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# This file is ~steger/donald/SU21/C18p3/D.txt
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CASE: $(C_{18},p=3)$

The BASE FIELDS:

  k = \QQ[\sqrt{6}]
  \ell = k[\zeta_3] = k[\sqrt{-3}]
       = \QQ[\sqrt{6},\sqrt{-3}] = \QQ[\sqrt{-2},\sqrt{-3}] 

$k$~ramifies over~$\QQ$ only at the $2$-adic and $3$-adic places.
$\ell$ doesn't ramify over~$k$ at any places.

The EXTENSION FIELD used to present the division algebra:

  m = \QQ[\sqrt{6},\zeta_9] = \QQ[\sqrt{-2},\zeta_9] = k[\zeta_9]
    = k[Z]/(Z^6+Z^3+1)

Here~$Z=\zeta_9$ is a primitive $9$'th root of unity.

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

  \{ Z\mapsto Z, \mapsto Z^2, \mapsto Z^4,
                 \mapsto Z^8, \mapsto Z^7, \mapsto Z^5 \}

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

  \sqrt{-3} = 1 + 2\zeta_3 = 1 + 2Z^3

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

  \phi(Z)     = Z^4     = \zeta_3 Z
  \phi(Z^j)   = Z^{4j}  = \zeta_3^j Z^j

  \phi^2(Z)   = Z^{16}  = \zeta_3^2 Z
  \phi^2(Z^j) = Z^{16j} = \zeta_3^{2j} Z^j

Specifically, it's

  \{ Z\mapsto Z, \mapsto Z^4, \mapsto Z^{16} \}

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

The intersection of~$m$ with the reals is generated by~$\sqrt{6}$ and

  W = Z + Z^{-1}

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

  W^3 - 3W + 1

The discriminant of $W^3-3W+1$ is~$81$.

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

  \phi^2(W)   = W^2-2
  \phi^2(W^2) = -W^2-W+4

Over~$\QQ$, $m$~ramifies at the $3$-adic place with ramification index
$e=6$; at the $2$-adic place it ramifies with ramification index~$e=2$

Over~$k$, $m$~ramifies only at the $3$-adic place with ramification
index $e=3$.

Over~$\ell$, $m$ ramifies only at the $3$-adic places with
ramification index $e=3$.

The ramification at the $3$-adic place is \emph{wild} ramification.

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

$\D$ is generated by~$m$ and~$\sigma$ where for $x\in m$

  \sigma x \sigma^{-1} = \phi(x)
  \sigma^3 = (\sqrt{-3} + \sqrt{6}) / 3 = 1/3 + 2 Z^3/3 + \sqrt{6}/3

Then

  (1') $\D\otimes\ell_{3\pm}$ has Hasse invariant $\pm 1$;
  (2') $\D\otimes\ell_v$ splits at every other place~$v$.

For any place~$v$ which is not $3$-adic, we know $\D\otimes\ell_v$
splits because $(\sqrt{-3}+\sqrt{6})/3$ has valuation zero at~$v$ and
because $m$~doesn't ramify over~$\ell$ at~$v$.

For the two $3$-adic place~$3\pm$ of~$\ell$, the calculations showing
that~$\D$ does not split are more complicated.  See Appendix~A.

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
                   (\sqrt{-3}+\sqrt{6})/3   0  0


For COMPLEX calculation:

   Z = 0.7660444431189780352023927 + 0.6427876096865393263226434 i
   W = Z + Z^{-1} = 1.5320888862379560704047853
   (\sqrt{-3}+\sqrt(6))/3
     = 0.8164965809277260327324280 + 0.5773502691896257645091488 i

PRELIMINARY INVOLUTION of the second kind:

  \bar Z               = Z^{-1}
  \bar W               = W
  \overline{\sqrt{-3}} = -\sqrt{-3}

  \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 $PU(3)\times PU(3)$ at the
real place, we need to adjust it.  We use

  T    = (3-3\sqrt{6}) + \sqrt{6} W + \sqrt{6} W^2
       = 3 + \sqrt{6} (-1-\zeta_9^4-\zeta_9^5)
  \phi(T) = (3+\sqrt{6}) - \sqrt{6} W^2
          = 3 + \sqrt{6} (-1+\zeta_9-\zeta_9^2+\zeta_9^4)
  === ERROR CORRECTED HERE ===>
  \phi^2(T) = (3-\sqrt{6}) - \sqrt{6} W
            = 3 + \sqrt{6} (-1-\zeta_9+\zeta_9^2+\zeta_9^5)

