Coverage for local/lib/python2.7/site-packages/sage/modular/pollack_stevens/padic_lseries.py : 78%
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# -*- coding: utf-8 -*- `p`-adic `L`-series attached to overconvergent eigensymbols
An overconvergent eigensymbol gives rise to a `p`-adic `L`-series, which is essentially defined as the evaluation of the eigensymbol at the path `0 \rightarrow \infty`. The resulting distribution on `\ZZ_p` can be restricted to `\ZZ_p^\times`, thus giving the measure attached to the sought `p`-adic `L`-series. All this is carefully explained in [PS]_.
""" #***************************************************************************** # Copyright (C) 2012 Robert Pollack <rpollack@math.bu.edu> # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
r""" The `p`-adic `L`-series associated to an overconvergent eigensymbol.
INPUT:
- ``symb`` -- an overconvergent eigensymbol - ``gamma`` -- topological generator of `1 + p\ZZ_p` (default: `1+p` or 5 if `p=2`) - ``quadratic_twist`` -- conductor of quadratic twist `\chi` (default: 1) - ``precision`` -- if None (default) is specified, the correct precision bound is computed and the answer is returned modulo that accuracy
EXAMPLES::
sage: E = EllipticCurve('37a') sage: p = 5 sage: prec = 4 sage: L = E.padic_lseries(p, implementation="pollackstevens", precision=prec) # long time sage: L[1] # long time 1 + 4*5 + 2*5^2 + O(5^3) sage: L.series(3) # long time O(5^4) + (1 + 4*5 + 2*5^2 + O(5^3))*T + (3 + O(5^2))*T^2 + O(T^3)
::
sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: E = EllipticCurve('20a') sage: phi = E.pollack_stevens_modular_symbol() sage: Phi = phi.p_stabilize_and_lift(3, 4) # long time sage: L = pAdicLseries(Phi) # long time sage: L.series(4) # long time 2*3 + O(3^4) + (3 + O(3^2))*T + (2 + O(3))*T^2 + O(3^0)*T^3 + O(T^4)
An example of a `p`-adic `L`-series associated to a modular abelian surface. This is not tested as it takes too long.::
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: A = ModularSymbols(103,2,1).cuspidal_submodule().new_subspace().decomposition()[0] sage: p = 19 sage: prec = 4 sage: phi = ps_modsym_from_simple_modsym_space(A) sage: ap = phi.Tq_eigenvalue(p,prec) sage: c1,c2 = phi.completions(p,prec) sage: phi1,psi1 = c1 sage: phi2,psi2 = c2 sage: phi1p = phi1.p_stabilize_and_lift(p,ap = psi1(ap), M = prec) # not tested - too long sage: L1 = pAdicLseries(phi1p) # not tested - too long sage: phi2p = phi2.p_stabilize_and_lift(p,ap = psi2(ap), M = prec) # not tested - too long sage: L2 = pAdicLseries(phi2p) # not tested - too long sage: L1[1]*L2[1] # not tested - too long 13 + 9*19 + 18*19^2 + O(19^3) """
r""" Initialize the class
EXAMPLES::
sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: E = EllipticCurve('11a3') sage: phi = E.pollack_stevens_modular_symbol() sage: p = 11 sage: prec = 3 sage: Phi = phi.lift(p, prec,eigensymbol=True) # long time sage: L = pAdicLseries(Phi) # long time sage: L.series(3) # long time O(11^3) + (2 + 5*11 + O(11^2))*T + (10 + O(11))*T^2 + O(T^3)
sage: TestSuite(L).run() # long time """
raise ValueError("Not a p-adic overconvergent modular symbol.")
gamma = 1 + 4 else:
r""" Return the `n`-th coefficient of the `p`-adic `L`-series
EXAMPLES::
sage: E = EllipticCurve('14a5') sage: L = E.padic_lseries(7,implementation="pollackstevens",precision=5) # long time sage: L[0] # long time O(7^5) sage: L[1] # long time 5 + 5*7 + 2*7^2 + 2*7^3 + O(7^4) """
else: # ap = symb.Tq_eigenvalue(p)
else: for j in range(M, len(lb))])
* self._basic_integral(a, j) for a in range(1, p))
r""" Compare ``self`` and ``other``.
EXAMPLES::
sage: E = EllipticCurve('11a') sage: L = E.padic_lseries(11,implementation="pollackstevens",precision=6) # long time sage: L == loads(dumps(L)) # indirect doctest long time True """ if not isinstance(other, pAdicLseries): return False
return (self._symb == other._symb and self._quadratic_twist == other._quadratic_twist and self._gamma == other._gamma and self._precision == other._precision)
r""" Compare ``self`` and ``other``.
EXAMPLES::
sage: E = EllipticCurve('11a') sage: L = E.padic_lseries(11,implementation="pollackstevens",precision=6) # long time sage: L != L # long time False """ return not self.__eq__(other)
r""" Return the overconvergent modular symbol
EXAMPLES::
sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: E = EllipticCurve('21a4') sage: phi = E.pollack_stevens_modular_symbol() sage: Phi = phi.p_stabilize_and_lift(2,5) # long time sage: L = pAdicLseries(Phi) # long time sage: L.symbol() # long time Modular symbol of level 42 with values in Space of 2-adic distributions with k=0 action and precision cap 15 sage: L.symbol() is Phi # long time True """
r""" Return the prime `p` as in `p`-adic `L`-series.
