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# This file is ~steger/donald/SU21/a15p2/D.txt
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CASE: $(a=15,p=2)$

The BASE FIELDS:

  k=\QQ
  \ell=\QQ[\sqrt(-15)]

$\ell$~ramifies over~$\QQ$ at the $3$-adic and $5$-adic places

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

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

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) =                        
    (-\sqrt{-15}/5) Z^2                 (\sqrt{-15}/5) Z^2
  + (-1/2 + 3\sqrt{-15}/10) Z         + (-1/2 - 3\sqrt{-15}/10) Z
  + (2\sqrt{-15}/5)                   + (-2\sqrt{-15}/5)
                                                                     
  \phi(Z^2) =                         \phi^2(Z^2) =                    
    (-1/2-3\sqrt{-15}/10) Z^2           (-1/2+3\sqrt{-15}/10) Z^2
  + (\sqrt{-15}/5) Z                  + (-\sqrt{-15}/5) Z                 
  + (3 + 3\sqrt{-15}/5)               + (3 - 3\sqrt{-15}/5)

The discriminant of $Z^3+3Z^2+3$ is $(-15)9$.

$m$~ramifies over~$\ell$ only at the $3$-adic place.

$m$~splits at the $5$-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 = 2.103803402735536533164947332829

INVOLUTION of the second kind:

  \bar Z                = Z
  \overline{\sqrt{-15}} = -\sqrt{-15}

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

On complex matrices:

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

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

which means unitary matrices ought to preserve the form:

  |z_1|^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{-15}^j Z^k \sigma^l

where the coefficients

  d_{jkl} \in \QQ

satisfy integrality conditions as follows:

  (.) For any prime~$v$ except 2, 3, or 5, 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 primes~3 and~5, one must use special conditions,
  as calculated in ~/donald/SU21.  For each of these two primes, there
  are two different conditions: Type~1 and Type~2.

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

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

DETERMINANTS:

An element of $U(\D,\iota)$ can have a determinant which is any power
of~$(1+\sqrt{-15})/4$.  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 $Z^3+3Z+3=0$ 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   
   \phi \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, $3$-adic, and $5$-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~$\sqrt{-7}$ in $\QQ_p$, and so
  $\ell_v=\QQ_p$. 

  (b) the case where $\ell_v=\QQ_p[\sqrt{-7}]$ 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) $z^3+3z^2+3$ has a root in~$\ell_v$, so $m_w=\ell_v$;

  (d) $z^3+3z^2+3$ 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 right even for~$p=2$, because the $2$-adic
building reduces to a single point.