The adjusted involution is:

  \iota(x) = T^-1 \iota_0(x) T

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

  \iota(\sigma)      = T^{-1} \sigma^{-1} T
                     = \phi^2(T)/T \sigma^{-1}
                     = (-1 + 4W - \sqrt{6} W^2) \sigma^{-1}
                     = ((-1-2\sqrt{6})+(4+\sqrt{6})Z+(-4-\Sqrt{6})Z^2
                        +\sqrt{6}Z^4-4Z^5) \sigma^-1

  \iota(\sigma^{-1}) = T^-1 \sigma T
                     = \phi(T)/T \sigma
                     = (-1 -\sqrt{6}W + 2W^2) \sigma
                     = (3+(-2-\sqrt{6})Z+(2+\sqrt{6})Z^2-2Z^4
                         +\sqrt{6}Z^5) \sigma

On complex matrices:

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

for

         T       0           0
   F =   0    \phi(T)        0
         0       0       \phi^2(T)

which means unitary matrices ought to preserve the form:

   T |v_1|^2 + \phi(T) |v_2|^2 + \phi^2(T) |v_3|^2

INTEGRALITY:

We want elements of the form

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

where the coefficients 

  c_{ijl}\in\QQ

satisfy integrality conditions as follows:

  (.) For any prime~$v$ except 2 or~3 the $c_{ijl}$ are integral in
  $\QQ_v$, i.e. no factors of~$v$ occur in the denominators.

  (.) Factors of~3 may appear in the denominators.  However, the power
  of~3 which appears is bounded above.  Almost certainly all such
  powers disappear if one multiplies the~$c_{ijl}$ by~729.

  (.) At the special prime~2 there are two possible conditions.  The
  Type~1 (or hyperspecial) condition is simple: the $c_{ijl}$ mustn't
  have any factors of two in their denominators.  For the Type~2 (or
  non-hyperspecial) condition one must use special conditions, as
  calculated in ~/donald/SU21/C18p3/padic.gap.

The Type~1 condition amounts to requiring that an element
of~$U(\D,\iota)$ fix a certain lattice in~$\ell_2^3$ 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~$\ell_2^3$, a pair which is also adjacent in the building
of~$PGL(3,\ell_2)$.

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 $\GG(\QQ_2)$ 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 $\GG(\QQ_2)$
belong to a certain non-hyperspecial parahoric.
  
DETERMINANTS:

Elements of $U(\D,\iota)$ can have determinant which is $1$,
$\zeta_3$, or $\zeta_3^2$ multiplied by any power
of~$(\sqrt{-3}+\sqrt{6})/3$.  In~$PU$, one can always choose a
representative where the power is~$0$ or~$\pm 1$.

APPENDIX A: NON-SPLITTING OF $\D$ AT THE 3-ADIC PLACES

We can write:

  \ell = \QQ[\sqrt{-2},\zeta_3]
  m    = \QQ[\sqrt{-2},\zeta_9]

As we do this, let us fix signs as follows and identify:

  \sqrt{-3} = 2\zeta_3+1
  \sqrt{6}  = -\sqrt{-3}\sqrt{-2}

For

  k'=\QQ[\sqrt{-2}]=\QQ[S']/(S'^2+2)

there are two $3$-adic places, $3+$ and~$3-$.  Both $k'_{3+}$ and
$k'_{3-}$ are copies of $\QQ_3$, and we can characterize the two
places, respectively, by the conditions $S'=4 \mod~9$ and $S'=5
\mod~9$.

For~$\ell$ there are likewise two $3$-adic places, still called $3+$
and~$3-$.  $\ell_{3+}$ (respectively~$\ell_{3-}$) is a copy of
$\QQ_3[\zeta_3]$, ramified of degree~$2$ over~$\QQ_3$.  For~$m$ there
are again two $3$-adic places, still called $3+$ and~$3-$.  $m_{3+}$
(respectively~$m_{3-}$) is a copy of $\QQ_3[\zeta_9]$, ramified of
degree~6 over $\QQ_3$ and of degree~3 over $\ell_{3+}$ (respectively
$\ell_{3-}$).