EXAMPLES::
sage: E = EllipticCurve('19a') sage: L = E.padic_lseries(19, implementation="pollackstevens",precision=6) # long time sage: L.prime() # long time 19 """
r""" Return the discriminant of the quadratic twist
EXAMPLES::
sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: E = EllipticCurve('37a') sage: phi = E.pollack_stevens_modular_symbol() sage: Phi = phi.lift(37,4) sage: L = pAdicLseries(Phi, quadratic_twist=-3) sage: L.quadratic_twist() -3 """
r""" Return the string representation
EXAMPLES::
sage: E = EllipticCurve('14a2') sage: L = E.padic_lseries(3, implementation="pollackstevens", precision=4) # long time sage: L._repr_() # long time '3-adic L-series of Modular symbol of level 42 with values in Space of 3-adic distributions with k=0 action and precision cap 8' """ return "%s-adic L-series of %s" % (self.prime(), self.symbol())
r""" Return the ``prec``-th approximation to the `p`-adic `L`-series associated to self, as a power series in `T` (corresponding to `\gamma-1` with `\gamma` the chosen generator of `1+p\ZZ_p`).
INPUT:
- ``prec`` -- (default 5) is the precision of the power series
EXAMPLES::
sage: E = EllipticCurve('14a2') sage: p = 3 sage: prec = 6 sage: L = E.padic_lseries(p,implementation="pollackstevens",precision=prec) # long time sage: L.series(4) # long time 2*3 + 3^4 + 3^5 + O(3^6) + (2*3 + 3^2 + O(3^4))*T + (2*3 + O(3^2))*T^2 + (3 + O(3^2))*T^3 + O(T^4)
sage: E = EllipticCurve("15a3") sage: L = E.padic_lseries(5,implementation="pollackstevens",precision=15) # long time sage: L.series(3) # long time O(5^15) + (2 + 4*5^2 + 3*5^3 + 5^5 + 2*5^6 + 3*5^7 + 3*5^8 + 2*5^9 + 2*5^10 + 3*5^11 + 5^12 + O(5^13))*T + (4*5 + 4*5^3 + 3*5^4 + 4*5^5 + 3*5^6 + 2*5^7 + 5^8 + 4*5^9 + 3*5^10 + O(5^11))*T^2 + O(T^3)
sage: E = EllipticCurve("79a1") sage: L = E.padic_lseries(2,implementation="pollackstevens",precision=10) # not tested sage: L.series(4) # not tested O(2^9) + (2^3 + O(2^4))*T + O(2^0)*T^2 + (O(2^-3))*T^3 + O(T^4) """
r""" Return the interpolation factor associated to self. This is the `p`-adic multiplier that which appears in the interpolation formula of the `p`-adic `L`-function. It has the form `(1-\alpha_p^{-1})^2`, where `\alpha_p` is the unit root of `X^2 - \psi(a_p) \chi(p) X + p`.
INPUT:
- ``ap`` -- the eigenvalue of the Up operator
- ``chip`` -- the value of the nebentype at p (default: 1)
- ``psi`` -- a twisting character (default: None)
OUTPUT: a `p`-adic number
EXAMPLES::
sage: E = EllipticCurve('19a2') sage: L = E.padic_lseries(3,implementation="pollackstevens",precision=6) # long time sage: ap = E.ap(3) # long time sage: L.interpolation_factor(ap) # long time 3^2 + 3^3 + 2*3^5 + 2*3^6 + O(3^7)
Comparing against a different implementation::
sage: L = E.padic_lseries(3) sage: (1-1/L.alpha(prec=4))^2 3^2 + 3^3 + O(3^5) """ M = self.symbol().precision_relative() p = self.prime() if p == 2: R = pAdicField(2, M + 1) else: R = pAdicField(p, M) if psi is not None: ap = psi(ap) ap = ap * chip sdisc = R(ap ** 2 - 4 * p).sqrt() v0 = (R(ap) + sdisc) / 2 v1 = (R(ap) - sdisc) / 2 if v0.valuation() > 0: v0, v1 = v1, v0 alpha = v0 return (1 - 1 / alpha) ** 2
r""" Return `\int_{a+pZ_p} (z-{a})^j d\Phi(0-infty)` -- see formula [Pollack-Stevens, sec 9.2]
INPUT:
- ``a`` -- integer in range(p) - ``j`` -- integer in range(self.symbol().precision_relative())
EXAMPLES::
sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: E = EllipticCurve('11a3') sage: L = E.padic_lseries(5, implementation="pollackstevens", precision=4) #long time sage: L._basic_integral(1,2) # long time 2*5^2 + 5^3 + O(5^4) """ raise PrecisionError("Too many moments requested") * ((a - ZZ(K.teichmuller(a))) ** (j - r)) * (p ** r) * symb_twisted.moment(r) for r in range(j + 1)) / ap )
r""" Return the list of coefficients in the power series expansion (up to precision `M`) of `\binom{\log_p(z)/\log_p(\gamma)}{n}`
INPUT:
- ``p`` -- prime - ``gamma`` -- topological generator, e.g. `1+p` - ``z`` -- variable - ``n`` -- nonnegative integer - ``M`` -- precision
OUTPUT:
The list of coefficients in the power series expansion of `\binom{\log_p(z)/\log_p(\gamma)}{n}`
EXAMPLES::
sage: R.<z> = QQ['z'] sage: from sage.modular.pollack_stevens.padic_lseries import log_gamma_binomial sage: log_gamma_binomial(5,1+5,z,2,4) [0, -3/205, 651/84050, -223/42025] sage: log_gamma_binomial(5,1+5,z,3,4) [0, 2/205, -223/42025, 95228/25845375] """ |