def coleman(self,t1,t2,E=None,method='moments',mult=False,delta=-1,level=0):
r"""
If ``self`` is a `p`-adic automorphic form that corresponds to a rigid modular form,
then this computes the coleman integral of this form between two points on
the boundary `\PP^1(\QQ_p)` of the `p`-adic upper half plane.
INPUT:
- ``t1``, ``t2`` - elements of `\PP^1(\QQ_p)` (the endpoints of integration)
- ``E`` - (Default: None). If specified, will not compute the covering adapted
to ``t1`` and ``t2`` and instead use the given one. In that case, ``E``
should be a list of matrices corresponding to edges describing the open
balls to be considered.
- ``method`` - string (Default: 'moments'). Tells which algorithm to use
(alternative is 'riemann_sum', which is unsuitable for computations
requiring high precision)
- ``mult`` - boolean (Default: False). Whether to use the multiplicative version.
- ``delta`` - integer (Default: -1)
- ``level`` - integer (Default: 0)
OUTPUT:
The result of the coleman integral
EXAMPLES:
This example illustrates ...
::
TESTS:
::
sage: p = 7
sage: lev = 2
sage: prec = 20
sage: X = BTQuotient(p,lev, use_magma = True) # optional - magma
sage: k = 2 # optional - magma
sage: M = HarmonicCocycles(X,k,prec) # optional - magma
sage: B = M.basis() # optional - magma
sage: f = 3*B[0] # optional - magma
sage: MM = pAutomorphicForms(X,k,prec,overconvergent=True) # optional - magma
sage: D = -11 # optional - magma
sage: X.is_admissible(D) # optional - magma
True
sage: K.<a> = QuadraticField(D) # optional - magma
sage: CM = X.get_CM_points(D,prec=prec) # optional - magma
sage: Kp = CM[0].parent() # optional - magma
sage: P=CM[0] # optional - magma
sage: Q=P.trace()-P # optional - magma
sage: F=MM.lift(f, verbose = False) # long time optional - magma
sage: J0=F.coleman(P,Q,mult=True) # long time optional - magma
sage: E=EllipticCurve([1,0,1,4,-6]) # optional - magma
sage: T=E.tate_curve(p) # optional - magma
sage: xx,yy=getcoords(T,J0,prec) # long time optional -magma
sage: P = E.base_extend(K).lift_x(algdep(xx,1).roots(QQ)[0][0]); P # long time optional - magma
(7/11 : 58/121*a - 9/11 : 1)
sage: p = 13 # optional - magma
sage: lev = 2 # optional - magma
sage: prec = 20 # optional - magma
sage: Y = BTQuotient(p,lev, use_magma = True) # optional - magma
sage: k = 2 # optional - magma
sage: M=HarmonicCocycles(Y,k,prec) # optional - magma
sage: B=M.basis() # optional - magma
sage: f = B[1] # optional - magma
sage: g = -4*B[0]+3*B[1] # optional - magma
sage: MM = pAutomorphicForms(Y,k,prec,overconvergent=True) # optional - magma
sage: D = -11 # optional - magma
sage: Y.is_admissible(D) # optional - magma
True
sage: K.<a> = QuadraticField(D) # optional - magma
sage: CM = Y.get_CM_points(D,prec=prec) # optional - magma
sage: Kp = parent(CM[0]) # optional - magma
sage: P = CM[0] # optional - magma
sage: Q = P.trace()-P # optional - magma
sage: F = MM.lift(f, verbose = False) # long time optional - magma
sage: J11 = F.coleman(P,Q,mult = True) # long time optional - magma
sage: E = EllipticCurve('26a2') # optional - magma
sage: T = E.tate_curve(p) # optional - magma
sage: xx,yy = getcoords(T,J11,prec) # long time optional - magma
sage: HP = E.base_extend(K).lift_x(algdep(xx,1).roots(QQ)[0][0]); HP # long time optional - magma
(-137/11 : 2/121*a + 63/11 : 1)
AUTHORS:
- Cameron Franc (2012-02-20)
- Marc Masdeu (2012-02-20)
"""
if(mult and delta>=0):
raise NotImplementedError, "Need to figure out how to implement the multiplicative part."
p=self._parent._X._p
K=t1.parent()
R=PolynomialRing(K,'x')
x=R.gen()
R1=LaurentSeriesRing(K,'r1')
r1=R1.gen()
if(E is None):
E=self._parent._X._BT.find_covering(t1,t2)
# print 'Got %s open balls.'%len(E)
value=0
ii=0
value_exp=K(1)
if(method=='riemann_sum'):
R1.set_default_prec(self._parent._U.weight()+1)
for e in E:
ii+=1
b=e[0,1]
d=e[1,1]
y=(b-d*t1)/(b-d*t2)
poly=R1(y.log()) #R1(our_log(y))
c_e=self.evaluate(e)
new=c_e.evaluate(poly)
value+=new
if mult:
value_exp *= K.teichmuller(y)**Integer(c_e[0].rational_reconstruction())
elif(method=='moments'):
R1.set_default_prec(self._parent._U.dimension())
for e in E:
ii+=1
f=(x-t1)/(x-t2)
a,b,c,d=e.list()
y0=f(R1([b,a])/R1([d,c])) #f( (ax+b)/(cx+d) )
y0=p**(-y0(0).valuation())*y0
mu=K.teichmuller(y0(0))
y=y0/mu-1
poly=R1(0)
ypow=y
for jj in range(1,R1.default_prec()+10):
poly+=(-1)**(jj+1)*ypow/jj
ypow*=y
if(delta>=0):
poly*=((r1-t1)**delta*(r1-t2)**(self._parent._n-delta))
c_e=self.evaluate(e)
new=c_e.evaluate(poly)
value+=new
if mult:
value_exp *= K.teichmuller((b-d*t1)/(b-d*t2))**Integer(c_e[0].rational_reconstruction())
else:
print 'The available methods are either "moments" or "riemann_sum". The latter is only provided for consistency check, and should not be used in practice.'
return False
if mult:
return value_exp * value.exp()
return value
