def katz_expansions(k0, p, ellp, mdash, n):
r"""
Returns a list e of q-expansions, and the Eisenstein series `E_{p-1} = 1 +
\dots`, all modulo `(p^\text{mdash},q^\text{ellp})`. The list e contains
the elements `e_{i,s}` in the Katz expansions basis in Step 3 of Algorithm
1 in [AGBL]_ when one takes as input to that algorithm p,m and k and define
``k0``, ``mdash``, n, ``ellp = ell*p`` as in Step 1.
INPUT:
- ``k0`` -- integer in range 0 to p-1.
- ``p`` -- prime at least 5.
- ``ellp,mdash,n`` -- positive integers.
OUTPUT:
- list of q-expansions and the Eisenstein series E_{p-1} modulo
`(p^\text{mdash},q^\text{ellp})`.
EXAMPLES::
sage: from sage.modular.overconvergent.hecke_series import katz_expansions
sage: katz_expansions(0,5,10,3,4)
([1 + O(q^10), q + 6*q^2 + 27*q^3 + 98*q^4 + 65*q^5 + 37*q^6 + 81*q^7 + 85*q^8 + 62*q^9 + O(q^10)],
1 + 115*q + 35*q^2 + 95*q^3 + 20*q^4 + 115*q^5 + 105*q^6 + 60*q^7 + 25*q^8 + 55*q^9 + O(q^10))
"""
S = Zmod(p**mdash)
Ep1 = eisenstein_series_qexp(p - 1, ellp, K=S, normalization="constant")
E4 = eisenstein_series_qexp(4, ellp, K=S, normalization="constant")
E6 = eisenstein_series_qexp(6, ellp, K=S, normalization="constant")
delta = delta_qexp(ellp, K=S)
h = delta / E6**2
hj = delta.parent()(1)
e = []
# We compute negative powers of E_(p-1) successively (this saves a great
# deal of time). The effect is that Ep1mi = Ep1 ** (-i).
Ep1m1 = ~Ep1
Ep1mi = 1
for i in xrange(0, n + 1):
Wi, hj = compute_Wi(k0 + i * (p - 1), p, h, hj, E4, E6)
for bis in Wi:
eis = p**floor(i / (p + 1)) * Ep1mi * bis
e.append(eis)
Ep1mi = Ep1mi * Ep1m1
return e, Ep1
def katz_expansions(k0,p,ellp,mdash,n):
r"""
Returns a list e of q-expansions, and the Eisenstein series `E_{p-1} = 1 +
\dots`, all modulo `(p^\text{mdash},q^\text{ellp})`. The list e contains
the elements `e_{i,s}` in the Katz expansions basis in Step 3 of Algorithm
1 in [AGBL]_ when one takes as input to that algorithm p,m and k and define
``k0``, ``mdash``, n, ``ellp = ell*p`` as in Step 1.
INPUT:
- ``k0`` -- integer in range 0 to p-1.
- ``p`` -- prime at least 5.
- ``ellp,mdash,n`` -- positive integers.
OUTPUT:
- list of q-expansions and the Eisenstein series E_{p-1} modulo
`(p^\text{mdash},q^\text{ellp})`.
EXAMPLES::
sage: from sage.modular.overconvergent.hecke_series import katz_expansions
sage: katz_expansions(0,5,10,3,4)
([1 + O(q^10), q + 6*q^2 + 27*q^3 + 98*q^4 + 65*q^5 + 37*q^6 + 81*q^7 + 85*q^8 + 62*q^9 + O(q^10)],
1 + 115*q + 35*q^2 + 95*q^3 + 20*q^4 + 115*q^5 + 105*q^6 + 60*q^7 + 25*q^8 + 55*q^9 + O(q^10))
"""
S = Zmod(p**mdash)
Ep1 = eisenstein_series_qexp(p-1, ellp, K=S, normalization="constant")
E4 = eisenstein_series_qexp(4, ellp, K=S, normalization="constant")
E6 = eisenstein_series_qexp(6, ellp, K=S, normalization="constant")
delta = delta_qexp(ellp, K=S)
h = delta / E6**2
hj = delta.parent()(1)
e = []
