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# This file is ~steger/donald/SU21/C10p2/D.txt
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CASE: $(C_{10},p=2)$

The BASE FIELDS:

  k = \QQ[\sqrt{2}]
  \ell = k[\sqrt{-5+2\sqrt{2}}] = \QQ[\sqrt{2},\sqrt{-5+2\sqrt{2}}]
       = \QQ[\sqrt{2},U] / (U^2-(1+\sqrt{2})U+2)
       = k[U] / (U^2-(1+\sqrt{2})U+2)

where

  U = ((1+\sqrt{2}) + \sqrt{-5+2\sqrt{2}}) / 2

$k$~ramifies over~$\QQ$ only at the $2$-adic place.
$k$~has two different $17$-adic places, denoted $17\pm$.

$\ell$ ramifies over~$k$ only at~$17-$.
$\ell$ is a Galois extension of~$k$, but is not Galois over~$\QQ$. 

Since $(-5+2\sqrt{2})(1-\sqrt{2})^2 = -7-4\sqrt{2}$,
$\ell=k[\sqrt{-7-4\sqrt{2}}]$.  In [Prasad--Yeung] $\ell$~is given as
$k[\sqrt{-7+4\sqrt{2}]$ which is isomorphic to our~$\ell$ via an
isomorphism which exchanges $\pm\sqrt{2}$.

The EXTENSION FIELD used to present the division algebra:

  m = \QQ[\sqrt{2},U,\zeta_9+\zeta_9^-1] = \ell[\zeta_9+\zeta_9^-1]
    = \ell[W]/(W^3-3W+1)

Here~$\zeta_9$ is a primitive 9'th root of unity, and $W=2\Re\zeta_9$
generates the real subfield of~$\QQ[\zeta_9]$.  The discriminant of
$W^3-3W+1$ is~$81$.

$m$~is a cylic Galois extension of degree~$3$ over~$\ell$ with Galois
group induced from the maps

  \{ \zeta_9\mapsto\zeta_9, \mapsto\zeta_9^4, \mapsto\zeta_9^2 \}

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

  \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;

The intersection of~$m$ with the reals is generated by~$\sqrt{2}$
and~$W$.

Over~$k$, $m$~ramifies at the~$3$-adic place with ramification index
$e=3$ and at the place~$17-$ with ramification index~$e=2$

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

  \sigma x \sigma^{-1} = \phi(x)     (for $x\in m$)
  \sigma^3 = ((1+\sqrt{2}) + \sqrt{-5+2\sqrt{2}}) / (2\sqrt{2})
           = U / \sqrt{2}
Then

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

For a place~$v$ which is neither $2$-adic nor $3$-adic, we know
$\D\otimes\ell_v$ splits because
$((1+\sqrt{2})+\sqrt{-5+2\sqrt{2}})/(2\sqrt{2})$ has valuation zero
at~$v$ and because $m$~doesn't ramify over~$\ell$ at~$v$.

Since there is no $\sqrt{2}$ in~$\QQ_2$, there is only one $2$-adic
place of~$k$, which we continue to denote by~$2$.  The
completion~$k_2=\QQ_2[\sqrt{2}]$ is totally ramified of degree~2
over~$\QQ_2$.  There is a square root of $-5+2\sqrt{2}$ in~$\k_2$.
Indeed:

  (3-5\sqrt{2})^2 - (-5+2\sqrt{2}) = 64-32\sqrt{2}

and this can be extended to an exact square root using Hensel's Lemma.
Thus, there are two 2-adic places of~$\ell$ denoted~ $2\pm$ and
characterized by:

  \sqrt{-5+2\sqrt{2}) = \pm(3 - 5\sqrt{2}) \mod 16\sqrt{2}

Each of~$\ell_{2\pm}$ is isomorphic to $k_2$.  Adjoining a primitive
9'th root of unity to~$k_2$, we obtain $\QQ_2[\sqrt{2},\zeta_9]$,
which is the unramified extension of degree~6 of~$k_2$.  Since
$[\QQ_2[\sqrt{2},\zeta_9]:\QQ_2[\sqrt{2},W]\leq 2$, it follows that
$m_{2\pm}=\ell_{2\pm}[W]=k_2[W]=k_2[\zeta_9]$ is unramified of
degree~3 over $\ell_{2\pm}$.

Thus an element of $\ell_{2\pm}^\times$ is a norm from
$m_{2\pm}^\times$ if and only if its valuation is a multiple of~3.
For $\ell_{2+}$ we find:

  \sigma^3 = ((1+\sqrt{2}) + \sqrt{-5+2\sqrt{2}})/(2\sqrt{2})
           = ((1+\sqrt{2})+(3-5\sqrt{2}))/(2\sqrt{2}) \mod 16\sqrt{2}
           = (4-4\sqrt{2})/(2\sqrt{2}) \mod 16\sqrt{2}
           = \sqrt{2}-2 \mod 8

which has valuation~$1$.  Similarly, for $\ell_{2-}$ we find that
$\sigma^3$ has valuation~$-1$  Thus, the Hasse invariants of~$\D$
at~$\ell_{2\pm}$ are nonzero and opposite.

