foundation, project 16-11-10002 (Secs. 5-8) for a financial support.
The second author is grateful to Laurent Lafforgue for invitation to IHES
where this paper was started. Discussions with Laurent Fargues and Alain
Genestier on the subject of these notes were very important. Particularly,
they indicated to the second author the analogy between Anderson varieties and
unitary Shimura varieties, and Alain Genestier attracted his attention to the
paper [F] where the notion of the dual Anderson module is defined, and
indicated the statement of Theorem 3.4.1a. Finally, the second author is
grateful to Nicholas Katz for a detailed explanation of his paper [K].

0\. Introduction. Let $X$ be a Shimura variety of PEL-type. Its points
parametrize abelian varieties with some fixed endomorphism rings, polarization
and level. There is a problem to describe relations between the rings
$\mathbb{H}_{p}(X)$ of $p$-Hecke correspondences of $X$ ($p$ is a prime) and
of $\tilde{X}_{p}$ \- the reduction of $X$ at $p$, particularly, to find the
characteristic polynomial of the Frobenius correspondence of $\tilde{X}_{p}$
over $\mathbb{H}_{p}(X)$. Also, we can ask what are geometric interpretations
of reductions of $p$-Hecke correspondences. There is also a problem of extreme
complexity — to prove Langlands theorems for Shimura varieties (relations
between $L$-functions of submotives of $X$ and of automorphic representations
of the corresponding reductive group $G$). It is solved only in a few cases of
$X$ of low dimension. Even exact statements of theorems giving these relations
is a complicated problem. Analogs of abelian varieties for the function field
case are Anderson t-motives (or Anderson modules — their categories are anti-
equivalent). It is natural to consider an object (function field analog of a
Shimura variety) whose points parametrize these t-motives (for example,
Anderson t-motives of a fixed dimension $n$, rank $r$, type of a nilpotent
operator $N$, see Section 2.4, and of an analog of PEL-type). We denote this
object by $X_{fun}=X_{fun}(r,n)$. Unfortunately, at the moment for $n>1$ these
objects are conjectural (only for the case $n=1$ we have a good theory of
moduli spaces of Drinfeld modules). For example, if we restrict ourselves by
pure uniformizable Anderson t-motives and assume that there is 1 – 1 (or near
1 – 1) correspondence between these t-motives and their lattices (which is
rather likely, see [GL17]), then the moduli variety of lattices would be the
quotient set of Siegel matrices by an (almost) action of
$GL_{r}(\mathbb{F}_{q}[\theta])$. But this (almost) action does not have
desired properties, see [GL17], Proposition 1.7.1. So, the whole contents of
the present paper concerns conjectural objects $X_{fun}(r,n)$ that we shall
call Anderson modular objects. Nevertheless, we can get some information about
them. Let us give more definitions. Naively, $X_{fun}(r,n)$ parametrize pure
abelian t-motives of rank $r$ and dimension $n$ whose nilpotent operator $N$
is 0. The corresponding reductive group $G$ is $GL_{r}$ and the dominant
coweight $\mu$ is $(1,\dots,1,0,\dots,0)$ ($n$ ones and $r-n$ zeroes). For
more information for any $G$ on $X_{fun}$ and $\mu$ see [V]. Let $\goth p$ be
a prime ideal of $\mathbb{F}_{q}[\theta]$ ($q$ is a power of $p$ and $\theta$
an abstract variable, see Section 2). The contents of the present paper is the
following. (1) We formulate (for some cases) in Section 2 the theorems
concerning reductions at $\goth p$ of Hecke correspondences $T_{\goth p,i}$
($i=0,\dots,r$) on $X_{fun}$, and their geometric interpretation. These
results are functional analogs of [FCh], Chapter 7, [BR], Chapter 6, and [W].
The case $n=1$ (Drinfeld varieties) is treated with more details and explicit
formulas. (2) Sketches of the proofs of these theorems are given in Section 3.
We see that the function field case — Anderson varieties of rank $r$ and
dimension $n$, $G_{fun}=GL_{r}(\mathbb{F}_{q}(\theta))$ is analogous to the
number field case where $G_{num}=GU(r-n,n)$ corresponds to Shimura varieties
(called unitary for brevity) of PEL-type parametrizing abelian $r$-folds with
multiplication by an imaginary quadratic field $K$, of signature $(r-n,n)$. We
indicate in Section 4 that really, properties of unitary Shimura varieties are
similar to the properties of $X_{fun}(r,n)$. By the way, this analogy is a
source of more results, see for example [GL21]. (3) Finally, in Section 5 we
state conjectural formulas for Hodge numbers (over fields generated by Hecke
correspondences) of submotives of middle cohomology of $X_{fun}$. They are
analogs of the corresponding formulas for Shimura varieties ([BR], SEction
4.3, p. 548). In Section 6 we consider the action of Hecke correspondences on
some non-ordinary Drinfeld modules. These results will be useful for a future
proof of analog of Kolyvagin’s theorem (finiteness of Tate-Shafarevich group)
for the case of Drinfeld varieties. In order to show the analogy between the
number field and the function fiels cases, we give in Section 1 some well-
known results on Hecke correspondences and their reductions for the case of
Siegel varieties. 1\. Definitions and results for the number field case
(Siegel varieties). 1.1. Reductions of correspondences. For a comparison with
the number field case, here we formulate well-known results for reductions of
Siegel modular varieties. References for the results of this section are:
[FCh], Section 7, and [W]. Let $X$ be a Siegel variety of genus $g$ (of any
fixed level), i.e. a quotient of the Siegel upper half plane by a congruence
subgroup of $GSp_{2g}(\mathbb{Z})$, or, equivalently, a set of principally
polarized abelian varieties of dimension $g$ together with some level
structure. Let the congruence subgroup be such that $X$ is defined over
$\mathbb{Q}$. We have $G=G_{X}=GSp_{2g}(\mathbb{Q})$ is the corresponding
reductive group. Let $p$ be a fixed prime which does not divide the level, and
$\tilde{X}$ the reduction of $X$ at $p$. Let $\hbox{ Corr}(X)=\hbox{
Corr}_{p}(X)$ (resp. $\hbox{ Corr}(\tilde{X})=\hbox{ Corr}_{p}(\tilde{X})$) be
the algebra of $p$-Hecke correspondences on $X$ (resp. $\tilde{X}$). We have
the Frobenius map on $\tilde{X}$; considering it as a correspondence we get an
element $fr_{X}\in\hbox{ Corr}(\tilde{X})$. There is a map $\gamma:\hbox{
Corr}(X)\to\hbox{ Corr}(\tilde{X})$ — the reduction of a correspondence at
$p$. There is a problem of description of $\gamma$ and of finding of the
characteristic polynomial of $fr$ over $im(\gamma)$. The answer is the
following. Let $M$ be the following block diagonal subgroup of $G$:

$M=\left\\{\left(\begin{matrix}A&0\\\
0&\lambda\cdot(A^{t})^{-1}\end{matrix}\right)\right\\}\subset G,$ $None$

(blocks have size $g$), and let $T$ be the subgroup of $M$ consisting of
diagonal matrices. The abstract $p$-Hecke algebras
$\mathbb{H}(G)=\mathbb{H}(G)(\mathbb{Q}_{p})$ (resp.
$\mathbb{H}(M)=\mathbb{H}(M)(\mathbb{Q}_{p})$,
$\mathbb{H}(T)=\mathbb{H}(T)(\mathbb{Q}_{p})$) consist of double cosets
$K\alpha K$, where $K=K_{G}=G(\mathbb{Z}_{p})$ (resp.
$K=K_{M}=M(\mathbb{Z}_{p})$, $K=K_{T}=T(\mathbb{Z}_{p})$) and $\alpha\in
G(\mathbb{Q}_{p})$, resp. $\alpha\in M(\mathbb{Q}_{p})$, $\alpha\in
T(\mathbb{Q}_{p})$. There are the Satake inclusions
$S^{G}_{M}:\mathbb{H}(G)\to\mathbb{H}(M)$,
$S^{M}_{T}:\mathbb{H}(M)\to\mathbb{H}(T)$. The Hecke algebra $\mathbb{H}(T)$
is the subalgebra of $\mathbb{Z}[U_{1}^{\pm 1},...,U_{g}^{\pm 1},V_{1}^{\pm
1},...,V_{g}^{\pm 1}]$ (here $U_{i}$, $V_{i}$ are abstract variables)
generated by $(U_{1}V_{1}^{-1})^{\pm 1},...,(U_{g}V_{g}^{-1})^{\pm 1}$ and
$(U_{1}\cdot...\cdot U_{g})^{\pm 1}$. The Weyl group $W_{G}$ of $G$ is the
semidirect product of the permutation group $S_{g}$ and of
$(\mathbb{Z}/2\mathbb{Z})^{g}=(\pm 1)^{g}$, where $S_{g}$ permutes coordinates
in $(\pm 1)^{g}$. There is a section $S_{g}\hookrightarrow W_{G}$, we denote
its image by $W_{G,M}$. Further, $W_{G}$ acts on $\mathbb{H}(T)$ in the
obvious manner: $S_{g}$ permutes indices and $(\pm 1)^{g}$ interchanges
$U_{*}$, $V_{*}$. We have:

$S^{M}_{T}(\mathbb{H}(M))=\mathbb{H}(T)^{W_{G,M}},\ \
S^{G}_{T}(\mathbb{H}(G))=\mathbb{H}(T)^{W_{G}}.$ $None$

It is known that there are surjections

$\beta_{1}:\mathbb{H}(G)\to\hbox{ Corr}(X),$ $None$
$\beta_{2}:\mathbb{H}(M)\to\hbox{ Corr}(\tilde{X})$ $None$

whose kernel is generated by the relation $KpK=id$, where $p=pI_{2g}$ is the
scalar matrix in both $G$ and $M$, and $id$ is the trivial correspondence on
both $X$, $\tilde{X}$. Theorem 1.1.5. There exists a commutative diagram:

