zbMATH — the first resource for mathematics

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)

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
Full Text: DOI