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CASE: $(a=7,p=2)$ (NEW VERSION)

The BASE FIELDS:

  k=\QQ
  \ell=\QQ[\sqrt{-7}]

$\ell$~ramifies over~$\QQ$ only at the $7$-adic place.

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

  m=\QQ[Z]/(Z^6+Z^5+Z^4+Z^3+Z^2+Z+1)

Here~$Z=\zeta_7$ is a primitive $7$'th root of unity.  (NOTA BENE:
this~$m$ and this~$Z$ are different from those used for the other
presentation of $\D$).

$m$~is a cylic Galois extension of degree~$6$ over~$\QQ$ with Galois
group

  \{ Z\mapsto Z, \mapsto Z^2, \mapsto Z^3,
                 \mapsto Z^4, \mapsto Z^5, \mapsto Z^6 \}

$m$~is also an extension of $\ell$, where we take:

  \sqrt{-7} = 1 + 2Z + 2Z^2 + 2Z^4
            = Z + Z^2 + Z^4 - Z^{-1} - Z^{-2} - Z^{-4} 

The irreducible polynomial of~$Z$ over~$\ell$ is:

  Z^3 + (1/2-\sqrt{-7}/2) Z^2 + (-1/2-\sqrt{-7}/2) Z - 1

The Galois group of~$m$ over~$\ell$ is of order~3 generated by~$\phi$:

  \phi(Z)     = Z^2    = +Z^2
  \phi(Z^2)   = Z^4    = -Z^2 - Z + (-1/2+\sqrt{-7}/2)

  \phi^2(Z)   = Z^4    = -Z^2 - Z + (-1/2+\sqrt{-7}/2)
  \phi^2(Z^2) = Z^{16} = +Z^2

That is, the Galois group of~$m$ over~$\ell$ is:

  \{ Z\mapsto Z, \mapsto Z^2, \mapsto Z^4 \}

which is the unique subgroup of order~$3$ of the Galois group of
$m$~over~$\QQ$.

The intersection of~$m$ with the reals is generated by

  W = Z + Z^{-1}

whose irreducible polynomial over~$\QQ$ (or over~$\ell$) is

  W^3 + W^2 - 2W -1

The discriminant of $W^3+W^2-2W-1$ is $49$.

  \phi(W)     = W^2-2
  \phi(W^2)   = -W^2-W+3

  \phi^2(W)   = -W^2-W+1
  \phi^2(W^2) = W+2

Over $\QQ$, $m$ ramifies at the $7$-adic place with ramification
index $e=6$.

Over $\ell$, $m$ ramifies at the $7$-adic place with ramification
index $e=3$.

This is all \emph{tame} ramification, which is easier to deal with
than \emph{wild} ramification, which is what happens when a degree
three extension is ramified at a $3$-adic place.

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

  \sigma x \sigma^{-1} = \phi(x)
  \sigma^3 = (3 + \sqrt{-7}) / 4 = 1 + 1/2 (Z + Z^2 + Z^4)

Then

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

For a place~$v$ which is neither $2$-adic nor $7$-adic, we know
$\D\otimes\ell_v$ splits because $(3+\sqrt{-7})/4$ has valuation zero
at~$v$ and because $m$~doesn't ramify over~$v$.  For the $7$-adic
place~$v$, we know that $\D\otimes\ell_v$ splits because
$(3+\sqrt{-7})/4$ has valuation zero there, and because the projection
of $(3+\sqrt{-7})/4$ to the residue field, $\ZZ/7$, is $3/4 = -1$,
which is a cube, hence a norm from the residue field of~$m$.

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
                   (3+\sqrt{-7})/4  0  0


For COMPLEX calculation:

   Z = 0.6234898018587335305300629 + 0.7818314824680298087044108 i
   W = Z + Z^{-1} = 1.2469796037174670610601259
   (3+\sqrt(-7))/4 = 0.75 + 0.6614378277661476476254039 i

PRELIMINARY INVOLUTION of the second kind:

  \bar Z               = Z^{-1}
  \bar W               = W
  \overline{\sqrt{-7}} = -\sqrt{-7}

  \iota_0(x)      = \bar x            (for $x\in m$)
  \iota_0(\sigma) = \sigma^{-1}

On complex matrices:

  \iota_0(A) = A^*

INVOLUTION of the second kind:

Because $\iota_0$ gives the compact group $PU(3)$ at the real place,
we need to adjust it.

  \iota(x) = W^{-1} \iota_0(x) W

  \iota(x)       = \bar x  (for $x\in m$)
  \iota(\sigma)  = W^{-1} \sigma^{-1} W = \phi^2(W)/W \sigma^{-1}
                 = (W^2-3) \sigma^{-1}
                 = (Z^5+Z^2-1) \sigma^{-1}
  \iota(\sigma)  = W^{-1} \sigma W = \phi(W)/W \sigma
                 = (-2*W^2-W+4) \sigma
                 = (-Z^5+Z^4+Z^3-Z^2+1) \sigma

On complex matrices:

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

for

       Z+Z^{-1}      0           0
   F =   0       Z^2+Z^{-2}      0
         0           0       Z^4+Z^{-4}

which means unitary matrices ought to preserve the form:

  (Z+Z^{-1}) |v_1|^2 + (Z^2+Z^{-2}) |v_2|^2 + (Z^4+Z^{-4}) |v_3|^2

INTEGRALITY:

We want elements of the form

  \sum_{j=0,1,2,3,4,5} \sum_{l=-1,0,1} c_{jl} Z^j \sigma^l

where the coefficients 

  c_{jl}\in\QQ

satisfy integrality conditions as follows:

  (.) For any prime~$v$ except 2, 3, or 5, or~7 the $c_{jl}$ 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, 5, and~7 one must use special
  conditions, as calculated in ~/donald/SU21/a7p2N/padic.gap.  For
  each of these primes there are two different conditions: Type~1 and
  Type~2.  For the prime~3 (respectively~5) the Type~1 condition is
  simply that there are no factors of~3 (respectively~5) in the
  denominators of the~$c_{jl}$.

The Type~1 condition amounts to requiring that an element
of~$U(\D,\iota)$ fix 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 an element
of~$U(\D,\iota)$ fix a certain pair of mutually dual lattices
in~$\QQ_v^3$, a pair which are also adjacent in the building
of~$PGL(3,\QQ_v)$.

At the primes~3 and~5, the Type~1 condition is the condition that the
element of~$U(\D,\iota)$ fix a certain hyperspecial vertex of the
tree, or in other words that its image in $\GG(\QQ_v)$ belong to a
certain hyperspecial parahoric subgroup.  Contrariwise, the Type~2
condition is the condition that the element fix a certain
non-hyperspecial vertex on the tree; the condition that its image in
$\GG(\QQ_v)$ belong to a certain non-hyperspecial parahoric.
  
DETERMINANTS:

Elements of $U(\D,\iota)$ can have determinant which is any power
of~$(3+\sqrt{-7})/4$.  In~$PU$, one can always choose a
representative where the power is~$0$ or~$\pm 1$.