$\begin{matrix}S^{G}_{M}:&\mathbb{H}(G)&\to&\mathbb{H}(M)\\\ \\\
&\beta_{1}\downarrow&&\beta_{2}\downarrow\\\ \\\ \gamma:&\hbox{
Corr}(X)&\to&\hbox{ Corr}(\tilde{X})\end{matrix}$ $None$

We denote by $\tau_{p}$ the matrix $\left(\begin{matrix}1&0\\\
0&p\end{matrix}\right)$, where entries are scalar $g\times g$-blocks, and we
denote the corresponding elements $K_{G}\tau_{p}K_{G}$ (resp.
$K_{M}\tau_{p}K_{M}$) of $\mathbb{H}(G)$ (resp. $\mathbb{H}(M)$) by $T_{p}$
(resp. $fr_{M}$). Theorem 1.1.7. $\beta_{2}(fr_{M})=fr_{X}$. Remark. Formulas
1.1.2 and theorems 1.1.5, 1.1.7 permit us to find the Hecke polynomial of $X$
(the characteristic polynomial of $fr_{X}$ over $\hbox{ Corr}(X)$ ). Really,
1.1.2 implies that $\mathbb{H}(M)$ is a free module over
$S^{G}_{M}(\mathbb{H}(G))$ of dimension $\\#(W_{G})/\\#(W_{G,M})=2^{g}$. An
explicit description of $fr_{M}\in\mathbb{H}(M)$ (see below) permits us to
find easily its characteristic polynomial over $\mathbb{H}(G)$. Theorem 1.1.5
implies that it is also the characteristic polynomial of the Frobenius
correspondence on $\tilde{X}$ over the algebra $\hbox{ Corr}X$. 1.2. Geometric
interpretation.For $i=0,\dots,g$ we consider diagonal matrices in a block form

$\varphi_{i}=\left(\begin{matrix}I_{i}&0&0&0\\\ 0&pI_{g-i}&0&0\\\
0&0&pI_{i}&0\\\ 0&0&0&I_{g-i}\end{matrix}\right)\in M$

(sizes of diagonal blocks are $i$, $g-i$, $i$, $g-i$), and we denote the
corresponding elements $K_{M}\varphi_{i}K_{M}\in\mathbb{H}(M)$ by $\Phi_{i}$.
Particularly, $\Phi_{g}=fr_{M}$. Let $I$ be a subset of $\\{1,...,g\\}$. We
denote $U_{I}:=\prod_{i\in I}U_{i}\prod_{i\not\in I}V_{i}\in\mathbb{H}(T)$. We
have

$S^{M}_{T}(\Phi_{i})=\sum_{\\#(I)=i}U_{I}$

and

$S^{G}_{M}(T_{p})=\Phi_{0}+\Phi_{1}+\dots+\Phi_{g}\in\mathbb{H}(M).$ $None$

We denote $\beta_{1}(T_{p})\in\hbox{ Corr}(X)$, $\beta_{2}(\Phi_{i})\in\hbox{
Corr}(\tilde{X})$ again by $T_{p}$, $\Phi_{i}$ respectively, so (1.2.1) and
Theorem 1.1.5 give us the following equality on $\hbox{ Corr}(\tilde{X})$:

$\gamma(T_{p})=\Phi_{0}+\Phi_{1}+\dots+\Phi_{g}.$ $None$

For any algebraic variety $Z$ there exists an involution on $\hbox{ Corr}(Z)$
(symmetry with respect the coordinates). Also, there exist involutions on
$\mathbb{H}(M)$, $\mathbb{H}(G)$ commuting with involutions on $\hbox{
Corr}(X)$, $\hbox{ Corr}(\tilde{X})$ with respect to (1.1.6). We denote these
involutions by hat; we have $\hat{T}_{p}=T_{p}$, $\hat{\Phi}_{i}=\Phi_{g-i}$.
The geometric interpretation of (1.2.2) is the following. Let $t\in
X(\bar{\mathbb{Q}})$ be a generic point, $A_{t}$ the corresponding principally
polarized abelian $g$-fold with a fixed polarization form and $(A_{t})_{p}$
the $\mathbb{F}_{p}$-space of its $p$-torsion points. The polarization on
$A_{t}$ defines a skew form on $(A_{t})_{p}$. $T_{p}(t)$ is a finite set of
points; we have: $t^{\prime}\in T_{p}(t)$ iff there exists an isogeny
$\alpha_{t,t^{\prime}}:A_{t}\to A_{t^{\prime}}$ of type $(1,...,1,p,...,p)$.
The kernel of $\alpha_{t,t^{\prime}}$ is an isotropic $g$-dimensional subspace
of $(A_{t})_{p}$. So, we have a Theorem 1.2.3. The set $T_{p}(t)$ is in 1–1
correspondence with the set of isotropic $g$-dimensional subspaces of
$(A_{t})_{p}$. Now let $t\in X(\mathbb{Q})$ be a generic point such that
$A_{t}$ has a good ordinary reduction at $p$. Let $(\tilde{A}_{t})_{p,points}$
be the set of closed points of $\tilde{A}_{t}$ of order $p$, and
$red:(A_{t})_{p}\to(\tilde{A}_{t})_{p,points}$ the reduction map. We denote by
$D=D_{Siegel}$ the kernel of $red$, it is an isotropic $g$-dimensional
subspace of $(A_{t})_{p}$. Let $t^{\prime}\in T_{p}(t)$ and
$\tilde{t}^{\prime}\in\tilde{X}$ its reduction. (1.2.2) shows that
$\tilde{t}^{\prime}$ belongs to one of $\Phi_{i}(\tilde{t})$. Theorem 1.2.4.
Number $i$ is defined as follows:

$i=\hbox{dim }(\hbox{Ker }(\alpha_{t,t^{\prime}})\cap D_{Siegel}).$ $None$

Particularly, $i=g\iff\tilde{t}^{\prime}=fr(\tilde{t})\iff\hbox{Ker
}(\alpha_{t,t^{\prime}})=D_{Siegel}$. Further, we have the following Theorem
1.2.6. Let $t^{\prime}$, $t^{\prime\prime}$ be 2 points of $T_{p}(t)$. Then

$\tilde{t}^{\prime}=\tilde{t}^{\prime\prime}\iff\hbox{Ker
}(\alpha_{t,t^{\prime}})\cap D_{Siegel}=\hbox{Ker
}(\alpha_{t,t^{\prime\prime}})\cap D_{Siegel}.$

Recall that any correspondence $C$ on $X$ has the bidergee $d_{1}(C),d_{2}(C)$
— the degrees of 2 projections $\pi_{1}$, $\pi_{2}$ of its graph $\Gamma(C)$
to $X$. By definition, $\pi_{i}(\hat{C})=\pi_{\hat{i}}(C)$ (here $\hat{1}=2,\
\hat{2}=1$). Further, $C$ has the separable (resp. non-separable) bidergee
$d^{s}_{1}(C),d^{s}_{2}(C)$ (resp. $d^{ns}_{1}(C),d^{ns}_{2}(C)$) — the
separable (resp. non-separable) degrees of $\pi_{1}$, $\pi_{2}$. We have
$d_{i}(C)=d^{s}_{i}(C)d^{ns}_{i}(C)$ and

$d_{i}^{*}(\hat{C})=d_{\hat{i}}^{*}(C),\ \ \ i=1,2,\ \ \ \ *=\emptyset,s,ns.$
$None$

We denote by $\goth g(k,l)$ the cardinality of the Grassmann variety
$Gr(k,l)(\mathbb{F}_{p})$:

$\goth g(k,l)=\prod_{i=1}^{k}\frac{p^{l}-p^{i-1}}{p^{k}-p^{i-1}}.$ $None$

Theorem 1.2.8.

$d_{1}^{s}(\Phi_{i})=\goth g(i,g),\ d_{1}^{ns}(\Phi_{i})=p^{(g+1-i)(g-i)/2},\
d_{2}^{s}(\Phi_{i})=\goth g(i,g),\ d_{2}^{ns}(\Phi_{i})=p^{(i+1)i/2}.\
\square$

2\. Definitions and statement of conjectures for the case of Anderson modular
objects.We use standard notations for Anderson t-motives. Let $q$ be a power
of a prime $p$, $\mathbb{F}_{q}$ the finite field of order $q$. The function
field analog of $\mathbb{Z}$ is the ring of polynomials
$\mathbb{F}_{q}[\theta]$ where $\theta$ is an abstract variable. The analog of
the archimedean valuation on $\mathbb{Q}$ is the valuation at infinity of the
fraction field $\mathbb{F}_{q}(\theta)$ of $\mathbb{F}_{q}[\theta]$; it is
denoted by $ord$, it is uniquely determined by the property $\hbox{ord
}(\theta)=-1$. The completion of an algebraic closure of the completion of
$\mathbb{F}_{q}(\theta)$ with respect the valuation ”ord” is the function
field analog of $\mathbb{C}$. It is denoted by $\mathbb{C}_{\infty}$. The
definition of a t-motive $\goth M$ is given in [G], Definitions 5.4.2, 5.4.12
(Goss uses another terminology: ”abelian t-motive” of [G] = ”t-motive” of the
present paper). Particularly, $\goth M$ is a free
$\mathbb{C}_{\infty}[T]$-module of dimension $r$ (this number $r$ is called
the rank of $\goth M$) endowed by a $\mathbb{C}_{\infty}$-skew-linear operator
$\tau$ satisfying some properties. Its dimension $n$ is defined in [G], Remark
5.4.13.2 (Goss denotes the dimension by $\rho$). A nilpotent operator
$N=N(\goth M)$ associated to a t-motive is defined in [G], Remark 5.4.3.2.
Condition $N=0$ implies $n\leq r$. Except Section 2.4, we shall consider only
the case $N=0$. As it was written in the introduction, the main object of the
present paper is conjectural. It is called an Anderson modular object, it is
denoted by $X_{fun}=X_{fun}(r,n)$, it is the function field analog of $X$.
Naively, it parametrizes Anderson t-motives of rank $r$ and dimension $n$. An
analog of $p$ of Section 1 is a valuation (distinct of ord) of
$\mathbb{F}_{q}(\theta)$ = a prime ideal of $\mathbb{F}_{q}[\theta]$. We
denote by $\goth p$ both its generator and the prime ideal itself, and we
denote $\goth q=\\#(\mathbb{F}_{q}[\theta]/\goth p)$. The corresponding
algebraic group $G_{fun}$ — the function field analog of
$GSp_{2g}(\mathbb{Q})$ — is $GL_{r}(\mathbb{F}_{q}(\theta))$. Hence, the
analogs of $\mathbb{Q}_{p}$, $\mathbb{Z}_{p}$ for the functional case are
$\mathbb{F}_{q}(\theta)_{\goth}p$, $\mathbb{F}_{q}[\theta]_{\goth}p$
respectively, and the analog of $K_{G}$ of Section 1 is
$GL_{r}(\mathbb{F}_{q}[\theta]_{\goth}p)$ (it will be denoted by $K_{G}$ as
well). In order to simplify the present version of the text, for the case
$n>1$ we state conjectures of Section 2.3 only for uniformizable Anderson
t-motives. Analytically, an uniformizable Anderson t-motives of rank $r$ and
dimension $n$ over $\mathbb{C}_{\infty}$ is the quotient
$\mathbb{C}_{\infty}/L$, where $L$ is a free $r$-dimensional
$\mathbb{F}_{q}[\theta]$-module. Since not all Anderson t-motives are
uniformizable, the exact statements of these conjectures must be slightly
changed, see Remark 2.3.4a.

