def get_embedding(self,prec):
r"""
Returns an embedding of the quaternion algebra
into the algebra of 2x2 matrices with coefficients in `\QQ_p`.
INPUT:
- prec -- Integer. The precision of the splitting.
"""
if self.F == QQ and self.discriminant == 1:
R = Qp(self.p,prec)
self._F_to_local = QQ.hom([R(1)])
def iota(q):
return q.change_ring(R)
self._prec = prec
else:
I,J,K = self.local_splitting(prec)
mats = [1,I,J,K]
def iota(q):
R=I.parent()
try:
q = q.coefficient_tuple()
except AttributeError:
q = q.list()
return sum(self._F_to_local(a)*b for a,b in zip(q,mats))
return iota
def get_embedding(self, prec):
r"""
Returns an embedding of the quaternion algebra
into the algebra of 2x2 matrices with coefficients in `\QQ_p`.
INPUT:
- prec -- Integer. The precision of the splitting.
"""
if prec == -1:
prec = self._prec
if self.F == QQ and self.discriminant == 1:
R = Qp(self.p, prec)
self._F_to_local = QQ.hom([R(1)])
def iota(q):
return q.change_ring(R)
if prec > self._prec: # DEBUG
self._prec = prec
else:
I, J, K = self.local_splitting(prec)
mats = [1, I, J, K]
def iota(q):
R = I.parent()
try:
q = q.coefficient_tuple()
except AttributeError:
q = q.list()
return sum(self._F_to_local(a) * b for a, b in zip(q, mats))
return iota
def _compute_padic_splitting(self, prec):
verbose('Entering compute_padic_splitting')
prime = self.p
if self.seed is not None:
self.magma.eval('SetSeed(%s)' % self.seed)
R = Qp(prime, prec + 10) #Zmod(prime**prec) #
B_magma = self.Gpn._get_B_magma()
a, b = self.Gpn.B.invariants()
if self._hardcode_matrices:
self._II = matrix(R, 2, 2, [1, 0, 0, -1])
self._JJ = matrix(R, 2, 2, [0, 1, 1, 0])
goodroot = self.F.gen().minpoly().change_ring(R).roots()[0][0]
self._F_to_local = self.F.hom([goodroot])
else:
verbose('Calling magma pMatrixRing')
if self.F == QQ:
_, f = self.magma.pMatrixRing(self.Gpn._Omax_magma,
prime *
self.Gpn._Omax_magma.BaseRing(),
Precision=20,
nvals=2)
self._F_to_local = QQ.hom([R(1)])
else:
_, f = self.magma.pMatrixRing(self.Gpn._Omax_magma,
sage_F_ideal_to_magma(
self.Gpn._F_magma,
self.ideal_p),
Precision=20,
nvals=2)
try:
self._goodroot = R(
f.Image(B_magma(
B_magma.BaseRing().gen(1))).Vector()[1]._sage_())
except SyntaxError:
raise SyntaxError(
"Magma has trouble finding local splitting")
self._F_to_local = None
for o, _ in self.F.gen().minpoly().change_ring(R).roots():
if (o - self._goodroot).valuation() > 5:
self._F_to_local = self.F.hom([o])
break
assert self._F_to_local is not None
verbose('Initializing II,JJ,KK')
v = f.Image(B_magma.gen(1)).Vector()
self._II = matrix(R, 2, 2, [v[i + 1]._sage_() for i in xrange(4)])
v = f.Image(B_magma.gen(2)).Vector()
self._JJ = matrix(R, 2, 2, [v[i + 1]._sage_() for i in xrange(4)])
v = f.Image(B_magma.gen(3)).Vector()
self._KK = matrix(R, 2, 2, [v[i + 1]._sage_() for i in xrange(4)])
self._II, self._JJ = lift_padic_splitting(self._F_to_local(a),
self._F_to_local(b),
self._II, self._JJ,
prime, prec)
self.Gn._F_to_local = self._F_to_local
if not self.use_shapiro():
self.Gpn._F_to_local = self._F_to_local
self._KK = self._II * self._JJ
self._prec = prec
return self._II, self._JJ, self._KK
def _compute_padic_splitting(self,prec):
verbose('Entering compute_padic_splitting')
prime = self.p
if self.seed is not None:
self.magma.eval('SetSeed(%s)'%self.seed)
R = Qp(prime,prec+10) #Zmod(prime**prec) #
B_magma = self.Gn._B_magma
a,b = self.Gn.B.invariants()
if self._matrix_group:
self._II = matrix(R,2,2,[1,0,0,-1])
self._JJ = matrix(R,2,2,[0,1,1,0])
goodroot = self.F.gen().minpoly().change_ring(R).roots()[0][0]
self._F_to_local = self.F.hom([goodroot])
else:
verbose('Calling magma pMatrixRing')
if self.F == QQ:
_,f = self.magma.pMatrixRing(self.Gn._O_magma,prime*self.Gn._O_magma.BaseRing(),Precision = 20,nvals = 2)
self._F_to_local = QQ.hom([R(1)])
else:
_,f = self.magma.pMatrixRing(self.Gn._O_magma,sage_F_ideal_to_magma(self.Gn._F_magma,self.ideal_p),Precision = 20,nvals = 2)
try:
self._goodroot = R(f.Image(B_magma(B_magma.BaseRing().gen(1))).Vector()[1]._sage_())
except SyntaxError:
raise SyntaxError("Magma has trouble finding local splitting")
self._F_to_local = None
for o,_ in self.F.gen().minpoly().change_ring(R).roots():
if (o - self._goodroot).valuation() > 5:
self._F_to_local = self.F.hom([o])
break
assert self._F_to_local is not None
verbose('Initializing II,JJ,KK')
v = f.Image(B_magma.gen(1)).Vector()
self._II = matrix(R,2,2,[v[i+1]._sage_() for i in xrange(4)])
v = f.Image(B_magma.gen(2)).Vector()
self._JJ = matrix(R,2,2,[v[i+1]._sage_() for i in xrange(4)])
v = f.Image(B_magma.gen(3)).Vector()
self._KK = matrix(R,2,2,[v[i+1]._sage_() for i in xrange(4)])
self._II , self._JJ = lift_padic_splitting(self._F_to_local(a),self._F_to_local(b),self._II,self._JJ,prime,prec)
self.Gn._F_to_local = self._F_to_local
if not self.use_shapiro():
self.Gpn._F_to_local = self._F_to_local
self._KK = self._II * self._JJ
self._prec = prec
return self._II, self._JJ, self._KK