# We compute negative powers of E_(p-1) successively (this saves a great
# deal of time). The effect is that Ep1mi = Ep1 ** (-i).
Ep1m1 = ~Ep1
Ep1mi = 1
for i in xrange(0,n+1):
Wi,hj = compute_Wi(k0 + i*(p-1),p,h,hj,E4,E6)
for bis in Wi:
eis = p**floor(i/(p+1)) * Ep1mi * bis
e.append(eis)
Ep1mi = Ep1mi * Ep1m1
return e,Ep1
def higher_level_katz_exp(p, N, k0, m, mdash, elldash, elldashp, modformsring,
bound):
r"""
Returns a matrix `e` of size ``ell x elldashp`` over the integers modulo
`p^\text{mdash}`, and the Eisenstein series `E_{p-1} = 1 + .\dots \bmod
(p^\text{mdash},q^\text{elldashp})`. The matrix e contains the coefficients
of the elements `e_{i,s}` in the Katz expansions basis in Step 3 of
Algorithm 2 in [AGBL]_ when one takes as input to that algorithm
`p`,`N`,`m` and `k` and define ``k0``, ``mdash``, ``n``, ``elldash``,
``elldashp = ell*dashp`` as in Step 1.
INPUT:
- ``p`` -- prime at least 5.
- ``N`` -- positive integer at least 2 and not divisible by p (level).
- ``k0`` -- integer in xrange 0 to p-1.
- ``m,mdash,elldash,elldashp`` -- positive integers.
- ``modformsring`` -- True or False.
- ``bound`` -- positive (even) integer.
OUTPUT:
- matrix and q-expansion.
EXAMPLES::
sage: from sage.modular.overconvergent.hecke_series import higher_level_katz_exp
sage: e,Ep1 = higher_level_katz_exp(5,2,0,1,2,4,20,true,6)
sage: e
[ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[ 0 1 18 23 19 6 9 9 17 7 3 17 12 8 22 8 11 19 1 5]
[ 0 0 1 11 20 16 0 8 4 0 18 15 24 6 15 23 5 18 7 15]
[ 0 0 0 1 4 16 23 13 6 5 23 5 2 16 4 18 10 23 5 15]
sage: Ep1
1 + 15*q + 10*q^2 + 20*q^3 + 20*q^4 + 15*q^5 + 5*q^6 + 10*q^7 +
5*q^9 + 10*q^10 + 5*q^11 + 10*q^12 + 20*q^13 + 15*q^14 + 20*q^15 + 15*q^16 +
10*q^17 + 20*q^18 + O(q^20)
"""
ordr = 1 / (p + 1)
S = Zmod(p**mdash)
Ep1 = eisenstein_series_qexp(p - 1,
prec=elldashp,
K=S,
normalization="constant")
n = floor(((p + 1) / (p - 1)) * (m + 1))
Wjs = complementary_spaces(N, p, k0, n, mdash, elldashp, elldash,
modformsring, bound)
Basis = []
for j in xrange(n + 1):
Wj = Wjs[j]
dimj = len(Wj)
Ep1minusj = Ep1**(-j)
for i in xrange(dimj):
wji = Wj[i]
b = p**floor(ordr * j) * wji * Ep1minusj
Basis.append(b)
# extract basis as a matrix
ell = len(Basis)
M = matrix(S, ell, elldashp)
for i in xrange(ell):
for j in xrange(elldashp):
M[i, j] = Basis[i][j]
ech_form(M, p) # put it into echelon form
return M, Ep1
def complementary_spaces(N, p, k0, n, mdash, elldashp, elldash, modformsring,
bound):
r"""
Returns a list ``Ws``, each element in which is a list ``Wi`` of
q-expansions modulo `(p^\text{mdash},q^\text{elldashp})`. The list ``Wi`` is
a basis for a choice of complementary space in level `\Gamma_0(N)` and
weight `k` to the image of weight `k - (p-1)` forms under multiplication by
the Eisenstein series `E_{p-1}`.
The lists ``Wi`` play the same role as `W_i` in Step 2 of Algorithm 2 in
[AGBL]_. (The parameters ``k0,n,mdash,elldash,elldashp = elldash*p`` are
defined as in Step 1 of that algorithm when this function is used in
:func:`hecke_series`.) However, the complementary spaces are computed in a
different manner, combining a suggestion of David Loeffler with one of John
Voight. That is, one builds these spaces recursively using random products
of forms in low weight, first searching for suitable products modulo
`(p,q^\text{elldash})`, and then later reconstructing only the required
products to the full precision modulo `(p^\text{mdash},q^{elldashp})`. The
forms in low weight are chosen from either bases of all forms up to weight
``bound`` or from a (tentative) generating set for the ring of all modular
forms, according to whether ``modformsring`` is ``False`` or ``True``.
INPUT:
- ``N`` -- positive integer at least 2 and not divisible by p (level).
- ``p`` -- prime at least 5.