Theorem 2.1. The analog of $M$ for this case is the group $M_{r-n,n}$ of block
diagonal matrices $\left(\begin{matrix}*&0\\\ 0&*\end{matrix}\right)\subset
GL_{r}(\mathbb{F}_{q}(\theta))$, sizes of blocks are $r-n$, $n$. We denote
$K_{M}:=M(\mathbb{F}_{q}[\theta]_{\goth}p)$. The analogs of Hecke algebras and
of algebras of correspondences are defined like in Section 1. The analogs of
Theorems 1.1.5, 1.1.7 hold for this case. Particularly, $\hbox{
Corr}(\tilde{X}_{fun})$ is the quotient of $\mathbb{H}(M)$ by the trivial
relation $K_{M}\goth pK_{M}=id$. 2.2. Description of $\mathbb{H}(G_{fun})$,
$\mathbb{H}(M)$ and of the Satake inclusions.Let like in Section 1, $T$ be the
subgroup of $G_{fun}$ of diagonal matrices. We have
$\mathbb{H}(T)=\mathbb{Z}[U_{1}^{\pm 1},...,U_{r}^{\pm 1}]$, and the Weyl
group $W_{G_{fun}}=S_{r}$, it acts on $\mathbb{H}(T)$ permuting indices. An
analog of $W_{G,M}$ is the subgroup $W_{G_{fun},M}=S_{r-n}\times
S_{n}\hookrightarrow S_{r}=W_{G_{fun}}$ with the obvious inclusion. Formulas
(1.1.2) hold for our case, explicit formulas are the following. 2.2.1.
$\mathbb{H}(G_{fun})$: For $i=0,\dots,r$ we denote by $\tau_{\goth p,i}$ the
diagonal matrix $\left(\begin{matrix}I_{r-i}&0\\\ 0&\goth
pI_{i}\end{matrix}\right)\in G_{fun}$, where sizes of blocks are $r-i$, $i$,
and we denote the corresponding elements $K_{G}\tau_{\goth
p,i}K_{G}\in\mathbb{H}(G_{fun})$ by $T_{\goth p,i}$. We have $T_{\goth
p,0}=1$, $T_{\goth p,r}$ is the trivial correspondence, and other $T_{\goth
p,i}$ are free generators of $\mathbb{H}(G_{fun})$. We have

$S^{G_{fun}}_{T}(T_{\goth p,i})=\goth
q^{-i(i-1)/2}\sigma_{i}(U_{1},...,U_{r}),$ $None$

where $\sigma_{i}$ is the $i$-th symmetric polynomial, and

$\hat{T}_{\goth p,i}=T_{\goth p,r-i}.$ $None$

2.2.3. $\mathbb{H}(M)$: (a) For $i=0,\dots,r-n$ we denote by $\varphi_{i}$ the
diagonal matrix $\left(\begin{matrix}I_{r-n-i}&0&0\\\ 0&\goth pI_{i}&0\\\
0&0&I_{n}\end{matrix}\right)\in M$ where sizes of blocks are $r-n-i$, $i$,
$n$, and we denote the corresponding elements
$K_{M}\varphi_{i}K_{M}\in\mathbb{H}(M)$ by $\Phi_{i}$.

(b) For $i=0,\dots,n$ we denote by $\psi_{i}$ the diagonal matrix
$\left(\begin{matrix}I_{r-n}&0&0\\\ 0&\goth pI_{i}&0\\\
0&0&I_{n-i}\end{matrix}\right)\in M$ where sizes of blocks are $r-n$, $i$,
$n-i$, and we denote the corresponding elements
$K_{M}\psi_{i}K_{M}\in\mathbb{H}(M)$ by $\Psi_{i}$. We have
$\Phi_{0}=\Psi_{0}=1$, $\Phi_{i}=0$ (resp. $\Psi_{i}=0$) if $i$ is out of the
range $0,\dots,r-n$ (resp. $0,\dots,n$) and other $\Phi_{i}$, $\Psi_{i}$ are
free generators of $\mathbb{H}(M)$. Obviously

$S^{M}_{T}(\Phi_{i})=\goth q^{-i(i-1)/2}\sigma_{i}(U_{1},...,U_{r-n}),\
S^{M}_{T}(\Psi_{i})=\goth q^{-i(i-1)/2}\sigma_{i}(U_{r-n+1},...,U_{r}).$
$None$

Formulas (2.2.2), (2.2.4) imply immediately that

$S^{G_{fun}}_{M}(T_{\goth p,j})=\sum_{i=0}^{j}\goth
q^{-i(j-i)}\Psi_{i}\Phi_{j-i}.$ $None$

Further, we have:

$\Psi_{n}\Phi_{r-n}=\goth q^{n(r-n)},$ $None$ $\hat{\Phi}_{i}=\goth
q^{-n(r-n-i)}\Psi_{n}\Phi_{r-n-i},$ $None$ $\hat{\Psi}_{i}=\goth
q^{-(n-i)(r-n)}\Psi_{n-i}\Phi_{r-n}.$ $None$

particularly, $\hat{\Phi}_{r-n}=\Psi_{n}$. Coefficients of (2.2.5b,c) can be
found from the property that equations (2.2.2a) and (2.2.5) are concordant
with respect to the duality. Remark. (2.2.5), (2.1), (1.1.5) imply that for
the case $n=1$ (Drinfeld modules) the explicit formulas are the following
($fr=\Psi_{1}$):

$\tilde{T}_{\goth p,1}=fr+\Phi_{1};$ $\tilde{T}_{\goth p,2}=\frac{1}{\goth
q}fr\ \Phi_{1}+\Phi_{2};$ $\dots$ $None$ $\tilde{T}_{\goth
p,r-1}=\frac{1}{\goth q^{r-2}}fr\ \Phi_{r-2}+\Phi_{r-1};$ $\tilde{T}_{\goth
p,r}=\frac{1}{\goth q^{r-1}}fr\ \Phi_{r-1}.$

$\mathbb{H}(M)$ is a free module over $S^{G_{fun}}_{M}(\mathbb{H}(G_{fun}))$
respectively the Satake inclusion. Its dimension is
$\\#(W_{G_{fun}})/\\#(W_{G_{fun},M})=\left(\begin{matrix}r\\\
n\end{matrix}\right)$.

We denote $\Psi_{n}$ by $fr_{M}$. Its characteristic polynomial over
$S^{G_{fun}}_{M}(\mathbb{H}(G_{fun}))$ (the Hecke polynomial) can be easily
found by elimination of $\Phi_{1},...,\Phi_{r-n},\Psi_{1},...,\Psi_{n-1}$ in
the system (2.2.5). We denote it by $P_{r,n}$, it belongs to
$\mathbb{H}_{\goth p}(G_{fun})[fr_{M}]$. For $n=1$ we have

$\begin{matrix}P_{r,1}=\sum_{i=0}^{r}(-1)^{i}\goth q^{i(i-1)/2}T_{\goth
p,i}fr^{r-i}\\\ \\\ =fr^{r}-T_{\goth p,1}fr^{r-1}+\goth qT_{\goth
p,2}fr^{r-2}\pm\dots+(-1)^{r}\goth q^{r(r-1)/2}T_{\goth p,r}.\end{matrix}$
$None$

2.3. Statements of results. Theorems 1.2.4, 1.2.6 can be rewritten almost
word-to-word as conjectures for the functional case. Let us do it. It is more
convenient to use Anderson modules ([G], 5.4.5; Goss calls them Anderson
$T$-modules) instead of Anderson t-motives. The categories of Anderson
t-motives and modules are anti-isomorphic, so there is no essential difference
which object to use.

