class pAutomorphicForms(Module):
Element=pAutomorphicForm
r"""
The module of (quaternionic) `p`-adic automorphic forms.
EXAMPLES:
::
AUTHORS:
- Cameron Franc (2012-02-20)
- Marc Masdeu (2012-02-20)
"""
def __init__(self,X,U,prec=None,t=None,R=None,overconvergent=False):
if(R is None):
if(not isinstance(U,Integer)):
self._R=U.base_ring()
else:
if(prec is None):
prec=100
self._R=Qp(X._p,prec)
else:
self._R=R
#U is a CoefficientModuleSpace
if(isinstance(U,Integer)):
if(t is None):
if(overconvergent):
t=prec-U+1
else:
t=0
self._U=OCVn(U-2,self._R,U-1+t)
else:
self._U=U
self._X=X
self._V=self._X.get_vertex_list()
self._E=self._X.get_edge_list()
self._prec=self._R.precision_cap()
self._n=self._U.weight()
Module.__init__(self,base=self._R)
self._populate_coercion_lists_()
def _repr_(self):
r"""
This returns the representation of self as a string.
EXAMPLES:
This example illustrates ...
::
"""
s='Space of automorphic forms on '+str(self._X)+' with values in '+str(self._U)
return s
def _coerce_map_from_(self, S):
r"""
Can coerce from other HarmonicCocycles or from pAutomorphicForms
"""
if(isinstance(S,HarmonicCocycles)):
if(S._k-2!=self._n):
return False
if(S._X!=self._X):
return False
return True
if(isinstance(S,pAutomorphicForms)):
if(S._n!=self._n):
return False
if(S._X!=self._X):
return False
return True
return False
def _element_constructor_(self,x):
r"""
"""
#Code how to coherce x into the space
#Admissible values of x?
if isinstance(x,(HarmonicCocycleElement,pAutomorphicForm)):
return pAutomorphicForm(self,x)
def _an_element_(self):
r"""
Returns an element of the module.
"""
return pAutomorphicForm(self,1)
def embed_quaternion(self,g):
r"""
Returns the image of ``g`` under the embedding
of the quaternion algebra into 2x2 matrices.
"""
return self._X.embed_quaternion(g,prec = self._prec)
def lift(self,f, verbose = True):
r"""
Lifts the harmonic cocycle ``f`` to an
overconvegent automorphic form, thus computing
all the moments.
"""
F=self.element_class(self,f)
F.improve(verbose = verbose)
return F
def _apply_Up_operator(self,f,scale=False, fix_lowdeg_terms = True):
r"""
Apply the Up operator to ``f``.
EXAMPLES:
::
sage: X = BTQuotient(3,11)
sage: M = HarmonicCocycles(X,4,30)
sage: A = pAutomorphicForms(X,4,10, overconvergent = True)
sage: F = A.lift(M.basis()[0], verbose = False); F
p-adic automorphic form on Space of automorphic forms on Quotient of the Bruhat T**s tree of GL_2(QQ_3) with discriminant 11 and level 1 with values in Overconvergent coefficient module of weight n=2 over the ring 3-adic Field with capped relative precision 10 and depth 10:
e | c(e)
