def _prec_for_solve_diff_eqn(M, p):
r"""
A helper function for determining the (absolute) precision of the input
to solve_diff_eqn required in order obtain an answer with (absolute)
precision ``M``. The parameter ``p`` is the prime.
Given input (absolute) precision `M_\text{in}`, the output of
solve_diff_eqn has (absolute) precision
.. MATH::
M = M_\text{in} - \lfloor\log_p(M_\text{in})\rfloor - 1.
::EXAMPLES:
sage: from sage.modular.pollack_stevens.modsym_OMS_space import _prec_for_solve_diff_eqn
sage: [_prec_for_solve_diff_eqn(M, p) for p in [3,7,11,29] for M in [1,3,10,20]]
[2, 5, 13, 23, 2, 4, 12, 22, 2, 4, 12, 22, 2, 4, 11, 21]
"""
if M < 1:
raise ValueError("Desired precision M(=%s) must be at least 1." % (M))
# A good guess to begin:
M_in = ZZ(1 + M + ZZ(M).exact_log(p))
# It looks like usually there are no iterations
while M > M_in - M_in.exact_log(p) - 1:
M_in += 1
#print("An iteration in _prec_solve_diff_eqn")
return M_in
def dirichlet_coefficients(self, ncoeffs=20):
from sage.rings.power_series_ring import PowerSeriesRing
from sage.misc.misc import srange
#from sage.misc.mrange import mrange
ncoeffs = ZZ(ncoeffs)
ps = prime_range(ncoeffs + 1)
num_ps = len(ps)
exps = [ncoeffs.exact_log(p) + 1 for p in ps]
M = ncoeffs.exact_log(2) + 1
#print ps, exps, M
PS = PowerSeriesRing(QQ, 'x', M)
euler_factors = dict([[ps[i], ~PS(self.euler_factor(ps[i]), exps[i] + 1)] for i in range(num_ps)])
ans = []
for n in srange(1, ncoeffs + 1):
ans.append(prod([euler_factors[p][e] for p, e in n.factor()]))
#ans = [0] * ncoeffs
#is_zeroes = True
#for e in mrange(exps):
# if is_zeroes:
# is_zeroes = False
# ans[0] = QQ.one()
# continue
# n = 1
# an = QQ.one()
# breaker = False
# for i in range(num_ps):
# if exps[i] == 0:
# continue
# n *= ps[i] ** e[i]
# if n > ncoeffs:
# breaker = True
# break
# an *= euler_factors[i][e[i]]
# #print n, an, e
# if breaker:
# continue
# ans[n-1] = an
return ans
def solve_diff_eqn(self):
r"""
Solves the difference equation.
See Theorem 4.5 and Lemma 4.4 of [PS].
INPUT:
- ``self`` - an overconvergent distribution `\mu` of absolute
precision `M`
OUTPUT:
- an overconvergent distribution `\nu` of absolute precision
`M - \lfloor\log_p(M)\rfloor - 1` such that
.. math::
\nu|\Delta = \mu,\text{ where }\Delta=\begin{pmatrix}1&1\\0&1
\end{pmatrix} - 1.
EXAMPLES::
sage: D = OverconvergentDistributions(0,7,base=ZpCA(7,5))
sage: D10 = D.change_precision(10)
sage: mu10 = D10((O(7^10), 4 + 6*7 + 5*7^3 + 2*7^4 + 5*7^5 + O(7^9), 5 + 7^3 + 5*7^4 + 6*7^5 + 7^6 + 6*7^7 + O(7^8), 2 + 7 + 6*7^2 + 6*7^4 + 7^5 + 7^6 + O(7^7), 3*7 + 4*7^2 + 4*7^3 + 3*7^4 + 3*7^5 + O(7^6), 5 + 3*7 + 2*7^2 + 7^3 + 3*7^4 + O(7^5), 1 + 7^2 + 7^3 + O(7^4), 6*7 + 6*7^2 + O(7^3), 2 + 3*7 + O(7^2), 1 + O(7)))
sage: nu10 = mu10.solve_diff_eqn()
sage: MS = OverconvergentModularSymbols(14, coefficients=D)
sage: MR = MS.source()
sage: Id = MR.gens()[0]
sage: nu10 * MR.gammas[Id] - nu10 - mu10
7^8 * ()
sage: D = OverconvergentDistributions(0,7,base=Qp(7,5))
sage: D10 = D.change_precision(10)