Let $t\in X_{fun}$ be such that the corresponding Anderson module $E_{t}$ is
uniformizable: $E_{t}=\mathbb{C}_{\infty}^{n}/L$ (as earlier, $n$ is the
dimension and $r$ is the rank). This condition is “closed under Hecke
correspondences”: if $t^{\prime}\in T_{\goth p,j}(t)$ then $E_{t^{\prime}}$ is
also uniformizable. Obviously, we have Theorem 2.3.1. $(E_{t})_{\goth p}$ —
the set of ${\goth p}$-torsion points of $E_{t}$ — is $\goth p^{-1}L/L$ and
hence is an $r$-dimensional $\mathbb{F}_{q}[\theta]/\goth p$-vector space.
Theorem 2.3.2. $t^{\prime}\in T_{\goth p,j}(t)$ iff there exists an isogeny
$\alpha_{t,t^{\prime}}:E_{t}\to E_{t^{\prime}}$ of type $(1,...,1,\goth
p,...,\goth p)$ ($r-j$ 1’s and $j$ ${\goth p}$’s). Theorem 2.3.3. The set
$T_{\goth p,j}(t)$ is in 1–1 correspondence with the set of $j$-dimensional
subspaces of $(E_{t})_{\goth p}$. Now we can formulate the conjecture on
reductions at ${\goth p}$. Let $(\tilde{E}_{t})_{\goth p,points}$ and
$red:(E_{t})_{\goth p}\to(\tilde{E}_{t})_{\goth p,points}$ be analogs of the
corresponding objects in Section 1.2. We denote by $D_{fun}$ the kernel of
$red$. Firstly, we have a Theorem 2.3.4. For a generic $t\in X_{fun}$
$D_{fun}$ is an $n$-dimensional subspace of $(E_{t})_{\goth p}$. Remark
2.3.4a. For an arbitrary $t\in X_{fun}$ (such that $E_{t}$ is not
uniformizable) statements of the above and below theorems and conjectures
require minor changes. For example, Theorem 2.3.1 becomes Theorem 2.3.4b.
$(E_{t})_{\goth p}$ is an $r$-dimensional $A/\goth p$-vector space. (we cannot
claim that it is $\goth p^{-1}L/L$ because $L$ does not exist). Geometric
interpretation.

Here we formulate analogs of (1.2.4) – (1.2.8) for the function field case.
Let $t^{\prime}\in T_{\goth p,j}(t)$ and
$\tilde{t}^{\prime}\in\tilde{X}_{fun}$ its reduction. Conjecture 2.1 and
(2.2.5) show that $\tilde{t}^{\prime}$ belongs to one of $\goth
q^{-i(j-i)}\Psi_{i}\Phi_{j-i}(\tilde{t})$. Theorem 2.3.5. Number $i$ is
defined as follows:

$i=\hbox{dim }(\hbox{Ker }(\alpha_{t,t^{\prime}})\cap D_{fun}).$ $None$

Particularly,
$j=i=n\iff\tilde{t}^{\prime}=fr_{X_{fun}}(\tilde{t})\iff\hbox{Ker
}(\alpha_{t,t^{\prime}})=D_{fun}$. Now $\goth g(k,l)$ will mean the
cardinality of Grassmann variety $Gr(k,l)$ over $\mathbb{F}_{q}[\theta]/\goth
p$, i.e. $p$ in (1.2.7) must be replaced by $\goth q$. Obviously we have

$d_{1}(T_{\goth p,i})=d_{2}(T_{\goth p,i})=\goth g(i,r).$

For the reader’s convenience, we formulate the following conjecture separately
for the case $n=1$. Theorem 2.3.7. For $n=1$ we have:

$d_{1}^{s}(\Psi_{1})=d_{1}^{ns}(\Psi_{1})=1,\ d_{2}^{s}(\Psi_{1})=1,\
d_{2}^{ns}(\Psi_{1})=\goth q^{r-1};$ $d_{1}^{s}(\Phi_{i})=\goth g(i,r-1),\
d_{1}^{ns}(\Phi_{i})=\goth q^{i},\ d_{2}^{s}(\Phi_{i})=\goth g(i,r-1),\
d_{2}^{ns}(\Phi_{i})=1.$

Corollary 2.3.7.1. For correspondences $\frac{1}{\goth q^{i}}fr\ \Phi_{i}$ —
summands in the right hand side of (2.2.6) — we have

$d_{1}^{s}(\frac{1}{\goth q^{i}}fr\ \Phi_{i})=\goth g(i,r-1),\ \
d_{1}^{ns}(\frac{1}{\goth q^{i}}fr\ \Phi_{i})=1,$ $d_{2}^{s}(\frac{1}{\goth
q^{i}}fr\ \Phi_{i})=\goth g(i,r-1),\ \ d_{2}^{ns}(\frac{1}{\goth q^{i}}fr\
\Phi_{i})=\goth q^{r-1-i}.$

An analog of the Theorem 1.2.6 is the following. Let $t^{\prime}$,
$t^{\prime\prime}$ be 2 points of $T_{\goth p,i}(t)$. Firstly we consider the
case when

$\hbox{Ker }(\alpha_{t,t^{\prime}})\cap D_{fun}=\hbox{Ker
}(\alpha_{t,t^{\prime\prime}})\cap D_{fun}=0,$

i.e. both $\tilde{t}^{\prime}$,
$\tilde{t}^{\prime\prime}\in\Phi_{i}(\tilde{t})$ (the second summand in
(2.2.6)). Theorem 2.3.8. Under this condition we have (for any $n$):
$\tilde{t}^{\prime}=\tilde{t}^{\prime\prime}$ as closed points iff the linear
spans coincide:

$<\hbox{Ker }(\alpha_{t,t^{\prime}}),D_{fun}>=<\hbox{Ker
}(\alpha_{t,t^{\prime\prime}}),D_{fun}>.$

If (for $n=1$)

$\hbox{Ker }(\alpha_{t,t^{\prime}})\supset D_{fun},\ \hbox{Ker
}(\alpha_{t,t^{\prime\prime}})\supset D_{fun},$

i.e. both $\tilde{t}^{\prime}$, $\tilde{t}^{\prime\prime}\in\frac{1}{\goth
q^{i-1}}\Psi_{1}\Phi_{i-1}(\tilde{t})=\frac{1}{\goth q^{i-1}}fr\
\Phi_{i-1}(\tilde{t})$ (the first summand in (2.2.6)) then (2.3.7.1) implies
that $\tilde{t}^{\prime}$, $\tilde{t}^{\prime\prime}$ are always different.
Now let us consider the case of arbitrary $n$. Theorem 2.3.10.

$d_{1}^{s}(\Psi_{i})=\goth g(i,n),\ d_{1}^{ns}(\Psi_{i})=1;$
$d_{1}^{s}(\Phi_{i})=\goth g(i,r-n),\ d_{1}^{ns}(\Phi_{i})=q^{in}.$

Remark 2.3.10a. Numbers $d_{2}^{*}(\Phi_{i})$, $d_{2}^{*}(\Psi_{i})$
($*=\emptyset,s,ns$) can be found from the above formulas using (1.2.6a),
(2.2.5b,c). Particularly, for the Frobenius $\Psi_{n}$ we have

$d_{2}^{s}(\Psi_{n})=1,\ d_{2}^{ns}(\Psi_{n})=q^{n(r-n)}.$

Corollary 2.3.11. For correspondences $\frac{1}{\goth
q^{i(j-i)}}\Psi_{i}\Phi_{j-i}$ — summands in the right hand side of (2.2.5) —
we have

$d_{1}^{s}(\frac{1}{\goth q^{i(j-i)}}\Psi_{i}\Phi_{j-i})=\goth g(i,n)\goth
g(j-i,r-n),\ d_{1}^{ns}(\frac{1}{\goth q^{i(j-i)}}\Psi_{i}\Phi_{j-i})=\goth
q^{(j-i)(n-i)}.$

Let $t^{\prime}$, $t^{\prime\prime}\in T_{\goth p,j}(t)$ be as above such that

$\hbox{dim }(\hbox{Ker }(\alpha_{t,t^{\prime}})\cap D_{fun})=\hbox{dim
}(\hbox{Ker }(\alpha_{t,t^{\prime\prime}})\cap D_{fun})=i.$

According Theorem 2.3.5, both $\tilde{t}^{\prime}$,
$\tilde{t}^{\prime\prime}\in\frac{1}{\goth q^{i(j-i)}}\Psi_{i}\Phi_{j-i}$.
Theorem 2.3.12. $\tilde{t}^{\prime}=\tilde{t}^{\prime\prime}$ as closed points
iff both intersections and linear spans coincide:

$\hbox{Ker }(\alpha_{t,t^{\prime}})\cap D_{fun}=\hbox{Ker
}(\alpha_{t,t^{\prime\prime}})\cap D_{fun};$ $<\hbox{Ker
}(\alpha_{t,t^{\prime}}),D_{fun}>=<\hbox{Ker
}(\alpha_{t,t^{\prime\prime}}),D_{fun}>.$

Remark 2.3.13. (a) If we fix an $i$-dimensional subspace $V_{i}\subset
D_{fun}$ and a $(n+j-i)$-dimensional overspace $V_{n+j-i}\supset D_{fun}$ then
the quantity of $j$-dimensional spaces $\alpha$ such that

$\alpha\cap D_{fun}=V_{i},\ \ <\alpha,D_{fun}>=V_{n+j-i},$

is equal to $\goth q^{(j-i)(n-i)}=d_{1}^{ns}(\frac{1}{\goth
q^{i(j-i)}}\Psi_{i}\Phi_{j-i})$, as it is natural to expect. (b) Thanks to
existence of skew pairing in the number case, we have

$<\hbox{Ker }(\alpha_{t,t^{\prime}}),D_{fun}>=(\hbox{Ker
}(\alpha_{t,t^{\prime}})\cap D_{fun})^{dual},$