0 | 3^2 + O(3^12) + O(3^32)*z + O(3^26)*z^2 + (2*3^2 + 3^3 + 2*3^5 + 3^7 + 3^8 + 2*3^9 + O(3^10))*z^3 + (2*3^5 + 2*3^6 + 2*3^7 + 3^9 + O(3^10))*z^4 + (3^2 + 3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 3^7 + 2*3^8 + 3^9 + O(3^10))*z^5 + (3^2 + 2*3^3 + 3^4 + 2*3^6 + O(3^10))*z^6 + (2*3^3 + 3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 3^8 + 3^9 + O(3^10))*z^7 + (3^2 + 3^3 + 2*3^6 + 3^7 + 3^8 + 3^9 + O(3^10))*z^8 + (2*3^2 + 2*3^3 + 2*3^5 + 2*3^7 + 3^8 + 2*3^9 + O(3^10))*z^9
1 | 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^11 + O(3^12) + (3^2 + O(3^12))*z + (2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^11 + O(3^12))*z^2 + (2*3^2 + 2*3^3 + 3^4 + 2*3^5 + 3^6 + 2*3^7 + 2*3^8 + O(3^10))*z^3 + (2*3^3 + 3^5 + 3^7 + 3^8 + O(3^10))*z^4 + (2*3^3 + 3^6 + 3^7 + 3^9 + O(3^10))*z^5 + (3^3 + 2*3^4 + 2*3^5 + 2*3^7 + 3^8 + 3^9 + O(3^10))*z^6 + (3^7 + 2*3^8 + 2*3^9 + O(3^10))*z^7 + (3^3 + 2*3^4 + 3^7 + O(3^10))*z^8 + (2*3^2 + 3^4 + 3^6 + 2*3^7 + 3^8 + 2*3^9 + O(3^10))*z^9
2 | 3^2 + 2*3^3 + 2*3^6 + 3^7 + 2*3^8 + O(3^12) + (3 + 2*3^2 + 2*3^3 + 3^5 + 2*3^6 + 3^7 + 3^10 + O(3^11))*z + (2*3 + 2*3^2 + 3^4 + 2*3^5 + 2*3^6 + 2*3^8 + 3^10 + O(3^11))*z^2 + (2*3 + 3^2 + 2*3^7 + 3^9 + O(3^10))*z^3 + (3 + 2*3^2 + 2*3^4 + 3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + O(3^10))*z^4 + (3 + 3^2 + 3^4 + 2*3^9 + O(3^10))*z^5 + (3^3 + 2*3^5 + 3^6 + 3^8 + 2*3^9 + O(3^10))*z^6 + (3^5 + 2*3^7 + 3^9 + O(3^10))*z^7 + (2*3 + 3^3 + 3^4 + 2*3^6 + O(3^10))*z^8 + (2*3 + 2*3^3 + 2*3^4 + 2*3^6 + O(3^10))*z^9
3 | 3^2 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^11 + O(3^12) + (3^3 + 2*3^4 + 2*3^8 + O(3^13))*z + (3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^6 + 3^7 + 2*3^8 + 2*3^9 + 2*3^10 + O(3^11))*z^2 + (3^2 + 2*3^3 + 3^4 + 3^7 + 3^8 + 2*3^9 + O(3^10))*z^3 + (3 + 2*3^2 + 2*3^3 + 3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 3^8 + O(3^10))*z^4 + (3 + 3^3 + 3^4 + 2*3^5 + 2*3^6 + 3^7 + 2*3^8 + 2*3^9 + O(3^10))*z^5 + (3 + 3^4 + 3^5 + 3^6 + 2*3^7 + O(3^10))*z^6 + (2*3 + 3^2 + 2*3^3 + 3^4 + 2*3^6 + 3^8 + 3^9 + O(3^10))*z^7 + (3 + 3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 3^7 + 2*3^9 + O(3^10))*z^8 + (2*3^2 + 3^4 + 3^5 + 3^8 + 3^9 + O(3^10))*z^9
"""
HeckeData=self._X._get_Up_data()
if scale == False:
factor=(self._X._p)**(self._U.weight()/2)
else:
factor=1
Tf=[]
for jj in range(2*len(self._E)):
tmp=self._U(0)
for d in HeckeData:
gg=d[0] # acter
u=d[1][jj] # edge_list[jj]
r=self._X._p**(-(u.power))*(u.t()*gg)
tmp+=f._F[u.label+u.parity*len(self._E)].r_act_by(r)
tmp *= factor
for ii in range(self._n+1):
tmp[ii] = f._F[jj][ii]
Tf.append(tmp)
return pAutomorphicForm(self,Tf,quick=True)
class pAutomorphicForms(Module):
Element = pAutomorphicFormElement
r"""
The module of (quaternionic) `p`-adic automorphic forms.