- ``k0`` -- integer in range 0 to ``p-1``.
- ``n,mdash,elldashp,elldash`` -- positive integers.
- ``modformsring`` -- True or False.
- ``bound`` -- positive (even) integer (ignored if ``modformsring`` is True).
OUTPUT:
- list of lists of q-expansions modulo
``(p^\text{mdash},q^\text{elldashp})``.
EXAMPLES::
sage: from sage.modular.overconvergent.hecke_series import complementary_spaces
sage: complementary_spaces(2,5,0,3,2,5,4,true,6) # random
[[1],
[1 + 23*q + 24*q^2 + 19*q^3 + 7*q^4 + O(q^5)],
[1 + 21*q + 2*q^2 + 17*q^3 + 14*q^4 + O(q^5)],
[1 + 19*q + 9*q^2 + 11*q^3 + 9*q^4 + O(q^5)]]
sage: complementary_spaces(2,5,0,3,2,5,4,false,6) # random
[[1],
[3 + 4*q + 2*q^2 + 12*q^3 + 11*q^4 + O(q^5)],
[2 + 2*q + 14*q^2 + 19*q^3 + 18*q^4 + O(q^5)],
[6 + 8*q + 10*q^2 + 23*q^3 + 4*q^4 + O(q^5)]]
"""
if modformsring == False:
LWB = random_low_weight_bases(N, p, mdash, elldashp, bound)
else:
LWB, bound = low_weight_generators(N, p, mdash, elldashp)
LWBModp = [[f.change_ring(GF(p)).truncate_powerseries(elldash) for f in x]
for x in LWB]
CompSpacesCode = complementary_spaces_modp(N, p, k0, n, elldash, LWBModp,
bound)
Ws = []
Epm1 = eisenstein_series_qexp(p - 1,
prec=elldashp,
K=Zmod(p**mdash),
normalization="constant")
for i in xrange(n + 1):
CompSpacesCodemi = CompSpacesCode[i]
Wi = []
for k in xrange(len(CompSpacesCodemi)):
CompSpacesCodemik = CompSpacesCodemi[k]
Wik = Epm1.parent()(1)
for j in xrange(len(CompSpacesCodemik)):
l = CompSpacesCodemik[j][0]
index = CompSpacesCodemik[j][1]
Wik = Wik * LWB[l][index]
Wi.append(Wik)
Ws.append(Wi)
return Ws
def higher_level_katz_exp(p,N,k0,m,mdash,elldash,elldashp,modformsring,bound):
r"""
Returns a matrix `e` of size ``ell x elldashp`` over the integers modulo
`p^\text{mdash}`, and the Eisenstein series `E_{p-1} = 1 + .\dots \bmod
(p^\text{mdash},q^\text{elldashp})`. The matrix e contains the coefficients
of the elements `e_{i,s}` in the Katz expansions basis in Step 3 of
Algorithm 2 in [AGBL]_ when one takes as input to that algorithm
`p`,`N`,`m` and `k` and define ``k0``, ``mdash``, ``n``, ``elldash``,
``elldashp = ell*dashp`` as in Step 1.
INPUT:
- ``p`` -- prime at least 5.
- ``N`` -- positive integer at least 2 and not divisible by p (level).
- ``k0`` -- integer in xrange 0 to p-1.
- ``m,mdash,elldash,elldashp`` -- positive integers.
- ``modformsring`` -- True or False.
- ``bound`` -- positive (even) integer.
OUTPUT:
- matrix and q-expansion.