($dual$ is with respect to the skew pairing), so in Theorem 1.2.6 it is
sufficient to claim only coincidence of intersections. (c) For $n=1$ (Theorem
2.3.8) intersections always coincide, so it is sufficient to claim only
coincidence of linear spans. 2.4. Case of $N\neq 0$. This is a subject of
further research. Here we do not even give statements of results, we indicate
only the discrete invariants of t-motives having $N\neq 0$ and explain the
methods how these statements can be obtained. Let $\goth M$ be a uniformizable
Anderson t-motive such that its $N$ is not (necessarily) 0, of dimension $n$
and of rank $r$. $\goth M$ is a $\mathbb{C}_{\infty}[T]$-module with a skew
map $\tau:\goth M\to\goth M$ such that $\goth M/\tau\goth M$ is annihilated by
a power of $T-\theta$. We have $\hbox{Lie}(\goth M)=\mathbb{C}_{\infty}^{n}$,
and $N$ is a nilpotent operator acting on $\hbox{Lie}(\goth M)$. Also, $T$
acts on $\hbox{Lie}(\goth M)$; we have $T=\theta I_{n}+N$ on $\hbox{End
}(\hbox{Lie}(\goth M))$. Particularly, we can consider $\hbox{Lie}(\goth M)$
as a $\mathbb{F}_{q}[T]$-module. The lattice $L(\goth M)$ of $\goth M$ is a
free $\mathbb{F}_{q}[T]$-submodule of $\hbox{Lie}(\goth M)$ considered as a
$\mathbb{F}_{q}[T]$-module. We have a natural inclusion
$\mathbb{F}_{q}[T]\hookrightarrow\mathbb{C}_{\infty}[[N]]$ ($T\mapsto
N+\theta$). Hence, the tautological inclusion $L(\goth
M)\hookrightarrow\hbox{Lie}(\goth M)$ defines a surjection

$L(\goth
M)\underset\mathbb{F}_{q}[T]\to{\otimes}\mathbb{C}_{\infty}[[N]]\twoheadrightarrow\hbox{Lie}(\goth
M).$

Its kernel is denoted by $\goth q_{\goth}M$; the exact sequence

$0\to\goth q_{\goth}M\to L(\goth
M)\underset\mathbb{F}_{q}[T]\to{\otimes}\mathbb{C}_{\infty}[[N]]\to\hbox{Lie}(\goth
M)\to 0$

(see [P]; [Gl18], (1.8.3); [HJ], Example 2.5, (2.2) for the case of Drinfeld
modules) is a particular case of a Hodge-Pink structure. Discrete invariants
of $\goth M$ are the discrete invariants of the pair of lattices $(\goth
q_{\goth}M;L(\goth
M)\underset\mathbb{F}_{q}[T]\to{\otimes}\mathbb{C}_{\infty}[[N]])$ over a
discrete valuation ring $\mathbb{C}_{\infty}[[N]]$. Let us give a description
of these invariants from [GL18], (3.3). Let $\nu$ be the minimal number such
that $N^{\nu}=0$. These invariants are numbers $k_{1}\geq
0,\dots,k_{\nu+1}\geq 0$ defined as follows. There exists a basis
$l_{1},\dots,l_{r}$ of $L(\goth M)$ over $\mathbb{F}_{q}[T]$ and its partition
on $\nu+1$ sets of lengths $k_{1},\dots,k_{\nu+1}$ (if some $k_{i}=0$ then the
$i$-th set is empty); the $i$-th set is denoted by $l_{i,1}\dots,l_{i,k_{i}}$,
having the following properties: $N^{\nu-1}(l_{\nu+1,i})$,
$i=1,\dots,k_{\nu+1}$, form a $\mathbb{C}_{\infty}$-basis of
$N^{\nu-1}\hbox{Lie}(\goth M)$ ([GL18], (3.6); $N^{\nu-2}(l_{\nu,i})$,
$i=1,\dots,k_{\nu}$, $N^{\nu-2}(l_{\nu+1,i})$, $i=1,\dots,k_{\nu+1}$,
$N^{\nu-1}(l_{\nu+1,i})$, $i=1,\dots,k_{\nu+1}$, form a
$\mathbb{C}_{\infty}$-basis of $N^{\nu-2}\hbox{Lie}(\goth M)$ ([GL18], (3.8);
$N^{\nu-3}(l_{\nu-1,i})$, $i=1,\dots,k_{\nu-1}$, $N^{\nu-3}(l_{\nu,i})$,
$N^{\nu-2}(l_{\nu,i})$, $i=1,\dots,k_{\nu}$, and $N^{\nu-3}(l_{\nu+1,i})$,
$N^{\nu-2}(l_{\nu+1,i})$, $N^{\nu-1}(l_{\nu+1,i})$, $i=1,\dots,k_{\nu+1}$,
form a $\mathbb{C}_{\infty}$-basis of $N^{\nu-3}\hbox{Lie}(\goth M)$ ([GL18],
(3.8); etc., until $l_{2,i},\ i=1,\dots,k_{2},\ \dots,\
N^{\nu-1}(l_{\nu+1,i})$, $i=1,\dots,k_{\nu+1}$, form a
$\mathbb{C}_{\infty}$-basis of $\hbox{Lie}(\goth M)$. See [GL18], (3.3) -
(3.10) for more details. Particularly, we have:

$r=k_{1}+...+k_{\nu+1};\ \ n=k_{2}+2k_{3}+3k_{4}+...+\nu k_{\nu+1}.$

([GL18], (3.10) and (3.5)). If $\nu=1$, i.e. $N=0$ — this is the case
considered above, then $k_{1}=r-n,\ k_{2}=n$. Therefore, numbers
$k_{1},\dots,k_{\nu+1}$ are $N\neq 0$-generalizations of numbers $r-n,\ n$.
Conjecture 2.4.1. The analog of the subgroup $M$ of $G_{fun}=GL_{r}$ for the
set of Anderson t-motives having $N\neq 0$ and invariants
$k_{1},\dots,k_{\nu+1}$ is the subgroup of $GL_{r}$ of block diagonal matrices
with block sizes $k_{1},\dots,k_{\nu+1}$. Remark 2.4.2. Some of $k_{i}$ can be
0. In this case we cannot distinguish between $M$ for different sets of
$k_{1},\dots,k_{\nu+1}$. Hence, maybe it is necessary to modify the statement
of Conjecture 2.4.1. As it was written, finding of analogs of statements of
Sections 2.2, 2.3 for the sets of Anderson t-motives having invariants
$k_{1},\dots,k_{\nu+1}$ is a subject of further research. Remark 2.4.3. Hartl
and Juschka use some other invariants of $\goth M$, see [HJ], Section 2.
First, they consider slightly more general objects, namely, their $\goth
q=\goth q_{\goth}M$ is a subset not of $L(\goth
M)\underset\mathbb{F}_{q}[T]\to{\otimes}\mathbb{C}_{\infty}[[N]]$ but of
$L(\goth M)\underset\mathbb{F}_{q}[T]\to{\otimes}\mathbb{C}_{\infty}((N))$
(also, they consider a weight filtration on $L(\goth M)$ ). Further, their
Hodge-Pink weights $\omega_{1},\dots,\omega_{r}$ are related with
$k_{1},\dots,k_{\nu+1}$ as follows: for all $i=1,\dots,\nu+1$ the number
$-i+1$ occurs $k_{i}$ times among $\omega_{1},\dots,\omega_{r}$ (i.e. among
$\omega_{1},\dots,\omega_{r}$ there are $k_{1}$ zeroes, $k_{2}$ minus ones
etc.). 3\. Proofs. We follow [FCh], Ch. 7, Section 4 using the same notations
if possible, and indicating results that are not completely analogous to the
number field case. Recall that $\goth p$ is a prime ideal of
$\mathbb{F}_{q}[\theta]$. We denote by $\mathbb{F}_{q}[\theta]_{\goth}p$,
$\mathbb{F}_{q}(\theta)_{\goth}p$ the completions at $\goth p$ of
$\mathbb{F}_{q}[\theta]$, $\mathbb{F}_{q}(\theta)$ respectively, and by
$\mathbb{F}_{q}[\theta]_{\goth}p^{nr}$ the ring of integers of the maximal
unramified extension of $\mathbb{F}_{q}(\theta)_{\goth}p$. As usual, bar means
an algebraic closure. There are maps
$\mathbb{F}_{q}[\theta]_{\goth}p^{nr}\hookrightarrow\overline{\mathbb{F}_{q}(\theta)_{\goth}p}$,
$\mathbb{F}_{q}[\theta]_{\goth}p^{nr}\twoheadrightarrow\overline{\mathbb{F}_{q}[\theta]/\goth
p}$. The corresponding maps of schemes $\hbox{Spec
}\overline{\mathbb{F}_{q}(\theta)_{\goth}p}\to\hbox{Spec
}\mathbb{F}_{q}[\theta]_{\goth}p^{nr}$, $\hbox{Spec
}\overline{\mathbb{F}_{q}[\theta]/\goth p}\to\hbox{Spec
}\mathbb{F}_{q}[\theta]_{\goth}p^{nr}$ are denoted by $\xi_{k}$, $\xi_{\goth
p}$ respectively. The inverse image $\xi_{\goth p}^{*}$ of an object (i.e. the
reduction of this object) is denoted by tilde. We fix $i$, and let $\Gamma$ be
the graph of $T_{\goth p,i}$ over $\hbox{Spec }\mathbb{F}_{q}[\theta]_{\goth
p}^{nr}$. It is known that it exists. For $t\in\Gamma$ (resp.
$t\in\tilde{\Gamma}$) let $\phi_{t}:E_{t}\to E^{\prime}_{t}$ be the
corresponding map of Anderson modules over $\hbox{Spec
}\mathbb{F}_{q}[\theta]_{\goth p}^{nr}$ (resp. $\hbox{Spec
}\overline{\mathbb{F}_{q}[\theta]/{\goth p}}$). We consider the ordinary locus
$\Gamma^{0}$ of $\Gamma$:

