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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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