\title{Minimum Co-volume of Lattices in Classical Chevalley Groups over $\mathbb F_q((1/t))$ }
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This talk is about minimum co-volume of lattices of a classical Chevalley group
over a local field with a positive characteristic. There are
a lot of ways to show that such value exists. One way is using a celebrated
finiteness theorem of Borel and Prasad. However none of these existence
theorems give us this value. A classical result of Siegel asserts that
if $G=SL_2(\bbr)$ and its Haar measure $\mu$ is appropriately normalized then
min$_{\Gamma}$$\mu(\Gamma \backslash G)=\pi/21$, where $\Gamma$ runs over all the lattices of $G$.
The minimum is obtained with a co-compact lattice. The case of two by two special
linear group over the complex numbers is unknown yet. In this case
Meyerhoff has shown that $SL_2(\mathcal{O})$ has the minimum co-volume between the
non-uniform lattices, where $\mathcal{O}$ is the ring of integers of
$\bbq(\sqrt{-3})$. Lubotzky considered this problem for non-Archimedean
fields. First he considered the positive characteristic case, and
surprisingly he observed that for odd primes minimum occurs only for
non-uniform lattices and he computed this value. In the second paper
Lubotzky and Weigel considered the characteristic zero case and calculated
the minimum co-volume. It is worth mentioning that in this case by a
theorem of Tamagawa all the lattices are uniform.
In my work, I studied lattices of classical Chevalley groups over
non-Archimedean local field F with positive characteristic. One
can see that in this case, for odd primes the minimum co-volume,
again occurs for non-uniform lattices. I managed to find this
value and characterize all the lattices whose co-volume is the
minimum one.\
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