EXAMPLES::
AUTHORS:
- Cameron Franc (2012-02-20)
- Marc Masdeu (2012-02-20)
"""
def __init__(self,domain,U,prec = None,t = None,R = None,overconvergent = False):
if(R is None):
if not isinstance(U,Integer):
self._R = U.base_ring()
else:
if prec is None:
prec = 20
self._R = Qp(domain._p,prec)
else:
self._R = R
#U is a CoefficientModuleSpace
if isinstance(U,Integer):
if t is None:
if overconvergent:
t = prec-U+1
else:
t = 0
self._U = OCVn(U-2,self._R,U-1+t)
else:
self._U = U
self._source = domain
self._list = self._source.get_list() # Contains also the opposite edges
self._prec = self._R.precision_cap()
self._n = self._U.weight()
self._p = self._source._p
Module.__init__(self,base = self._R)
self._populate_coercion_lists_()
def prime(self):
return self._p
def _repr_(self):
r"""
This returns the representation of self as a string.
EXAMPLES::
"""
s = 'Space of automorphic forms on '+str(self._source)+' with values in '+str(self._U)
return s
def _coerce_map_from_(self, S):
r"""
Can coerce from other HarmonicCocycles or from pAutomorphicForms
"""
if isinstance(S,HarmonicCocycles):
if S.weight()-2 != self._n:
return False
if S._source != self._source:
return False
return True
if isinstance(S,pAutomorphicForms):
if S._n != self._n:
return False
if S._source != self._source:
return False
return True
return False
def _element_constructor_(self,x):
r"""
"""
#Code how to coherce x into the space
#Admissible values of x?
if isinstance(x,(HarmonicCocycleElement,pAutomorphicFormElement)):
return pAutomorphicFormElement(self,x)
def _an_element_(self):
r"""
Returns an element of the module.
"""
return pAutomorphicFormElement(self,1)
def lift(self,f, verbose = True):
r"""
Lifts the harmonic cocycle ``f`` to an
overconvegent automorphic form, thus computing
all the moments.
"""
F = self.element_class(self,f)
F.improve(verbose = verbose)
return F
def _make_invariant(self, F):
r"""
EXAMPLES::
"""
S = self._source.get_stabilizers()
M = [e.rep for e in self._list]
coeff_module_class = self._U.element_class
newF = []
for ii in range(len(S)):
Si = S[ii]
x = coeff_module_class(F[ii].parent(),F[ii]._val,quick = True)
if(any([v[2] for v in Si])):
newFi = coeff_module_class(F[ii].parent(),0)
s = QQ(0)
m = M[ii]
for v in Si:
s += 1
# self._source should has a embed method that, given stabilizers,
# returns 2x2 matrices that can act on the distributions.
newFi += x.r_act_by(m.adjoint()*self._source.embed(v[0],prec = self._prec)*m)
newF.append((1/s)*newFi)
else:
newF.append(x)
return newF
def _apply_Up_operator(self,f,scale = False):
r"""
Apply the Up operator to ``f``.