Let~$v=3+$ or~$3-$.  Which elements of~$\ell_v$ are norms from~$m_v$?
That is, what is $N_{m_v/\ell_v) (m_v)$?  From general theory we know
that
  
  [\ell_v^\times : N_{m_v/\ell_v) (m_v^\times)] = [m_v : \ell_v] = 3

One element of minimal positive valuation in~$\ell_v$
is~$\pi=1-\zeta_3$, which satisfies $\pi^2-3\pi+3=0$.  Here are some
particular norms:

  N_{m_v/\ell_v}(1-\zeta_9)
    = (1 - \zeta_9) (1 - \zeta_9\zeta_3) (1 - \zeta_9\zeta_3^2)
    = 1 - \zeta_9^3
    = 1 -\zeta_3
    = \pi
  N_{m_v/\ell_v}(-1) = (-1)^3
    = -1
  N_{m_v/\ell_v}(\zeta_9)
    = \zeta_9 \zeta_9\zeta_3 \zeta_9\zeta_3^2
    = \zeta_9^3
    = \zeta_3
    = 1 - \pi
  N_{m_v/\ell_v} (1-\zeta_9\pi})
    = (1-\zeta_9\pi) (1-\zeta_9\zeta_3\pi) (1-\zeta_9\zeta_3^2\pi)
    = 1-(\zeta_9\pi)^3
    = 1-\zeta_3\pi^3
    = 1-\pi^3 \mod 9

With the first three of these one sees that any element in
$\ell_v^\times$ can be multiplied by norms to obtain

  first an element of valuation zero, 

  next an element $=1 \mod \pi$,

  and finally an element $=1\mod 3$.

If the final element is $=1 \mod 3\pi$ then it is a norm in~$\ell_v$.
Indeed, using $N_{m_v/\ell_v} (1-\zeta_9\pi)$ one can further reduce
it to an element $=1\mod 9$.  The reduced element is now a cube
in~$\ell_v$ --- use

  (1+3)^3 = 1+9 \mod 27
  (1+3\pi)^3 = 1+9\pi \mod 81
  (1+9)^3 = 1+27 \mod 243
  (1+9\pi)^3 = 1+27\pi \mod 729
  \dots

Not all elements are norms, which implies that while elements $=1 \mod
3\pi$ are norms, elements $=4$ or $=7 \mod 3\pi$ are not.

The element of interest is $\sigma^3 = (\sqrt{-3}+\sqrt{6})/3$.
Multiply first by the norm $\pi=1-\zeta_3=(3-\sqrt{-3})/2$ to get

  \sigma^3 \pi = (\sqrt{-3}+\sqrt{6})/3 (3-\sqrt{-3}) / 2
               = (1-\sqrt{-2}) \sqrt{-3} (3-\sqrt{-3} / 6
               = (1-\sqrt{-2}) (1+\sqrt{-3}) / 2 
               = (1-\sqrt{-2}) (2-\pi)

Let us analyse this for~$v=3-$ where $\sqrt{-2}=5 \mod 9$, whence

  \sigma^3 \pi = (1-5) (2-\pi) = 1+4\pi \mod 9
               = 1+\pi \mod 3\pi

Multiplying this by the norm $1-\pi$ we get

  \sigma^3 \pi (1-\pi) = 1-\pi^2 = 1-(1-\zeta_3)^2
     = 1-(1-2\zeta_3+\zeta_3^2) = 1+3\zeta_3
     = 1+3(1-\pi) = 4 \mod 3\pi

According to the previous analysis $4 \mod 3\pi$ is not in fact a norm
from~$m_v$ to~$\ell_v$.

A similar analysis for $v=3+$ is doable, but it can be avoided.
Indeed, the only difference in passing from~$3-$ to~$3+$ is a
change in sign for $\sqrt{-2}$, which is equivalent to
replacing~$\sigma^3$ with its inverse.  Consequently, for~$v=3+$,
$\sigma^3$ will be in the same $N_{m_v/\ell_v} m_v^\times$ coset as
$1/4 = 7 \mod 9$.