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# This file is ~steger/donald/SU21/a23p2/D.txt
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CASE: $(a=23,p=2)$

The BASE FIELDS:

  k=\QQ
  \ell=\QQ[\sqrt(-23)]

$\ell$~ramifies over~$\QQ$ only at the $23$-adic place

PRESENTATION of $\D$, degree three division algebra, central
over~$\ell$:

  m=\ell[Z]/(Z^3-Z-1)

This is a cylic Galois extension of~$\ell$ with Galois group

  \rangle 1, \phi, \phi^2 \rangle

with $\phi$ given by:

  \phi(Z) =                           \phi^2(Z) =                        
    (-3\sqrt{-23}/23) Z^2               (3\sqrt{-23}/23) Z^2
  + (-1/2 + 9\sqrt{-23}/46) Z         + (-1/2 - 9\sqrt{-23}/46) Z
  + (2\sqrt{-23}/23)                  + (-2\sqrt{-23}/23)

  \phi(Z^2) =                         \phi^2(Z^2) =                    
    (-1/2 - 9\sqrt{-23}/46 ) Z^2        (-1/2 + 9\sqrt{-23}/46 ) Z^2
  + (\sqrt{-23}/23) Z                 + (-\sqrt{-23}/23) Z           
  + (1 + 3\sqrt{-23}/23)              + (1 - 3\sqrt{-23}/23)        

The discriminant of $Z^3-Z-1$ is $-23$.

$m$~doesn't ramifies over~$\ell$ at any place.

$m$~splits at the $23$-adic place of~$\ell$.

$\D$ is generated by~$m$ and~$\sigma$ where

  \sigma x \sigma^{-1} = \phi(x)
  \sigma^3 = 2 

Then

  (1') $\D\otimes\ell_{2\pm}$ has invariant $\pm 1$;
  (2') $\D\otimes\ell_v$ splits at every other place~$v$.

EMBEDDING of $\D$ in $\Mat_{3\time 3}(\CC)$:

               x   0         0
   x \mapsto   0 \phi(x)     0        (for $x\in m$)
               0   0     \phi^2(x)


                   0  1  0   
   \sigma \mapsto  0  0  1
                   2  0  0


For REAL calculation:

  Z = 1.324717957244746025960908854478

INVOLUTION of the second kind:

  \bar Z                = Z
  \overline{\sqrt{-23}} = -\sqrt{-23}

  \iota(x)      = \bar x            (for $x\in m$)
  \iota(\sigma) = \sigma

On complex matrices:

  \iota(A) = F^{-1} A^* F

for
       2  0  0
   F = 0  0  1
       0  1  0

which means unitary matrices ought to preserve the form:

  2|z_1|^2 + 2\Re (z_1 \bar z_2)

INTEGRALITY:

We want elements of the form

  \sum_{j=0,1} \sum_{k=0,1,2} \sum_{k=0,1,2}
       d_{jkl} \sqrt{-23}^j Z^k \sigma^l

where the coefficients

  d_{jkl} \in \QQ

satisfy integrality conditions as follows:

  (.) For any prime~$v$ except 2 or 23 the $d_{jkl}$ are integral
  in $\QQ_v$, i.e. no factors of~$v$ occur in the denominators.

  (.) Factors of~2 may appear in the denominators.  However, the power
  of~2 which appears is bounded above.  Almost certainly all such
  powers disappear if one multiplies the~$d_{jkl}$ by~256.

  (.) At the special prime~23 one must use special conditions, as
  calculated in ~/donald/SU21/a2p3.  For~23 there are two
  different conditions: Type~1 and Type~2.

  The Type~1 condition amounts to requiring that the element
  of~$U(\D,\iota)$ fix a certain lattice in~$\QQ_{23}^3$ which is
  self-dual relative to the sesquilinear form associated to~$\iota$.

  The Type~2 condition amounts to requiring that the element
  of~$U(\D,\iota)$ fix a certain pair of mutually dual lattices
  in~$\QQ_{23}^3$, a pair which is also adjacent in the building
  of~$PGL(3,\QQ_{23})$.

DETERMINANTS:

Elements of $U(\D,\iota)$ can have determinant which is any power
of~$(7+3\sqrt{-23})/16$.  In~$PU$, one can always choose a
representative where the power is~$0$ or~$\pm 1$.

Action on BUILDINGS and TREES:

Let $v$ be any place of~$\ell$ and let~$w$ be some place of~$m$ lying
above~$v$.  If the equation $Z^3-Z-1=0$ is solvable in $\ell_v$, there
are three such places corresponding to the three solutions; for each
of the three places $m_w=\ell_v$.  If there is no solution, then there
is only one choice for~$w$, and $m_w$~is obtained from~$\ell_v$ by
adjoining a solution.

Now, consider the map $\D \to \Mat_{3\times 3}(m_w)$ given by

               x   0         0
   x \mapsto   0 \phi(x)     0        (for $x\in m$)
               0   0     \phi^2(x)


                   0  1  0   
   \sigma \mapsto  0  0  1
                   2  0  0

The map to $\Mat_{3\times 3}(\CC)$ was given by identical formulas.
Indeed, the map to $\Mat_{3\times 3}(\CC)$ is the special case of this
one for the situation where $v=v_\infty$, the real place of~$\ell$.

Except at the $2$-adic and $23$-adic places, the integrality condition
for an element $\gamma\in\D$ corresponds to the condition that at each
place~$v$, the matrix for $\gamma$ has entries which are integral
in~$m_w$.

Suppose $v$~lies over the rational prime~$p$.  One must analyse
separately:

  (a) the case where there exists a~$\sqrt{-23}$ in $\QQ_p$, and so
  $\ell_v=\QQ_p$. 

  (b) the case where $\ell_v=\QQ_p[\sqrt{-23}]$ is of degree two
  over~$\QQ_p$. 

In case~(a), the building of the unitary group at the prime~$p$ is of
type~$\tilde A_2$.  In case~(b) the building is a tree.

This division into cases is inherent in the algebraic group we want to
study, namely $U(\D,\iota)$, considered as an algebraic group
over~$\QQ$.

Vice versa, the specific presentation of~$\D$ which we're using is an
arbitrary choice on our part.  There are many fields which could be
used in place of~$m$.  The actual choice of~$m$ was the one that
appeared ``simplest''.  Nonetheless, having chosen~$m$, we must
further divide each of cases~(a) and~(b) according to one of:

  (c) $Z^3-2Z-2$ has a root in~$\ell_v$, so $m_w=\ell_v$;

  (d) $Z^3-2Z-2$ has no roots in~$\ell_v$, so $m_w$~is of degree~3
  over~$\ell_v$, obtained by adjoining a root.

Nonetheless, considering separately cases~(ac), (ad), (bc), and~(bd),
one sees that unitary elements $\gamma\in\D$ satisfying our
integrality condition are those which fix a hyperspecial vertex in
each of the sequence of buildings associated to the rational
primes~$p$.  This is correct also for~$p=2$, because the $2$-adic
building reduces to a single point.