On the semisimplicity of reductions and adelic openness for 𝐸-rational compatible systems over global function fields
Authors: Böckle Gebhard , Gajda Wojciech , Petersen Sebastian 
Published: January 2019

Let

X
X

be a normal geometrically connected variety over a finite field

κ
\kappa

of characteristic 

p
p

. Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis rho Subscript lamda Baseline colon pi 1 left-parenthesis upper X right-parenthesis right-arrow GL Subscript n Baseline left-parenthesis upper E Subscript lamda Baseline right-parenthesis right-parenthesis Subscript lamda">

(

ρ
λ

:

π
1

(
X
)
→

GL

n

(

E
λ

)

)
λ

(\rho _\lambda \colon \pi _1(X)\to \textrm {GL}_n(E_\lambda ) )_\lambda

be any semisimple

E
E

-rational compatible system, where

E
E

is a number field and

λ
\lambda

ranges over the finite places of

E
E

not above

p
p

. We derive new properties on the monodromy groups of such systems for almost all

λ
\lambda

and give natural criteria for the corresponding geometric adelic representation to have open image in an appropriate sense. Key inputs to our results are automorphic methods and the Langlands correspondence over global function fields proved by L. Lafforgue.

To say more, let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis rho overbar Subscript lamda Baseline colon pi 1 left-parenthesis upper X right-parenthesis right-arrow GL Subscript n Baseline left-parenthesis k Subscript lamda Baseline right-parenthesis right-parenthesis Subscript lamda">

(

ρ
¯

λ

:

π
1

(
X
)
→

GL

n

(

k
λ

)

)
λ

(\overline {\rho }_\lambda \colon \pi _1(X)\to \textrm {GL}_n(k_\lambda ) )_\lambda

be the corresponding mod-

λ
\lambda

system, where for every

λ
\lambda

by

O

λ

{\mathcal O}_\lambda

and

k
λ

k_\lambda

we denote the valuation ring and the residue field of

E
λ

E_\lambda

, and where the reduction is done with respect to some

π
1

(
X
)

\pi _1(X)

-stable

O

λ

{\mathcal O}_\lambda

-lattice

Λ
λ

\Lambda _\lambda

of

E
λ
n

E_\lambda ^n

. Let also

G
λ

g
e
o

G_\lambda ^{\mathrm {geo}}

be the Zariski closure of

ρ
λ

(

π
1

(

X

κ
¯

)
)

\rho _\lambda (\pi _1(X_{\overline {\kappa }}))

in

GL

n
,
E

\textrm {GL}_{n, E}

, and let

G

λ

g
e
o

\mathcal {G}_\lambda ^{\mathrm {geo}}

be its schematic closure in

Aut

O

λ

(

Λ
λ

)

\textrm {Aut}_{{\mathcal O}_\lambda }(\Lambda _\lambda )

. Assume in the following that the algebraic groups

G
λ

g
e
o

G_\lambda ^{\mathrm {geo}}

are connected.

We prove that for almost all

λ
\lambda

the group scheme

G

λ

g
e
o

\mathcal {G}_\lambda ^{\mathrm {geo}}

is semisimple over

O

λ

{\mathcal O}_\lambda

, and its special fiber agrees with the Nori envelope of

ρ
¯

λ

(

π
1

(

X

κ
¯

)
)

\overline {\rho }_\lambda (\pi _1(X_{\overline {\kappa }}))

. A comparable result under different hypotheses was proved by A. Cadoret, C.-Y. Hui, and A. Tamagawa using other methods. As an intermediate result, we show for

X
X

a curve that any potentially tame compatible system of mod-

λ
\lambda

representations can be lifted to a compatible system over a number field; this implies for almost all

λ
\lambda

the semisimplicity of the restriction

ρ
¯

λ

|

π
1

(

X

κ
¯

)

\overline {\rho }_\lambda |_{\pi _1(X_{\overline {\kappa }})}

. Finally, we establish adelic openness for

(

ρ
λ

|

π
1

(

X

κ
¯

)

)
λ

(\rho _\lambda |_{\pi _1(X_{\overline \kappa })} )_\lambda

in the sense of C. Y. Hui and M. Larsen, for

E
=

Q

E={\mathbb {Q}}

in general, and for

E
⊋

Q

E\supsetneq {\mathbb {Q}}

under additional hypotheses.