# Element class for Pollack-Stevens’ Modular Symbols¶

This is the class of elements in the spaces of Pollack-Steven’s modular symbols as described in [PS2011].

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol(); phi
Modular symbol of level 11 with values in Sym^0 Q^2
sage: phi.weight() # Note that weight k=2 of a modular form corresponds here to weight 0
0
sage: phi.values()
[-1/5, 1, 0]
sage: phi.is_ordinary(11)
True
sage: phi_lift = phi.lift(11, 5, eigensymbol = True) # long time
O(11^5) + (10 + 3*11 + 6*11^2 + 9*11^3 + O(11^4))*T + (6 + 3*11 + 2*11^2 + O(11^3))*T^2 + (2 + 2*11 + O(11^2))*T^3 + (5 + O(11))*T^4 + O(T^5)

sage: A = ModularSymbols(Gamma1(8),4).decomposition()[0].plus_submodule().new_subspace()
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
sage: phi = ps_modsym_from_simple_modsym_space(A)
sage: phi.values()
[(-1, 0, 0), (1, 0, 0), (-9, -6, -4)]

class sage.modular.pollack_stevens.modsym.PSModSymAction(actor, MSspace)

Create the action

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: g = phi._map._codomain._act._Sigma0(matrix(ZZ,2,2,[1,2,3,4]))
sage: phi * g # indirect doctest
Modular symbol of level 11 with values in Sym^0 Q^2

class sage.modular.pollack_stevens.modsym.PSModularSymbolElement(map_data, parent, construct=False)

Initialize a modular symbol

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: phi = E.pollack_stevens_modular_symbol()

Tq_eigenvalue(q, p=None, M=None, check=True)

Eigenvalue of $$T_q$$ modulo $$p^M$$

INPUT:

• q – prime of the Hecke operator

• p – prime we are working modulo (default: None)

• M – degree of accuracy of approximation (default: None)

• check – check that self is an eigensymbol

OUTPUT:

• Constant $$c$$ such that $$self|T_q - c * self$$ has valuation greater than or equal to $$M$$ (if it exists), otherwise raises ValueError

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.values()
[-1/5, 1, 0]
sage: phi_ord = phi.p_stabilize(p = 3, ap = E.ap(3), M = 10, ordinary = True)
sage: phi_ord.Tq_eigenvalue(2,3,10) + 2
O(3^10)

sage: phi_ord.Tq_eigenvalue(3,3,10)
2 + 3^2 + 2*3^3 + 2*3^4 + 2*3^6 + 3^8 + 2*3^9 + O(3^10)
sage: phi_ord.Tq_eigenvalue(3,3,100)
Traceback (most recent call last):
...
ValueError: result not determined to high enough precision

diagonal_valuation(p)

Return the minimum of the diagonal valuation on the values of self

INPUT:

• p – a positive integral prime

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.values()
[-1/5, 1, 0]
sage: phi.diagonal_valuation(2)
0
sage: phi.diagonal_valuation(3)
0
sage: phi.diagonal_valuation(5)
-1
sage: phi.diagonal_valuation(7)
0

dict()

Return dictionary on the modular symbol self, where keys are generators and values are the corresponding values of self on generators

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: Set([x.moment(0) for x in phi.dict().values()]) == Set([-1/5, 1, 0])
True

evaluate_twisted(a, chi)

Return $$\Phi_{\chi}(\{a/p\}-\{\infty\})$$ where $$\Phi$$ is self and $$\chi$$ is a quadratic character

INPUT:

• a – integer in the range range(p)

• chi – the modulus of a quadratic character.

OUTPUT:

The distribution $$\Phi_{\chi}(\{a/p\}-\{\infty\})$$.

