\(p\)-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 [PS2011].

sage.modular.pollack_stevens.padic_lseries.log_gamma_binomial(p, gamma, n, M)[source]¶

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. \(1+p\)
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

sage: from sage.modular.pollack_stevens.padic_lseries import log_gamma_binomial
sage: log_gamma_binomial(5,1+5,2,4)
[0, -3/205, 651/84050, -223/42025]
sage: log_gamma_binomial(5,1+5,3,4)
[0, 2/205, -223/42025, 95228/25845375]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.padic_lseries import log_gamma_binomial
>>> log_gamma_binomial(Integer(5),Integer(1)+Integer(5),Integer(2),Integer(4))
[0, -3/205, 651/84050, -223/42025]
>>> log_gamma_binomial(Integer(5),Integer(1)+Integer(5),Integer(3),Integer(4))
[0, 2/205, -223/42025, 95228/25845375]

Sage Live

from sage.modular.pollack_stevens.padic_lseries import log_gamma_binomial
log_gamma_binomial(5,1+5,2,4)
log_gamma_binomial(5,1+5,3,4)

class sage.modular.pollack_stevens.padic_lseries.pAdicLseries(symb, gamma=None, quadratic_twist=1, precision=None)[source]¶

Bases: SageObject

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

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)

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a')
>>> p = Integer(5)
>>> prec = Integer(4)
>>> L = E.padic_lseries(p, implementation='pollackstevens', precision=prec) # long time
>>> L[Integer(1)]                # long time
1 + 4*5 + 2*5^2 + O(5^3)
>>> L.series(Integer(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 Live

E = EllipticCurve('37a')
p = 5
prec = 4
L = E.padic_lseries(p, implementation='pollackstevens', precision=prec) # long time
L[1]                # long time
L.series(3)    # long time

Sage

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)

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
>>> E = EllipticCurve('20a')
>>> phi = E.pollack_stevens_modular_symbol()
>>> Phi = phi.p_stabilize_and_lift(Integer(3), Integer(4)) # long time
>>> L = pAdicLseries(Phi)                # long time
>>> L.series(Integer(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)

Sage Live

from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
E = EllipticCurve('20a')
phi = E.pollack_stevens_modular_symbol()
Phi = phi.p_stabilize_and_lift(3, 4) # long time
L = pAdicLseries(Phi)                # long time
L.series(4)                          # long time

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
>>> E = EllipticCurve('20a')
>>> phi = E.pollack_stevens_modular_symbol()
>>> Phi = phi.p_stabilize_and_lift(Integer(3), Integer(4)) # long time
>>> L = pAdicLseries(Phi)                # long time
>>> L.series(Integer(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)

Sage Live

from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
E = EllipticCurve('20a')
phi = E.pollack_stevens_modular_symbol()
Phi = phi.p_stabilize_and_lift(3, 4) # long time
L = pAdicLseries(Phi)                # long time
L.series(4)                          # long time

An example of a \(p\)-adic \(L\)-series associated to a modular abelian surface. This is not tested as it takes too long.:

Sage

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)

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
>>> from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
>>> A = ModularSymbols(Integer(103),Integer(2),Integer(1)).cuspidal_submodule().new_subspace().decomposition()[Integer(0)]
>>> p = Integer(19)
>>> prec = Integer(4)
>>> phi = ps_modsym_from_simple_modsym_space(A)
>>> ap = phi.Tq_eigenvalue(p,prec)
>>> c1,c2 = phi.completions(p,prec)
>>> phi1,psi1 = c1
>>> phi2,psi2 = c2
>>> phi1p = phi1.p_stabilize_and_lift(p,ap = psi1(ap), M = prec) # not tested - too long
>>> L1 = pAdicLseries(phi1p)                                     # not tested - too long
>>> phi2p = phi2.p_stabilize_and_lift(p,ap = psi2(ap), M = prec) # not tested - too long
>>> L2  = pAdicLseries(phi2p)                                    # not tested - too long
>>> L1[Integer(1)]*L2[Integer(1)]                                                  # not tested - too long
13 + 9*19 + 18*19^2 + O(19^3)

Sage Live

from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
A = ModularSymbols(103,2,1).cuspidal_submodule().new_subspace().decomposition()[0]
p = 19
prec = 4
phi = ps_modsym_from_simple_modsym_space(A)
ap = phi.Tq_eigenvalue(p,prec)
c1,c2 = phi.completions(p,prec)
phi1,psi1 = c1
phi2,psi2 = c2
phi1p = phi1.p_stabilize_and_lift(p,ap = psi1(ap), M = prec) # not tested - too long
L1 = pAdicLseries(phi1p)                                     # not tested - too long
phi2p = phi2.p_stabilize_and_lift(p,ap = psi2(ap), M = prec) # not tested - too long
L2  = pAdicLseries(phi2p)                                    # not tested - too long
L1[1]*L2[1]                                                  # not tested - too long

interpolation_factor(ap, chip=1, psi=None)[source]¶

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

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)