EXAMPLES::
sage: X = BTQuotient(3,11)
sage: M = HarmonicCocycles(X,4,20)
sage: A = pAutomorphicForms(X,4,10, overconvergent = True)
sage: F = A.lift(M.basis()[0], verbose = False)
sage: print F.values()
[3^2 + O(3^12) + O(3^22)*z + O(3^16)*z^2 + (2*3^2 + O(3^10))*z^3 + (3^4 + O(3^10))*z^4 + (3^2 + 3^3 + O(3^10))*z^5 + (3^2 + 2*3^3 + 2*3^4 + O(3^10))*z^6 + (2*3^3 + O(3^10))*z^7 + (3^2 + 3^3 + 2*3^4 + 3^5 + O(3^10))*z^8 + (2*3^2 + 2*3^3 + O(3^10))*z^9, 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^11 + O(3^12) + (3^2 + O(3^12))*z + (2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^11 + O(3^12))*z^2 + (2*3^2 + 3^4 + 3^6 + 3^8 + O(3^10))*z^3 + (3^4 + 2*3^5 + 2*3^6 + 3^8 + 2*3^9 + O(3^10))*z^4 + (3^3 + 2*3^4 + 2*3^7 + 3^8 + O(3^10))*z^5 + (2*3^4 + 2*3^6 + 3^7 + 3^8 + 2*3^9 + O(3^10))*z^6 + (3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 3^9 + O(3^10))*z^7 + (2*3^4 + 3^7 + O(3^10))*z^8 + (2*3^2 + 3^6 + 2*3^7 + 2*3^8 + O(3^10))*z^9, 3^2 + 2*3^3 + 2*3^6 + 3^7 + 2*3^8 + O(3^12) + (3 + 2*3^2 + 2*3^3 + 3^5 + 2*3^6 + 3^7 + 3^10 + O(3^11))*z + (2*3 + 2*3^2 + 3^4 + 2*3^5 + 2*3^6 + 2*3^8 + 3^10 + O(3^11))*z^2 + (2*3 + 3^2 + 3^4 + 3^5 + 2*3^7 + 2*3^8 + O(3^10))*z^3 + (3 + 2*3^2 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + O(3^10))*z^4 + (3 + 3^2 + 3^4 + 2*3^5 + 2*3^6 + 3^7 + 2*3^8 + 3^9 + O(3^10))*z^5 + (3^3 + 3^4 + 3^7 + O(3^10))*z^6 + (2*3^4 + 3^5 + 3^6 + 2*3^9 + O(3^10))*z^7 + (2*3 + 3^3 + 3^5 + 3^6 + 3^7 + 3^8 + 3^9 + O(3^10))*z^8 + (2*3 + 2*3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + O(3^10))*z^9, 3^2 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^11 + O(3^12) + (3^3 + 2*3^4 + 2*3^8 + O(3^12))*z + (3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^6 + 3^7 + 2*3^8 + 2*3^9 + O(3^10))*z^2 + (3^2 + 2*3^3 + 3^4 + 3^5 + 2*3^8 + 3^9 + O(3^10))*z^3 + (3 + 2*3^2 + 2*3^3 + 3^4 + 2*3^5 + 2*3^9 + O(3^10))*z^4 + (3 + 3^3 + 2*3^4 + 2*3^5 + 2*3^7 + 3^8 + 3^9 + O(3^10))*z^5 + (3 + 3^4 + 3^5 + 2*3^6 + 3^7 + 2*3^8 + 3^9 + O(3^10))*z^6 + (2*3 + 3^2 + 2*3^3 + 2*3^5 + 3^7 + 3^8 + 3^9 + O(3^10))*z^7 + (3 + 3^3 + 2*3^4 + 2*3^5 + 3^6 + 3^7 + 3^9 + O(3^10))*z^8 + (2*3^2 + 3^4 + 2*3^5 + 2*3^6 + 3^7 + 2*3^8 + O(3^10))*z^9]
"""
HeckeData = self._source._get_Up_data()
if scale == False:
factor = (self._p)**(self._n/2)
else:
factor = 1
Tf = []
for jj in range(len(self._list)):
tmp = self._U(0)
for d in HeckeData:
gg = d[0] # acter
u = d[1][jj] # edge_list[jj]
r = self._p**(-u.power) * (u.t(2*self._prec+1)*gg)
tmp += f._value[u.label].r_act_by(r)
tmp *= factor
for ii in range(self._n+1):
tmp[ii] = f._value[jj][ii]
Tf.append(tmp)
return pAutomorphicFormElement(self,Tf,quick = True)