EXAMPLES:

sage: E = EllipticCurve('17a1')
sage: L = E.padic_lseries(5, implementation="pollackstevens", precision=4) #long time
sage: D = L.quadratic_twist()          # long time
sage: L.symbol().evaluate_twisted(1,D) # long time
(1 + 5 + 3*5^2 + 5^3 + O(5^4), 5^2 + O(5^3), 1 + O(5^2), 2 + O(5))

sage: E = EllipticCurve('40a4')
sage: L = E.padic_lseries(7, implementation="pollackstevens", precision=4) #long time
sage: D = L.quadratic_twist()          # long time
sage: L.symbol().evaluate_twisted(1,D) # long time
(4 + 6*7 + 3*7^2 + O(7^4), 6*7 + 6*7^2 + O(7^3), 6 + O(7^2), 1 + O(7))

hecke(ell, algorithm='prep')

Return self | $$T_{\ell}$$ by making use of the precomputations in self.prep_hecke()

INPUT:

• ell – a prime

• algorithm – a string, either ‘prep’ (default) or ‘naive’

OUTPUT:

• The image of this element under the Hecke operator $$T_{\ell}$$

ALGORITHMS:

• If algorithm == 'prep', precomputes a list of matrices that only depend on the level, then uses them to speed up the action.

• If algorithm == 'naive', just acts by the matrices defining the Hecke operator. That is, it computes sum_a self | [1,a,0,ell] + self | [ell,0,0,1], the last term occurring only if the level is prime to ell.

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.values()
[-1/5, 1, 0]
sage: phi.hecke(2) == phi * E.ap(2)
True
sage: phi.hecke(3) == phi * E.ap(3)
True
sage: phi.hecke(5) == phi * E.ap(5)
True
sage: phi.hecke(101) == phi * E.ap(101)
True

sage: all(phi.hecke(p, algorithm='naive') == phi * E.ap(p) for p in [2,3,5,101]) # long time
True

is_Tq_eigensymbol(q, p=None, M=None)

Determine if self is an eigenvector for $$T_q$$ modulo $$p^M$$

INPUT:

• q – prime of the Hecke operator

• p – prime we are working modulo

• M – degree of accuracy of approximation

OUTPUT:

• True/False

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.values()
[-1/5, 1, 0]
sage: phi_ord = phi.p_stabilize(p = 3, ap = E.ap(3), M = 10, ordinary = True)
sage: phi_ord.is_Tq_eigensymbol(2,3,10)
True
sage: phi_ord.is_Tq_eigensymbol(2,3,100)
False
sage: phi_ord.is_Tq_eigensymbol(2,3,1000)
False
sage: phi_ord.is_Tq_eigensymbol(3,3,10)
True
sage: phi_ord.is_Tq_eigensymbol(3,3,100)
False

is_ordinary(p=None, P=None)

Return true if the $$p$$-th eigenvalue is a $$p$$-adic unit.

INPUT:

• p - a positive integral prime, or None (default None)

• P - a prime of the base ring above $$p$$, or None. This is ignored unless the base ring is a number field.

OUTPUT:

• True/False

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.is_ordinary(2)
False
sage: E.ap(2)
-2
sage: phi.is_ordinary(3)
True
sage: E.ap(3)
-1
sage: phip = phi.p_stabilize(3,20)
sage: phip.is_ordinary()
True


A number field example. Here there are multiple primes above $$p$$, and $$\phi$$ is ordinary at one but not the other.:

sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
sage: f = Newforms(32, 8, names='a')[1]
sage: phi = ps_modsym_from_simple_modsym_space(f.modular_symbols(1))
sage: (p1, _), (p2, _) = phi.base_ring().ideal(3).factor()
sage: phi.is_ordinary(p1) != phi.is_ordinary(p2)
True
sage: phi.is_ordinary(3)
Traceback (most recent call last):
...
TypeError: P must be an ideal

minus_part()

Return the minus part of self – i.e. self - self | [1,0,0,-1]

Note that we haven’t divided by 2. Is this a problem?

OUTPUT:

• self – self | [1,0,0,-1]

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.values()
[-1/5, 1, 0]
sage: (phi.plus_part()+phi.minus_part()) == phi * 2
True

plus_part()

Return the plus part of self – i.e. self + self | [1,0,0,-1].

Note that we haven’t divided by 2. Is this a problem?

