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Lattices of minimal covolume in $$\mathrm{SL}_ 2$$: a nonarchimedean analogue of Siegel’s theorem $$\mu\geq \pi /21$$. (English) Zbl 0731.22009
Let $$F$$ be a locally compact field of characteristic $$p>0$$ with residue field of order $$q=p^{\alpha}$$. Hence $$F$$ is isomorphic to $$F_ q((1/t))$$, the field of Laurent formal power series in $$1/t$$.
Theorem. Let $$G=\mathrm{SL}_ 2(F)$$ and $$\mu$$ be its Haar measure normalized so that $$\mu(K)=1$$ for all maximal compact subgroups of $$G$$. Then $\min \{\mu (\Gamma \setminus G)\mid \Gamma \text{ a lattice in } G\}=(q- 1)^{-2}(q+1)^{-1}.$
This minimum is obtained for $$\Gamma_ 0=\mathrm{SL}_ 2(F_ q[t])$$, the characteristic $$p$$ modular group.
Reviewer: K. Riives (Tartu)

##### MSC:
 22E40 Discrete subgroups of Lie groups 20G25 Linear algebraic groups over local fields and their integers 22E20 General properties and structure of other Lie groups
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