$t\in\Gamma^{0}\iff\xi_{k}(E_{t}),\ \xi_{k}(E^{\prime}_{t})$

are ordinary. Lemma 3.1. $\tilde{\Gamma}^{0}$ is dense in $\tilde{\Gamma}$.
$\square$ 3.1a. Now let $\tau_{\goth p}\in G_{fun}$ be any diagonal matrix,
$T_{G,\goth p}$ the element of Hecke algebra $\mathbb{H}(G_{fun})$
corresponding to the double coset $K_{G}\tau_{\goth p}K_{G}$, $\Gamma$ the
graph of $T_{G,\goth p}$ over $\hbox{Spec }\mathbb{F}_{q}[\theta]_{\goth
p}^{nr}$, and $\tilde{\Gamma}^{0}$ for this $\Gamma$ is defined as earlier.
Let $c$ be the highest power of ${\goth p}$ that appears in the diagonal
entries of $\tau_{\goth p}$ (for example, if $\tau_{\goth p}=\tau_{\goth p,i}$
then $c=1$). Let $s\in\tilde{\Gamma}^{0}$ and $E_{s}$, $E^{\prime}_{s}$ the
corresponding Anderson t-motives over $\hbox{Spec
}\overline{\mathbb{F}_{q}[\theta]/{\goth p}}$. This means that we have a
direct sum decomposition of the finite $\mathbb{F}_{q}[\theta]$-module scheme
$(E_{s})[{\goth p}^{c}]$ over $\hbox{Spec
}\overline{\mathbb{F}_{q}[\theta]/{\goth p}}$ on its multiplicative and etale
part:

$(E_{s})[{\goth p}^{c}]=(E_{s})[{\goth p}^{c}]_{mult}\oplus(E_{s})[{\goth
p}^{c}]_{et},$ $None$

where

$(E_{s})[{\goth p}^{c}]_{mult}=(\mu_{\goth p^{c}})^{n},$ $None$
$(E_{s})[{\goth p}^{c}]_{et}=(\hbox{Spec }\mathbb{F}_{q}[\theta]/{\goth
p}^{c})^{r-n}.$ $None$

We can restrict $\phi_{s}$ to $(E_{s})[{\goth p}^{c}]$ getting a map

$(\phi_{s})[{\goth p}^{c}]:(E_{s})[{\goth p}^{c}]\to(E^{\prime}_{s})[{\goth
p}^{c}].$ $None$

In its turn, this map is restricted to both etale and multiplicative parts:

$(\phi_{s})[{\goth p}^{c}]_{mult}:(E_{s})[{\goth
p}^{c}]_{mult}\to(E^{\prime}_{s})[{\goth p}^{c}]_{mult}$ $None$

and

$(\phi_{s})[{\goth p}^{c}]_{et}:(E_{s})[{\goth
p}^{c}]_{et}\to(E^{\prime}_{s})[{\goth p}^{c}]_{et}.$ $None$

Taking into consideration (3.1.2), (resp, (3.1.3)), we see that $\phi_{s}$
defines elements in $\mathbb{H}(GL_{n})$ (resp. $\mathbb{H}(GL_{r-n})$). In
concordance of notations of [FCh], we denote them by $a$ (resp. $d$). This
pair ($a$, $d$) defines us an element of $\mathbb{H}(M)$. It is called the
type of $s$. Remark. Unlike in the number case, here the elements $a$, $d$ are
independent. In order to formulate the below proposition 3.4, we need the
following notations: 3.2. Let $\delta:\goth E_{1}\to\goth E_{2}$ be a map of
Anderson modules over $\hbox{Spec }\mathbb{F}_{q}[\theta]/{\goth p}$ of type
$\Psi_{i}\Phi_{j}$. This means that $c$ of (3.1.1) is 1, and kernels of the
map (3.1.5) (resp. (3.1.6)) is isomorphic to $(\mu_{\goth p})^{i}$ (resp.
$(\hbox{Spec }\mathbb{F}_{q}[\theta]/{\goth p})^{j}$). We denote them by
$\tilde{K}_{m}$, $\tilde{K}_{e}$ respectively. Further, let $E_{1}$ be a
Anderson module over $\hbox{Spec }\mathbb{F}_{q}[\theta]_{\goth p}$ such that
$\tilde{E}_{1}=\goth E_{1}$. Lemma 3.3. We can identify $\tilde{K}_{m}$ (resp.
$\tilde{K}_{e}$) with some $i$ (resp. $j$)-dimensional subspaces in
$D_{fun}(E_{1})$ (resp. $(E_{1})_{\goth p}/D_{fun}(E_{1})$ ). We denote these
subspaces by $K_{m}$, $K_{e}$ respectively. Now let us consider the set of
pairs $(\phi,E_{2})$ where $\phi:E_{1}\to E_{2}$ is a map of Anderson modules
over $\hbox{Spec }\mathbb{F}_{q}[\theta]_{\goth p}$, such that
$\tilde{\phi}=\delta$ (and hence $\tilde{E}_{2}=\goth E_{2}$). Proposition
3.4. The set of the above $(\phi,E_{2})$ is isomorphic to the set of subspaces
$\Cal{W}\subset(E_{1})_{\goth p}$ such that

$\Cal{W}\cap D_{fun}(E_{1})=K_{m},\ \
\Cal{W}+D_{fun}(E_{1})/D_{fun}(E_{1})=K_{e}.$ $None$

Proof. We need the function field analog of [K], Th. 2.1. Let $R$ be an
Artinian local ring with residue field
$\overline{\mathbb{F}_{q}[\theta]/{\goth p}}$ and the maximal ideal $\goth m$.
We consider only the case $R=R_{\eta}=\mathbb{F}_{q}[\theta]_{\goth
p}^{nr}/{\goth p}^{\eta}$ for some $\eta$. Let $\Cal{E}$ be an ordinary
Anderson module over $\overline{\mathbb{F}_{q}[\theta]/{\goth p}}$. Let us
consider (3.1.1) for $\Cal{E}$, and let $T_{\goth p}(\Cal{E})$ be the Tate
module of the etale part: $T_{\goth p}(\Cal{E})=\underset
c\to\infty\to{\hbox{invlim }}\Cal{E}[{\goth p}^{c}]_{et}.$ The dual Anderson
module $\Cal{E}^{t}$ is defined in [L], [F], it is of rank $r$ and dimension
$n$. Let $\Cal{E}_{R}$ be an Anderson module over $R$ such that its reduction
to $\overline{\mathbb{F}_{q}[\theta]/{\goth p}}$ is $\Cal{E}$ (a lift of
$\Cal{E}$ on $R$). The function field analog of [K], Th. 2.1, (1) is the
following: Theorem 3.4.1a. The set of $\Cal{E}_{R}$ is in 1 – 1 correspondence
with the set of maps

$\hbox{Hom }(T_{\goth p}(\Cal{E})\otimes T_{\goth p}(\Cal{E}^{t}),\goth m),$
$None$

where Hom is of $\mathbb{F}_{q}[T]/\goth p$-modules. Notation. For a fixed
$\Cal{E}_{R}$ we denote this map by $q_{\Cal{E}_{R}}$. Idea of the proof.
First, we define the analog of the map $\varphi_{\mathbb{A}/R}$, [K], p. 151
for the present situation. Here it is
$\varphi_{\Cal{E}/R}:T_{\goth}p(\Cal{E})\to\goth m^{\oplus n}$. Recall that
$\eta$ satisfies $\goth m^{\eta}=0$. We choose $k$ such that $q^{k}\geq\eta$,
and we consider formulas of multiplication by $\goth p^{k}$ for $\Cal{E}$:

$\goth p^{k}(X)=\sum_{i=k}^{\eta}C_{i}X^{q^{i}}$ $None$

where $X\in\mathbb{C}_{\infty}^{\oplus n}$ is a column vector and $C_{i}\in
M_{n\times n}(\mathbb{C}_{\infty})$. Condition $X\in\Cal{E}[\goth p^{k}]_{et}$
means that $\sum_{i=k}^{\eta}C_{i}X^{q^{i}}=0$.

Let $\tilde{X}\in R^{\oplus n}$ be a lift of $X\in\Cal{E}[\goth p^{k}]_{et}$.
Since for the first term $C_{k}X^{q^{k}}$ of 3.4.2a we have $q^{k}\geq N$, we
get that
$\varphi_{\Cal{E}/R}(X):=\sum_{i=k}^{\eta}C_{i}\tilde{X}^{q^{i}}\in\goth
m^{\oplus n}$ does not depend on the choice of $\tilde{X}$. $\square$ Now we
need the function field analog of [K], Th. 2.1, (4). Let $\Cal{E}_{1}$,
$\Cal{E}_{2}$ be ordinary Anderson modules over
$\overline{\mathbb{F}_{q}[\theta]/{\goth p}}$,
$\alpha:\Cal{E}_{1}\to\Cal{E}_{2}$ a map and $\Cal{E}_{1,R}$, $\Cal{E}_{2,R}$
lifts of $\Cal{E}_{1}$, $\Cal{E}_{2}$ on $R$. We denote by

$T_{\goth p}(\alpha):T_{\goth p}(\Cal{E}_{1})\to T_{\goth p}(\Cal{E}_{2})$
$None$

$T_{\goth p}(\alpha^{t}):T_{\goth p}(\Cal{E}_{2}^{t})\to T_{\goth
p}(\Cal{E}_{1}^{t})$ $None$

the maps obtained by functoriality. Lemma 3.4.5. A map
$\alpha_{R}:\Cal{E}_{1,R}\to\Cal{E}_{2,R}$ such that its reduction is $\alpha$
exists iff for any $x\in T_{\goth p}(\Cal{E}_{1})$, $y\in T_{\goth
p}(\Cal{E}_{2}^{t})$ we have