Python

>>> from sage.all import *
>>> E = EllipticCurve('19a2')
>>> L = E.padic_lseries(Integer(3),implementation='pollackstevens',precision=Integer(6))  # long time
>>> ap = E.ap(Integer(3))               # long time
>>> L.interpolation_factor(ap) # long time
3^2 + 3^3 + 2*3^5 + 2*3^6 + O(3^7)

Sage Live

E = EllipticCurve('19a2')
L = E.padic_lseries(3,implementation='pollackstevens',precision=6)  # long time
ap = E.ap(3)               # long time
L.interpolation_factor(ap) # long time

Comparing against a different implementation:

Sage

sage: L = E.padic_lseries(3)
sage: (1-1/L.alpha(prec=4))^2
3^2 + 3^3 + O(3^5)

Python

>>> from sage.all import *
>>> L = E.padic_lseries(Integer(3))
>>> (Integer(1)-Integer(1)/L.alpha(prec=Integer(4)))**Integer(2)
3^2 + 3^3 + O(3^5)

Sage Live

L = E.padic_lseries(3)
(1-1/L.alpha(prec=4))^2

prime()[source]¶

Return the prime \(p\) as in \(p\)-adic \(L\)-series.

EXAMPLES:

Sage

sage: E = EllipticCurve('19a')
sage: L = E.padic_lseries(19, implementation='pollackstevens',precision=6) # long time
sage: L.prime()                   # long time
19

Python

>>> from sage.all import *
>>> E = EllipticCurve('19a')
>>> L = E.padic_lseries(Integer(19), implementation='pollackstevens',precision=Integer(6)) # long time
>>> L.prime()                   # long time
19

Sage Live

E = EllipticCurve('19a')
L = E.padic_lseries(19, implementation='pollackstevens',precision=6) # long time
L.prime()                   # long time

quadratic_twist()[source]¶

Return the discriminant of the quadratic twist.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
>>> E = EllipticCurve('37a')
>>> phi = E.pollack_stevens_modular_symbol()
>>> Phi = phi.lift(Integer(37),Integer(4))
>>> L = pAdicLseries(Phi, quadratic_twist=-Integer(3))
>>> L.quadratic_twist()
-3

Sage Live

from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
E = EllipticCurve('37a')
phi = E.pollack_stevens_modular_symbol()
Phi = phi.lift(37,4)
L = pAdicLseries(Phi, quadratic_twist=-3)
L.quadratic_twist()

series(prec=5)[source]¶

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) the precision of the power series

EXAMPLES:

Sage

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)

Python

>>> from sage.all import *
>>> E = EllipticCurve('14a2')
>>> p = Integer(3)
>>> prec = Integer(6)
>>> L = E.padic_lseries(p,implementation='pollackstevens',precision=prec) # long time
>>> L.series(Integer(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)

>>> E = EllipticCurve("15a3")
>>> L = E.padic_lseries(Integer(5),implementation='pollackstevens',precision=Integer(15))  # long time
>>> L.series(Integer(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)

>>> E = EllipticCurve("79a1")
>>> L = E.padic_lseries(Integer(2),implementation='pollackstevens',precision=Integer(10)) # not tested
>>> L.series(Integer(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)

Sage Live

E = EllipticCurve('14a2')
p = 3
prec = 6
L = E.padic_lseries(p,implementation='pollackstevens',precision=prec) # long time
L.series(4)          # long time
E = EllipticCurve("15a3")
L = E.padic_lseries(5,implementation='pollackstevens',precision=15)  # long time
L.series(3)            # long time
E = EllipticCurve("79a1")
L = E.padic_lseries(2,implementation='pollackstevens',precision=10) # not tested
L.series(4)  # not tested

symbol()[source]¶

Return the overconvergent modular symbol.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
>>> E = EllipticCurve('21a4')
>>> phi = E.pollack_stevens_modular_symbol()
>>> Phi = phi.p_stabilize_and_lift(Integer(2),Integer(5))   # long time
>>> L = pAdicLseries(Phi)                 # long time
>>> 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
>>> L.symbol() is Phi                       # long time
True

Sage Live

from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
E = EllipticCurve('21a4')
phi = E.pollack_stevens_modular_symbol()
Phi = phi.p_stabilize_and_lift(2,5)   # long time
L = pAdicLseries(Phi)                 # long time
L.symbol()                              # long time
L.symbol() is Phi                       # long time