OUTPUT:

• self + self | [1,0,0,-1]

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.values()
[-1/5, 1, 0]
sage: (phi.plus_part()+phi.minus_part()) == 2 * phi
True

valuation(p=None)

Return the valuation of self at $$p$$.

Here the valuation is the minimum of the valuations of the values of self.

INPUT:

• p - prime

OUTPUT:

• The valuation of self at $$p$$

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.values()
[-1/5, 1, 0]
sage: phi.valuation(2)
0
sage: phi.valuation(3)
0
sage: phi.valuation(5)
-1
sage: phi.valuation(7)
0
sage: phi.valuation()
Traceback (most recent call last):
...
ValueError: you must specify a prime

sage: phi2 = phi.lift(11, M=2)
sage: phi2.valuation()
0
sage: phi2.valuation(3)
Traceback (most recent call last):
...
ValueError: inconsistent prime
sage: phi2.valuation(11)
0

values()

Return the values of the symbol self on our chosen generators.

The generators are listed in self.dict().

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.values()
[-1/5, 1, 0]
sage: sorted(phi.dict())
[
[-1 -1]  [ 0 -1]  [1 0]
[ 3  2], [ 1  3], [0 1]
]
sage: sorted(phi.values()) == sorted(phi.dict().values())
True

weight()

Return the weight of this Pollack-Stevens modular symbol.

This is $$k-2$$, where $$k$$ is the usual notion of weight for modular forms!

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.weight()
0

class sage.modular.pollack_stevens.modsym.PSModularSymbolElement_dist(map_data, parent, construct=False)

Return the $$p$$-adic L-series of this modular symbol.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: L = phi.lift(37, M=6, eigensymbol=True).padic_lseries(); L  # long time
37-adic L-series of Modular symbol of level 37 with values in Space of 37-adic distributions with k=0 action and precision cap 7
sage: L.series(2) # long time
O(37^6) + (4 + 37 + 36*37^2 + 19*37^3 + 21*37^4 + O(37^5))*T + O(T^2)

precision_relative()

Return the number of moments of each value of self

EXAMPLES:

sage: D = OverconvergentDistributions(0, 5, 10)
sage: M = PollackStevensModularSymbols(Gamma0(5), coefficients=D)
sage: f = M(1)
sage: f.precision_relative()
1

reduce_precision(M)

Only hold on to $$M$$ moments of each value of self

EXAMPLES:

sage: D = OverconvergentDistributions(0, 5, 10)
sage: M = PollackStevensModularSymbols(Gamma0(5), coefficients=D)
sage: f = M(1)
sage: f.reduce_precision(1)
Modular symbol of level 5 with values in Space of 5-adic distributions with k=0 action and precision cap 10

specialize(new_base_ring=None)

Return the underlying classical symbol of weight $$k$$ - i.e., applies the canonical map $$D_k \to Sym^k$$ to all values of self.

EXAMPLES:

sage: D = OverconvergentDistributions(0, 5, 10);  M = PollackStevensModularSymbols(Gamma0(5), coefficients=D); M
Space of overconvergent modular symbols for Congruence Subgroup Gamma0(5) with sign 0
and values in Space of 5-adic distributions with k=0 action and precision cap 10
sage: f = M(1)
sage: f.specialize()
Modular symbol of level 5 with values in Sym^0 Z_5^2
sage: f.specialize().values()
[1 + O(5), 1 + O(5), 1 + O(5)]
sage: f.values()
[1 + O(5), 1 + O(5), 1 + O(5)]
sage: f.specialize().parent()
Space of modular symbols for Congruence Subgroup Gamma0(5) with sign 0 and values in Sym^0 Z_5^2
sage: f.specialize().parent().coefficient_module()
Sym^0 Z_5^2
sage: f.specialize().parent().coefficient_module().is_symk()
True
sage: f.specialize(Qp(5,20))
Modular symbol of level 5 with values in Sym^0 Q_5^2

class sage.modular.pollack_stevens.modsym.PSModularSymbolElement_symk(map_data, parent, construct=False)
completions(p, M)

If $$K$$ is the base_ring of self, this function takes all maps $$K\to \QQ_p$$ and applies them to self return a list of (modular symbol,map: $$K\to \QQ_p$$) as map varies over all such maps.

