symplectic group Sp(p,q) and reduction modulo a prime

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ognjenvukadin@yahoo.com - 03 Oct 2008 12:25 GMT

Hi!
Does anyone know a book or a reference where one can find something
about the reduction modulo a prime of certain algebraic groups, i.p.
f.e. if we have a group G defined over a totally real number field F,
with the help of a quaternion division algebra D/F and a hermitian
form on D^{n+1} s.t.
G(R) ~ Sp(n,1) (\times ...),
and a prime ideal p of O_F, what group do we get by reduction G_p of G
modulo p, i.e.
to what group is G_p ( O/p ) isomorphic? (is it in the series of
finite classical groups,...?)

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J.S. Milne - 03 Oct 2008 14:43 GMT

The "reduction modulo a prime" of an algebraic group is not well-
defined since it depends on the choice of a model over the ring of
integers. In down-to-earth terms, an embedding into GLn defines a
reduction, but a different embedding may give a different reduction.

Models of semisimple groups over rings of integers in local fields
(hence reductions) were studied in a long series of long papers by
Bruhat and Tits. For example, in your situation, a smooth semisimple
model exists if and only if the group is unramified (quasi-split and
splits over an unramified extension).

I don't know of any very friendly introduction to Bruhat-Tits theory.
There is a summary of it in
Tits, J. Reductive groups over local fields. Automorphic forms,
representations and $L$-functions (Proc. Sympos. Pure Math., Oregon
State Univ., Corvallis, Ore., 1977), Part 1, pp. 29--69, Proc.
Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.
MR0546588

J.S. Milne

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Math Forum / Mathematics / Research / October 2008

symplectic group Sp(p,q) and reduction modulo a prime