$q_{\Cal{E}_{2},R}(T_{\goth p}(\alpha)(x),y)=q_{\Cal{E}_{1},R}(x,T_{\goth
p}(\alpha^{t})(y)).$ $None$

and moreover if this condition is satisfied then $\alpha_{R}$ is unique.
$\square$ Lemma 3.4.6a. (Conjectural statement). To define $E_{2}$ over
$\mathbb{F}_{q}[\theta]_{\goth p}$ is the same as to define a concordant
system of $(E_{2})_{\eta}$ over $R_{\eta}$ (the concordance condition is
clear). Remark. Obviously $E_{2}$ defines a concordant system of $(\goth
E_{2})_{\eta}$. But is the inverse really true? Maybe non-trivial
automorphisms of $(\goth E_{2})_{\eta}$ give obstacles? Now we return to the
proof of Proposition 3.4. We fix $\eta$, we take $\Cal{E}_{1}=\goth E_{1}$,
$\Cal{E}_{2}=\goth E_{2}$. According (3.2), there exist bases
$e_{1}^{t},\dots,e_{n}^{t}$, $e_{n+1},\dots,e_{r}$,
$f_{1}^{t},\dots,f_{n}^{t}$, $f_{n+1},\dots,f_{r}$ of $T_{\goth p}(\goth
E_{1}^{t})$, $T_{\goth p}(\goth E_{1})$, $T_{\goth p}(\goth E_{2}^{t})$,
$T_{\goth p}(\goth E_{2})$ respectively such that the maps $T_{\goth
p}(\delta)$, $T_{\goth p}(\delta^{t})$ in these bases are the following: 7
$T_{\goth p}(\delta)(e_{n+1})=\goth pf_{n+1}$

& … (maps of type 1)

$T_{\goth p}(\delta)(e_{n+j})=\goth pf_{n+j}$, $T_{\goth
p}(\delta)(e_{n+j+1})=f_{n+j+1}$

… (maps of type 2)

$T_{\goth p}(\delta)(e_{r})=f_{r}$, $T_{\goth p}(\delta^{t})(f_{1}^{t})=\goth
pe_{1}^{t}$

… (maps of type 3)

$T_{\goth p}(\delta^{t})(f_{i}^{t})=\goth pe_{i}^{t}$, $T_{\goth
p}(\delta^{t})(f_{i+1}^{t})=e_{i+1}^{t}$

… (maps of type 4)

$T_{\goth p}(\delta^{t})(f_{n}^{t})=e_{n}^{t}$. Now we apply formula (3.4.6)
to these formulas. We consider 4 types of $x$, $y$:

Type 13. $x$ of type 1, $y$ of type 3 ($\lambda\in[n+1,\cdots,n+j]$,
$\mu\in[1,\cdots,i])$: We get:

$\goth p\cdot q_{\goth E_{2},R}(f_{\lambda},f_{\mu}^{t})=\goth p\cdot q_{\goth
E_{1},R}(e_{\lambda},e_{\mu}^{t}).$ $None$

If $\goth m$ had no $\goth p$-torsion then we can divide the above equality by
$\goth p$ and to get

$q_{\goth E_{2},R}(f_{\lambda},f_{\mu}^{t})=q_{\goth
E_{1},R}(e_{\lambda},e_{\mu}^{t}),$ $None$

this means that $q_{\goth E_{2},R}$ on these $f_{\lambda}$, $f_{\mu}^{t}$ is
defined uniquely. We think that in order to prove that we can really divide by
$\goth p$, we must consider not one fixed $\eta$, but all the values of them.
The similar problem exists for the next type: Type 14. $x$ of type 1, $y$ of
type 4 ($\lambda\in[n+1,\cdots,n+j]$, $\mu\in[i+1,\cdots,n])$: We get:

$\goth p\cdot q_{\goth E_{2},R}(f_{\lambda},f_{\mu}^{t})=q_{\goth
E_{1},R}(e_{\lambda},e_{\mu}^{t}).$ $None$

If $\goth m$ were $\goth p$-divisible and had the $\goth p$-torsion isomorphic
to $\mathbb{F}_{q}[\theta]/\goth p$ then OK: we have $\goth q^{(n-i)j}$
possibilities for $(\goth E_{2})_{R}$ as it should be. For other types of $x$,
$y$ there is no such problem. Really: Type 23. $x$ of type 2, $y$ of type 3
($\lambda\in[n+j+1,\cdots,r]$, $\mu\in[1,\cdots,i])$: We get:

$q_{\goth E_{2},R}(f_{\lambda},f_{\mu}^{t})=\goth p\cdot q_{\goth
E_{1},R}(e_{\lambda},e_{\mu}^{t}).$ $None$

This means that $q_{\goth E_{2},R}$ on these $f_{\lambda}$, $f_{\mu}^{t}$ is
defined uniquely; Type 24. $x$ of type 2, $y$ of type 4
($\lambda\in[n+j+1,\cdots,r]$, $\mu\in[i+1,\cdots,n])$: We get:

$q_{\goth E_{2},R}(f_{\lambda},f_{\mu}^{t})=q_{\goth
E_{1},R}(e_{\lambda},e_{\mu}^{t}).$ $None$

This means that $q_{\goth E_{2},R}$ on these $f_{\lambda}$, $f_{\mu}^{t}$ is
defined uniquely; 3.4.12. We get that we have $\goth q^{(n-i)j}$ modules
$E_{2}$, this number is equal to the quantity of $\Cal{W}$ satisfying (3.4.1).
3.4.13. Now we need to prove that these $\Cal{W}$ really satisfy (3.4.1).
$\square$ Now we can define the map $\beta_{2}:\mathbb{H}(M)\to\hbox{
Corr}(\tilde{X})$ from (1.1.4). Idea of the definition: let $\tau_{\goth p}$
have the form $\left(\begin{matrix}\goth p^{A}&0\\\ 0&\goth
p^{B}\end{matrix}\right)$ where $A=(a_{1},\dots,a_{r-n})$,
$B=(b_{1},\dots,b_{n})$, $\goth p^{A}=\hbox{ diag }(\goth
p^{a_{1}},\dots,\goth p^{a_{r-n}})$, $\goth p^{b}=\hbox{ diag }(\goth
p^{b_{1}},\dots,\goth p^{b_{n}})$. We denote by $T_{M,\goth p}=T_{M,\goth
p}(A,B)$ the element of Hecke algebra $\mathbb{H}(M)$ corresponding to the
double coset $K_{M}\tau_{\goth p}K_{M}$. Explicit formula for
$S^{G_{fun}}_{M}$ shows that

$S^{G_{fun}}_{M}(T_{G,\goth p})=\goth q^{-m_{A,B}}T_{M,\goth p}+\hbox{ other
terms,}$ $None$

where these other terms are linear combinations of $T_{M,\goth
p}(A^{\prime},B^{\prime})$ for pairs $(A^{\prime},B^{\prime})$ distinct from
$(A,B)$. Coefficient $m_{A,B}\geq 0$ can be easily found explicitly; for
$T_{M,\goth p}(A,B)=\Psi_{i}\Phi_{j}$ we have $m_{A,B}=ij$. Now we consider
the reduction of the correspondence $\beta_{1}(T_{G,\goth p})$. Let
$\Gamma_{irr}$ be an irreducible component of its graph, $\phi:E_{1}\to E_{2}$
a map of Anderson modular objects over $A/\goth p$ corresponding to a point of
$\Gamma_{irr}$, and $\goth t\in\mathbb{H}(M)$ its type. $\goth t$ depends only
on $\Gamma_{irr}$ but not on $\phi:E_{1}\to E_{2}$ because it is a discrete
invariant, so we can call it the type of $\Gamma_{irr}$. First, we denote by
$\goth C(A,B)$ the correspondence on $\tilde{X}$ whose graph is the sum of all
the irreducible components of the graph of reduction of the correspondence
$\beta_{1}(T_{G,\goth p})$ whose type is $T_{M,\goth p}(A,B)$ (really, for
each $(A,B)$ there exists only one such component). By abuse of notations we
denote by $\goth C(\Psi_{i}\Phi_{j})$ the $\goth C(A,B)$ where $A$, $B$ are
from 2.2.3 a,b. Finally, we define

$\beta_{2}(T_{M,\goth p}(A,B))=\goth q^{m_{A,B}}\goth C(A,B),$ $None$

hence

$\beta_{2}(\Psi_{i}\Phi_{j})=\goth q^{ij}\goth C(\Psi_{i}\Phi_{j}).$ $None$

(3.5) and (3.6) show immediately that the function field analog of the diagram
(1.1.6) is commutative. Corollary 3.8. $d_{1}^{ns}(\goth
C(\Psi_{i}\Phi_{j}))=\goth q^{(n-i)j}$. Proof. Follows immediately from
3.4.12. $\square$ 3.7 and 3.8 imply that

$d_{1}^{ns}(\beta_{2}(\Psi_{i}\Phi_{j}))=\goth q^{nj}.$ $None$

Proposition 3.10. $\beta_{2}$ is a ring homomorphism. Idea of the proof. Let
$(A,B)$, $(A^{\prime},B^{\prime})$ be 2 pairs of multiindices as above and let
$T_{M,\goth p}(A,B)\cdot T_{M,\goth
p}(A^{\prime},B^{\prime})=\sum_{i}\kappa_{i}T_{M,\goth p}(A_{i},B_{i})$ for
some pairs $(A_{i},B_{i})$ and coefficients $\kappa_{i}$.

Lemma 3.10.1. For all $i$ we have

$\goth q^{m_{A,B}}d_{1}^{ns}(\goth C(A,B))\cdot\goth
q^{m_{A^{\prime},B^{\prime}}}d_{1}^{ns}(\goth
C(A^{\prime},B^{\prime}))=\kappa_{i}\goth q^{m_{A_{i},B_{i}}}d_{1}^{ns}(\goth
C(A_{i},B_{i})).$

Proof. Explicit calculation. For a particular case corresponding to
$(A,B)=\Psi_{i}$, $(A^{\prime},B^{\prime})=\Phi_{j}$,
$(A_{i},B_{i})=\Psi_{i}\Phi_{j}$ this follows from the above results.

