$p$--adic random matrix theory

Can $p$-adic random matrix theory explain the arithmetic factors that appear in the asymptotic formulas for the mean values of $L$-functions?

It would be interesting to evaluate moments of the $p$-adic absolute value of the characteristic polynomial of matrices averaged over the classical groups (over ${\mathbb Z}_p$). In other words, evaluate

\int_{GL_n({\mathbb Z}_p)}
\vert\det (1-u)\vert _p^s\, du .

(More information is sought. It was suggested that these calculations may have already been done by Anderson(?) or they may be in Macdonald's book?)

Back to the main index for L-functions and Random Matrix Theory.