def completions(self, p, M):
r"""
If `K` is the base_ring of self, this function takes all maps
`K-->Q_p` and applies them to self return a list of
(modular symbol,map: `K-->Q_p`) as map varies over all such maps.
.. NOTE::
This only returns all completions when `p` splits completely in `K`
INPUT:
- ``p`` -- prime
- ``M`` -- precision
OUTPUT:
- A list of tuples (modular symbol,map: `K-->Q_p`) as map varies over all such maps
EXAMPLES::
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
sage: D = ModularSymbols(67,2,1).cuspidal_submodule().new_subspace().decomposition()[1]
sage: f = ps_modsym_from_simple_modsym_space(D)
sage: f.completions(41,10)
[(Modular symbol with values in Sym^0 Q_41^2, Ring morphism:
From: Number Field in alpha with defining polynomial x^2 + 3*x + 1
To: 41-adic Field with capped relative precision 10
Defn: alpha |--> 5 + 22*41 + 19*41^2 + 10*41^3 + 28*41^4 + 22*41^5 + 9*41^6 + 25*41^7 + 40*41^8 + 8*41^9 + O(41^10)), (Modular symbol with values in Sym^0 Q_41^2, Ring morphism:
From: Number Field in alpha with defining polynomial x^2 + 3*x + 1
To: 41-adic Field with capped relative precision 10
Defn: alpha |--> 33 + 18*41 + 21*41^2 + 30*41^3 + 12*41^4 + 18*41^5 + 31*41^6 + 15*41^7 + 32*41^9 + O(41^10))]
"""
K = self.base_ring()
f = K.defining_polynomial()
R = Qp(p,M+10)['x']
x = R.gen()
v = R(f).roots()
if len(v) == 0:
L = Qp(p,M).extension(f,names='a')
a = L.gen()
V = self.parent().change_ring(L)
Dist = V.coefficient_module()
psi = K.hom([K.gen()],L)
embedded_sym = self.__class__(self._map.apply(psi,codomain=Dist, to_moments=True),V, construct=True)
ans = [embedded_sym,psi]
return ans
#raise ValueError, "No coercion possible -- no prime over p has degree 1"
else:
roots = [r[0] for r in v]
ans = []
V = self.parent().change_ring(Qp(p, M))
Dist = V.coefficient_module()
for r in roots:
psi = K.hom([r],Qp(p,M))
embedded_sym = self.__class__(self._map.apply(psi, codomain=Dist, to_moments=True), V, construct=True)
ans.append((embedded_sym,psi))
return ans