We have:

$T_{G,\goth p}(A,B)\cdot T_{G,\goth
p}(A^{\prime},B^{\prime})=\sum_{i}\kappa_{i}T_{G,\goth p}(A_{i},B_{i})+\hbox{
other terms}.$ $None$

Since the reduction is a ring homomorphism, we see that: (a) (3.5) applied to
the pairs $(A,B)$, $(A^{\prime},B^{\prime})$, $(A_{i},B_{i})$; (b) (3.10.2)
and Lemma 3.10.1; (c) Commutativity of the the function field analog of the
diagram (1.1.6) imply that

$\beta_{2}(T_{M,\goth p}(A,B))\cdot\beta_{2}(T_{M,\goth
p}(A^{\prime},B^{\prime}))+\hbox{ other terms}=$
$=\sum_{i}\kappa_{i}\beta_{2}(T_{M,\goth p}(A_{i},B_{i}))+\hbox{ other
terms.}$

3.10.3. Now naive considerations show us that “other terms” in both sides of
the above equality are equal. Really, let us denote by $S\Gamma(A,B)$ the
support of the graph $\Gamma(\beta_{2}(T_{M,\goth
p}(A,B)))\subset\tilde{X}\times\tilde{X}$, and analogically for the pairs
$(A^{\prime},B^{\prime})$, $(A_{i},B_{i})$. We have:

$(t_{1},t_{2})\in S\Gamma(A,B)\iff\hbox{ there is a map }E_{t_{1}}\to
E_{t_{2}}\hbox{ of type }T_{M,\goth p}(A,B).$

By definition of the product of correspondences,

$(t_{1},t_{3})\in\cup_{i}S\Gamma(A_{i},B_{i})\iff\hbox{ there exists
}t_{2}\hbox{ such that }$ $(t_{1},t_{2})\in S\Gamma(A,B),\ \ (t_{2},t_{3})\in
S\Gamma(A^{\prime},B^{\prime}).$

Since the type of the composition of maps of Anderson varieties is concordant
with the multiplication in $\mathbb{H}(M)$, we get 3.10.3. $\square$ 3a.
Conjectural form of Langlands correspondence. According Langlands, L-function
$L(\Cal{M},s)$ of an irreducible submotive $\Cal{M}$ of a Shimura variety is
related with $L(\pi,\goth r,s)$, where $\pi$ is an automorphic representation
of $G(\mathbb{A}_{\mathbb{Q}})$ and $\goth r:$ ${}^{L}G\to GL(\goth W)$ a
finite-dimensional representation of ${}^{L}G$:

$L(\Cal{M},s)\sim L(\pi,\goth r,s).$ $None$

Conjectural construction of $\goth r$ is given for example in [BR], Section
5.1, p. 550. Let us formulate an analog of this result for Anderson modular
objects. For this case an analog of $G(\mathbb{A}_{\mathbb{Q}})$ is
$G_{fun}(\mathbb{A}_{\mathbb{F}_{q}(\theta)})$. Theorem 3a.1. If an analog of
(3a.0) is true for Anderson modular objects $X_{fun}(r,n)$ of any level then
the restriction of $\goth r$ to $\hat{G}_{fun}$ is the $n$-th skew power
representation of $GL_{r}$. This theorem follows from the below Theorem 3a.3.
Let $\pi=\otimes\pi_{\goth l}$ be a representation of
$G_{fun}(\mathbb{A}_{\mathbb{F}_{q}(\theta)})$ corresponding (according
Langlands) to an irreducible submotive of an Anderson modular object, and
$\theta_{\goth p}\in\ ^{L}G$ a Langlands element of $\pi_{\goth p}$ (we
consider the case of $\goth p$ such that $\pi_{\goth p}$ is non-ramified). Let
$\alpha_{i}$, $i=1,...r$, be eigenvalues of $\theta_{\goth p}$ and $a_{i}$ the
eigenvalues of $T_{\goth p,i}$ (analogs of Fourier coefficients of an
automorphic form for the classical case). Standard formalism of Langlands
elements for $GL_{r}$ in the non-ramified case together with (2.2.2) shows
that

$a_{i}=\goth q^{-i(i-1)/2}\sigma_{i}(\alpha_{*}).$ $None$

We denote by $P^{\prime}_{r,n}$ the characteristic polynomial of $\goth
r(\theta_{\goth p})$, it belongs to $\mathbb{Z}[a_{1},...,a_{r}][T]$ where $T$
is an abstract variable. The following theorem follows immediately from
(2.2.2), (3a.2) (like in the number case): Theorem 3a.3.
$P_{r,n}=P^{\prime}_{r,n}$ (after identification of $T$ and $fr$, $a_{i}$ and
$T_{\goth p,i}$). $\square$ 4\. Unitary Shimura varieties. We consider abelian
varieties with multiplication by an imaginary quadratic field (abbreviation:
MIQF). Let $K$ be such field, $X_{num}$ the corresponding Shimura variety
parametrizing abelian $r$-folds with multiplication by $K$, of signature
$(r-n,n)$. We shall call them unitary Shimura varieties. The corresponding
reductive group over $\mathbb{Q}$ is $G=G_{num}=GU(r-n,n)$. We have $\hbox{dim
}X_{num}=(r-n)n$. Let $p$ be a prime inert in $K$; we shall consider $p$-Hecke
correspondences and the reduction at $p$. Theorem 4.1. $M$ for this case is
the same as in Theorem 2.1. Corollary 4.2. Satake maps for this case coincide
with the ones for the functional case (formulas (2.2.4), (2.2.5)). Let $A_{t}$
be as in Subsection 1.2. $(A_{t})_{p}$ is an $r$-dimensional vector space over
$\mathbb{F}_{p^{2}}$. Let $D=D_{unitary}$ be as in Subsection 1.2. Theorem
4.3. $D_{unitary}$ is a vector space over $\mathbb{F}_{p^{2}}$ of dimension
$max\ (r-n,n)$. Remark 4.4. There exists a symmetry between $n$ and $r-n$.
Nevertheless, here the analogy between functional and unitary case apparently
is not complete. Theorem 4.5. Analog of the Theorem 2.3.5 (i.e. formula 2.3.6)
holds for the unitary case (dimension is taken over $\mathbb{F}_{p^{2}}$).
Theorem 4.6. ([BR], Section 5.1, p. 550, example (b)). Restriction of $\goth
r$ on $\hat{G}\subset\ ^{L}G$ is the same as in Theorem 3a.1. We think that
analogs of Theorems 2.3.7, 2.3.8, 2.3.10 also hold for this case. 5\.
Conjectural values of Hodge numbers.There are conjectural formulas for values
of Hodge numbers $h^{ij}$ (over the fields of multiplications coming from
Hecke correspondences) of irreducible submotives of Shimura varieties (see,
for example, [BR], Section 4.3, p. 548). For example, for the case of Siegel
modular varieties of genus $g$ (their dimension is $d_{g}=g(g+1)/2$) and for a
generic pure submotive of weight $d_{g}$ they are the following: Theorem 5.1.
$h^{i,d_{g}-i}=$ {the quantity of subsets $(j_{1},\dots,j_{\alpha})$ of the
set $1,2,...,g$ such that $j_{1}+...+j_{\alpha}=i$}, where $\alpha$ is
arbitrary. For other types of submotives the formulas for $h^{ij}$ are similar
but more long. For example, for the case of unitary Shimura variety of Section
4 and for the same type of submotives the formula is the following: Theorem
5.2. $h^{i,(r-n)n-i}=$ {the quantity of subsets $(j_{1},...j_{n})$ of the set
$(1,2,...,r)$ such that $j_{1}+...+j_{n}-(1+...+n)=i$}. By analogy between
functional and unitary case we can conjecture that the same formula holds for
the functional case. 6\. Non-ordinary Drinfeld modules. For further
applications we shall state two problems and give their conjectural answers.
Let us restrict ourselves by the case $n=1$ and the correspondence $T_{\goth
p,1}$. These problems are related with the description of intersection of two
irreducible components of the graph of $T_{\goth p,1}$ in characteristic
$\goth p$. Let $E$ be a Drinfeld module such that its reduction is a generic
non-ordinary, i.e. $\hbox{dim }_{\mathbb{F}_{\goth p}}(D_{fun}(E))=2$ is the
least possible. Let $t$ be the point on $X_{fun}$ corresponding to $E$ and
$t^{\prime},t^{\prime\prime}\in T_{\goth p,1}(t)$. Question 6.1. Formulate
analogs of conjectures 2.3.5, 2.3.8 for this $t$. Conjectural answer. (a)
$\tilde{t}^{\prime}$ is (the only) closed point of $\Psi_{1}(\tilde{t})$ iff
$\hbox{Ker }(\alpha_{t,t^{\prime}})\subset D_{fun}$. All such
$\tilde{t}^{\prime}$ also belong to $\Phi_{1}(\tilde{t})$. (b)
$\tilde{t}^{\prime}=\tilde{t}^{\prime\prime}$ as closed points iff the
$\mathbb{F}_{\goth p}$-linear spans coincide:

$<\hbox{Ker }(\alpha_{t,t^{\prime}}),D_{fun}>=<\hbox{Ker
}(\alpha_{t,t^{\prime\prime}}),D_{fun}>.$

Now we consider a more special situation. Let $r$ be even, $L$ a quadratic
extension of $\mathbb{F}_{q}(\theta)$ such that $\goth p$ inert in
$L/\mathbb{F}_{q}(\theta)$, and $E$ a generic Drinfeld module with
multiplication by $L$. In this case $D_{fun}(E)$ is a 1-dimensional
$\mathbb{F}_{\goth p^{2}}$-vector space. Let $t,t^{\prime},t^{\prime\prime}$
be as above. Question 6.2. Formulate analogs of conjectures 2.3.5, 2.3.8 for
this $t$.

Conjectural answer. (a) is the same as above, and in (b) we consider
$\mathbb{F}_{\goth p^{2}}$-linear spans:

$\tilde{t}^{\prime}=\tilde{t}^{\prime\prime}\iff<\hbox{Ker
}(\alpha_{t,t^{\prime}}),D_{fun}>_{\mathbb{F}_{\goth p^{2}}}=<\hbox{Ker
}(\alpha_{t,t^{\prime\prime}}),D_{fun}>_{\mathbb{F}_{\goth p^{2}}}.$

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