sage: mu10 = D10((O(7^10), 4 + 6*7 + 5*7^3 + 2*7^4 + 5*7^5 + O(7^9), 5 + 7^3 + 5*7^4 + 6*7^5 + 7^6 + 6*7^7 + O(7^8), 2 + 7 + 6*7^2 + 6*7^4 + 7^5 + 7^6 + O(7^7), 3*7 + 4*7^2 + 4*7^3 + 3*7^4 + 3*7^5 + O(7^6), 5 + 3*7 + 2*7^2 + 7^3 + 3*7^4 + O(7^5), 1 + 7^2 + 7^3 + O(7^4), 6*7 + 6*7^2 + O(7^3), 2 + 3*7 + O(7^2), 1 + O(7)))
sage: nu10 = mu10.solve_diff_eqn()
sage: MS = OverconvergentModularSymbols(14, coefficients=D);
sage: MR = MS.source();
sage: Id = MR.gens()[0]
sage: nu10 * MR.gammas[Id] - nu10 - mu10
7^8 * ()
sage: R = ZpCA(5, 5); D = OverconvergentDistributions(0,base=R);
sage: nu = D((R(O(5^5)), R(5 + 3*5^2 + 4*5^3 + O(5^4)), R(5 + O(5^3)), R(2*5 + O(5^2), 2 + O(5))));
sage: nu.solve_diff_eqn()
5 * (1 + 3*5 + O(5^2), O(5))
Check input of relative precision 2::
sage: from sage.modular.pollack_stevens.coeffmod_OMS_element import CoeffMod_OMS_element
sage: R = ZpCA(3, 9)
sage: D = OverconvergentDistributions(0, base=R, prec_cap=4)
sage: V = D.approx_module(2)
sage: nu = CoeffMod_OMS_element(V([R(0, 9), R(2*3^2 + 2*3^4 + 2*3^7 + 3^8 + O(3^9))]), D, ordp=0, check=False)
sage: mu = nu.solve_diff_eqn()
sage: mu
3 * ()
"""
#RH: see tests.sage for randomized verification that this function works correctly
p = self.parent().prime()
abs_prec = ZZ(self.precision_absolute())
if self.is_zero():
mu = self.parent()(0)
mu.ordp = abs_prec - abs_prec.exact_log(p) - 1
return mu
if self._unscaled_moment(0) != 0:
raise ValueError("Distribution must have total measure 0 to be in image of difference operator.")
M = ZZ(len(self._moments))
## RP: This should never happen -- the distribution must be 0 at this point if M==1
if M == 1:
return self.parent()(0)
if M == 2:
if p == 2:
raise ValueError("Not enough accuracy to return anything")
else:
out_prec = abs_prec - abs_prec.exact_log(p) - 1
if self.ordp >= out_prec:
mu = self.parent()(0)
mu.ordp = out_prec
return mu
mu = self.parent()(self._unscaled_moment(1))
mu.ordp = self.ordp
return mu
R = self.parent().base_ring()
K = R.fraction_field()
bern = [bernoulli(i) for i in range(0,M-1,2)]
minhalf = ~K(-2) #bernoulli(1)
# bernoulli(1) = -1/2; the only nonzero odd bernoulli number
v = [minhalf * self.moment(m) for m in range(M-1)] #(m choose m-1) * B_1 * mu[m]/m
for m in range(1,M):
scalar = K(self.moment(m)) * (~K(m))
for j in range(m-1,M-1,2):
v[j] += binomial(j,m-1) * bern[(j-m+1)//2] * scalar
ordp = min(a.valuation() for a in v)
#Is this correct in ramified extensions of QQp?
verbose("abs_prec: %s, ordp: %s"%(abs_prec, ordp), level=2)
if ordp != 0:
new_M = abs_prec - 1 - (abs_prec).exact_log(p) - ordp
verbose("new_M: %s"%(new_M), level=2)
V = self.parent().approx_module(new_M)
v = V([R(v[i] >> ordp) for i in range(new_M)])
else:
new_M = abs_prec - abs_prec.exact_log(p) - 1
verbose("new_M: %s"%(new_M), level=2)
V = self.parent().approx_module(new_M)
v = V([R(v[i]) for i in range(new_M)])
v[new_M-1] = v[new_M-1].add_bigoh(1) #To force normalize to deal with this properly. May not be necessary any more.
mu = CoeffMod_OMS_element(v, self.parent(), ordp=ordp, check=False)
verbose("mu.ordp: %s, mu._moments: %s"%(mu.ordp, mu._moments), level=2)
return mu.normalize()