Because there is no~$\sqrt{2}$ modulo~$3$, there is a unique 3-adic
place of~$k$, and $k_3=\QQ_3[\sqrt{2}]$ is the degree~2 unramified
extension of~$\QQ_3$.  One can check that in $\FF_9$ there is no
square root of $-5+2\sqrt{2}$.  Consequently, there is a unique 3-adic
place of~$\ell$ and $\ell_3=\QQ_3[\sqrt{2},\sqrt{-5+2\sqrt{2}}]$ is
the degree~4 unramified extension of~$\QQ_3$.

Since we have calculated the Hasse invariants at all the other places,
and since those sum to zero, the Hasse invariant at the 3-adic place
must also be zero.  In ~donald/SU21/C10p2/padic.gap there is an
explicit calculation showing that~$\sigma^3\in\ell_3$ is a norm
from~$m_3$.


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 
                ((1+\sqrt{2})+\sqrt{-5+2\sqrt{2}))/(2\sqrt{2})   0   0


            0        1  0
      =     0        0  1         
         U/\sqrt{2}  0  0

For COMPLEX calculation:

   \sqrt{2}  = 1.414213562373095048801688724209

   \sqrt{-5+2\sqrt{2}} = 1.4736257582079005931550091 i
   \sqrt{-5-2\sqrt{2}} = 4.3020528829540906907583866 i
   U  = (1+\sqrt{2}+\sqrt{-5+2\sqrt{2}})/2
      =  1.2071067811865475244008444 + 0.7368128791039502965775046 i 
   U' = (1-\sqrt{2}+\sqrt{-5-2\sqrt{2}})/2
      = -0.2071067811865475244008444 + 1.3989663259659067020315405 i
   U/\sqrt{2} = ((1+\sqrt{2})+\sqrt{-5+2\sqrt{2}))/(2\sqrt{2})
     = 0.8535533905932737622004222 + 0.5210053832799870771404151 i
   U'/(-\sqrt{2}) = ((1-\sqrt{2})+\sqrt{-5-2\sqrt{2}))/(-2\sqrt{2})
     = 0.1464466094067262377995778 - 0.9892185757421227086489334 i

   W         =  1.5320888862379560704047853
   \phi(W)   = -1.8793852415718167681082186
   \phi^2(W) =  0.3472963553338606977034332

PRELIMINARY INVOLUTION of the second kind:

  \overline{\sqrt{2}}            = \sqrt{2}
  \overline{\sqrt{-5+2\sqrt{2}}} = -\sqrt{-5+2\sqrt{2}}
  \bar W                         = W

  \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         = -\sqrt{2}+(1-\sqrt{2})W+W^2
  \phi(T)   = (4-3\sqrt{2})+\sqrt{2}W+(-1+\sqrt{2})W^2
  \phi^2(T) = (2+\sqrt{2})-W-\sqrt{2}W^2

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}
    = (-\sqrt{2}+(-3+\sqrt{2})W+(1-\sqrt{2})W^2) \sigma^{-1}

  \iota(\sigma^{-1}) = T^-1 \sigma T = \phi(T)/T \sigma
     = ((1-2\sqrt{2})+(1+2\sqrt{2})W+(1+\sqrt{2})W^2) \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:

Recall that

  U = ((1+\sqrt{2}) + \sqrt{-5+2\sqrt{2}}) / 2

We want elements of the form

  \sum_{i=0,1} \sum_{j=0,1} \sum_{k=0,1,2} \sum_{l=-1,0,1}
     c_{ijkl} \sqrt{2}^i U^j W^k \sigma^l 

where the coefficients 

  c_{ijkl}\in\QQ

satisfy integrality conditions as follows:

  (.) For any prime~$v$ except~$2$, $3$, or~$17$ the $c_{ijkl}$ 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~$c_{ijl}$ by~$256$.

  (.) At the primes~$3$ and~$17$ one must use special conditions, as
  calculated in ~/donald/SU21/C10p2/padic.gap.

  (.) For the prime~$3$, one always uses a Type~1, hyperspecial
  condition.

  (.) $\ell$~has two $17$-adic places, denoted $17\pm$.  At~$17+$ we
  always wish to use a Type~1, hyperspecial condition.  If we wish to
  use a Type~1 condition for~$17-$ too, then the $17$-adic condition
  on the $c_{ijkl}$ is the simple one --- no factors of~$17$ should
  appear in the denominators.

  (.) If, on the other hand, we wish to use a Type~2 condition
  for~$17-$, then we must use a pair of condition matrices.  Each
  matrix of the pair is $18\times 36$.  The first matrix corresponds
  to the Type~1 condition at $17+$ and the second to the Type~2
  condition at~$17-$.

The Type~1 conditions (correctly called the hyperspecial conditions
for~$3$ and for $17+$) amount to requiring that an element
of~$U(\D,\iota)$ fix a certain lattice in~$\ell_3^3
or~$\ell_{17\pm}^3$, a lattice which is self-dual relative to the
sesquilinear form associated to~$\iota$.

The Type~2 conditions amount to requiring that an element
of~$U(\D,\iota)$ fix a certain pair of mutually dual lattices
in~$\ell_{17-}^3$, a pair which is also adjacent in the building
of~$PGL(3,\ell_{17-})$.

DETERMINANTS:

Elements of $U(\D,\iota)$ can have determinant which is any power
of~$U/\sqrt{2} = ((1+\sqrt{2})+\sqrt{-5+2\sqrt{2}})/(2\sqrt{2})$.
In~$PU$, one can always choose a representative where the power is~$0$
or~$\pm 1$.