## Rigid local systems on $$\mathbb G_m$$ with finite monodromy.(English)Zbl 1193.22001

Katz has constructed some rigid local systems of rank 7 on $$\mathbb G_m$$ in finite characteristics, whose monodromy is a finite subgroup $$G$$ of a complex Lie group $$M$$ of type $$G_2$$.
Among the five subgroups $$G$$ of $$M$$ which appear in his work, four occur in their natural characteristic: $\mathrm{SL}_2(8) \text{ in }\operatorname{char} 2,\quad \mathrm{PU}_3(3) \text{ in } \operatorname{char} 3, \quad \mathrm{PGL}_2(7) \text{ in }\operatorname{char} 7, \quad \mathrm{PSL}_2(13) \text{ in }\operatorname{char} 13.$ In this paper, using the theory of Deligne-Lusztig curves, the author gives another construction of these four local systems on $$\mathbb G_m$$, and finds similar local systems with finite monodromy in other complex Lie groups. Some examples of finite groups which occur in exceptional complex Lie groups $$M$$, analogous to the ones above are:
\begin{alignedat}{2}2 G &= \mathrm{PSL}_2(27) \text{ in }\operatorname{char}3,\quad& M&=F_4,\\ G &= \mathrm{PSU}_3(8) \text{ in }\operatorname{char}2,\quad& M&=E_7,\\ G &=\mathrm{PGL}_2(31) \text{ in }\operatorname{char}31,\quad& M&=E_8,\\ G &= \mathrm{PSL}_2(61) \text{ in }\operatorname{char}61,\quad& M&=E_8. \end{alignedat}
There are interesting families in the classical groups. For example, when $$q=p^f>2$$, we have $G=\mathrm{PU}_3(q) \text{ in }\operatorname{char}p,\quad M=\mathrm{Sp}_{2n},\quad 2n=q(q-1).$ As a bonus, the author obtains simple wild parameters for the local field $$k((1/t))$$, by restricting the local systems to the decomposition group at $$t=\infty$$. These parameters are representations of the local Galois group into $$M$$, which have no inertial invariants on the adjoint representation and have Swan conductor equal to the rank of $$M$$.

### MSC:

 20G40 Linear algebraic groups over finite fields 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14L30 Group actions on varieties or schemes (quotients) 20G05 Representation theory for linear algebraic groups
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### References:

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