EXAMPLES::
sage: from sage.modular.overconvergent.hecke_series import higher_level_katz_exp
sage: e,Ep1 = higher_level_katz_exp(5,2,0,1,2,4,20,true,6)
sage: e
[ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[ 0 1 18 23 19 6 9 9 17 7 3 17 12 8 22 8 11 19 1 5]
[ 0 0 1 11 20 16 0 8 4 0 18 15 24 6 15 23 5 18 7 15]
[ 0 0 0 1 4 16 23 13 6 5 23 5 2 16 4 18 10 23 5 15]
sage: Ep1
1 + 15*q + 10*q^2 + 20*q^3 + 20*q^4 + 15*q^5 + 5*q^6 + 10*q^7 +
5*q^9 + 10*q^10 + 5*q^11 + 10*q^12 + 20*q^13 + 15*q^14 + 20*q^15 + 15*q^16 +
10*q^17 + 20*q^18 + O(q^20)
"""
ordr = 1/(p+1)
S = Zmod(p**mdash)
Ep1 = eisenstein_series_qexp(p-1,prec=elldashp,K=S, normalization="constant")
n = floor(((p+1)/(p-1))*(m+1))
Wjs = complementary_spaces(N,p,k0,n,mdash,elldashp,elldash,modformsring,bound)
Basis = []
for j in xrange(n+1):
Wj = Wjs[j]
dimj = len(Wj)
Ep1minusj = Ep1**(-j)
for i in xrange(dimj):
wji = Wj[i]
b = p**floor(ordr*j) * wji * Ep1minusj
Basis.append(b)
# extract basis as a matrix
ell = len(Basis)
M = matrix(S,ell,elldashp)
for i in xrange(ell):
for j in xrange(elldashp):
M[i,j] = Basis[i][j]
ech_form(M,p) # put it into echelon form
return M,Ep1
def complementary_spaces(N,p,k0,n,mdash,elldashp,elldash,modformsring,bound):
r"""
Returns a list ``Ws``, each element in which is a list ``Wi`` of
q-expansions modulo `(p^\text{mdash},q^\text{elldashp})`. The list ``Wi`` is
a basis for a choice of complementary space in level `\Gamma_0(N)` and
weight `k` to the image of weight `k - (p-1)` forms under multiplication by
the Eisenstein series `E_{p-1}`.
The lists ``Wi`` play the same role as `W_i` in Step 2 of Algorithm 2 in
[AGBL]_. (The parameters ``k0,n,mdash,elldash,elldashp = elldash*p`` are
defined as in Step 1 of that algorithm when this function is used in
:func:`hecke_series`.) However, the complementary spaces are computed in a
different manner, combining a suggestion of David Loeffler with one of John
Voight. That is, one builds these spaces recursively using random products
of forms in low weight, first searching for suitable products modulo
`(p,q^\text{elldash})`, and then later reconstructing only the required
products to the full precision modulo `(p^\text{mdash},q^{elldashp})`. The
forms in low weight are chosen from either bases of all forms up to weight
``bound`` or from a (tentative) generating set for the ring of all modular
forms, according to whether ``modformsring`` is ``False`` or ``True``.
INPUT:
- ``N`` -- positive integer at least 2 and not divisible by p (level).
- ``p`` -- prime at least 5.
- ``k0`` -- integer in range 0 to ``p-1``.
- ``n,mdash,elldashp,elldash`` -- positive integers.
- ``modformsring`` -- True or False.
- ``bound`` -- positive (even) integer (ignored if ``modformsring`` is True).
OUTPUT:
- list of lists of q-expansions modulo
``(p^\text{mdash},q^\text{elldashp})``.
EXAMPLES::
sage: from sage.modular.overconvergent.hecke_series import complementary_spaces
sage: complementary_spaces(2,5,0,3,2,5,4,true,6) # random
[[1],
[1 + 23*q + 24*q^2 + 19*q^3 + 7*q^4 + O(q^5)],
[1 + 21*q + 2*q^2 + 17*q^3 + 14*q^4 + O(q^5)],
[1 + 19*q + 9*q^2 + 11*q^3 + 9*q^4 + O(q^5)]]
sage: complementary_spaces(2,5,0,3,2,5,4,false,6) # random
[[1],
[3 + 4*q + 2*q^2 + 12*q^3 + 11*q^4 + O(q^5)],
[2 + 2*q + 14*q^2 + 19*q^3 + 18*q^4 + O(q^5)],
[6 + 8*q + 10*q^2 + 23*q^3 + 4*q^4 + O(q^5)]]
"""
if modformsring == False:
LWB = random_low_weight_bases(N,p,mdash,elldashp,bound)
else:
LWB,bound = low_weight_generators(N,p,mdash,elldashp)
LWBModp = [ [ f.change_ring(GF(p)).truncate_powerseries(elldash) for f in x] for x in LWB]
CompSpacesCode = complementary_spaces_modp(N,p,k0,n,elldash,LWBModp,bound)
Ws = []
Epm1 = eisenstein_series_qexp(p-1, prec=elldashp, K = Zmod(p**mdash), normalization="constant")
for i in xrange(n+1):
CompSpacesCodemi = CompSpacesCode[i]
Wi = []
for k in xrange(len(CompSpacesCodemi)):
CompSpacesCodemik = CompSpacesCodemi[k]
Wik = Epm1.parent()(1)
for j in xrange(len(CompSpacesCodemik)):
l = CompSpacesCodemik[j][0]
index = CompSpacesCodemik[j][1]
Wik = Wik*LWB[l][index]
Wi.append(Wik)
Ws.append(Wi)
return Ws