Note

This only returns all completions when $$p$$ splits completely in $$K$$

INPUT:

• p – prime

• M – precision

OUTPUT:

• A list of tuples (modular symbol,map: $$K\to \QQ_p$$) as map varies over all such maps

EXAMPLES:

sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
sage: D = ModularSymbols(67,2,1).cuspidal_submodule().new_subspace().decomposition()[1]
sage: f = ps_modsym_from_simple_modsym_space(D)
sage: S = f.completions(41,10); S
[(Modular symbol of level 67 with values in Sym^0 Q_41^2, Ring morphism:
From: Number Field in alpha with defining polynomial x^2 + 3*x + 1
To:   41-adic Field with capped relative precision 10
Defn: alpha |--> 5 + 22*41 + 19*41^2 + 10*41^3 + 28*41^4 + 22*41^5 + 9*41^6 + 25*41^7 + 40*41^8 + 8*41^9 + O(41^10)), (Modular symbol of level 67 with values in Sym^0 Q_41^2, Ring morphism:
From: Number Field in alpha with defining polynomial x^2 + 3*x + 1
To:   41-adic Field with capped relative precision 10
Defn: alpha |--> 33 + 18*41 + 21*41^2 + 30*41^3 + 12*41^4 + 18*41^5 + 31*41^6 + 15*41^7 + 32*41^9 + O(41^10))]
sage: TestSuite(S[0][0]).run(skip=['_test_category'])

lift(p=None, M=None, alpha=None, new_base_ring=None, algorithm=None, eigensymbol=False, check=True)

Return a ($$p$$-adic) overconvergent modular symbol with $$M$$ moments which lifts self up to an Eisenstein error

Here the Eisenstein error is a symbol whose system of Hecke eigenvalues equals $$\ell+1$$ for $$T_\ell$$ when $$\ell$$ does not divide $$Np$$ and 1 for $$U_q$$ when $$q$$ divides $$Np$$.

INPUT:

• p – prime

• M – integer equal to the number of moments

• alpha$$U_p$$ eigenvalue

• new_base_ring – change of base ring

• algorithm – ‘stevens’ or ‘greenberg’ (default ‘stevens’)

• eigensymbol – if True, lifts to Hecke eigensymbol (self must be a $$p$$-ordinary eigensymbol)

(Note: eigensymbol = True does not just indicate to the code that self is an eigensymbol; it solves a wholly different problem, lifting an eigensymbol to an eigensymbol.)

OUTPUT:

An overconvergent modular symbol whose specialization equals self, up to some Eisenstein error if eigensymbol is False. If eigensymbol = True then the output will be an overconvergent Hecke eigensymbol (and it will lift the input exactly, the Eisenstein error disappears).

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: f = E.pollack_stevens_modular_symbol()
sage: g = f.lift(11,4,algorithm='stevens',eigensymbol=True)
sage: g.is_Tq_eigensymbol(2)
True
sage: g.Tq_eigenvalue(3)
10 + 10*11 + 10*11^2 + 10*11^3 + O(11^4)
sage: g.Tq_eigenvalue(11)
1 + O(11^4)


We check that lifting and then specializing gives back the original symbol:

sage: g.specialize() == f
True


Another example, which showed precision loss in an earlier version of the code:

sage: E = EllipticCurve('37a')
sage: p = 5
sage: prec = 4
sage: phi = E.pollack_stevens_modular_symbol()
sage: Phi = phi.p_stabilize_and_lift(p,prec, algorithm='stevens', eigensymbol=True) # long time
sage: Phi.Tq_eigenvalue(5,M = 4) # long time
3 + 2*5 + 4*5^2 + 2*5^3 + O(5^4)


Another example:

sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
sage: E = EllipticCurve('37a')
sage: p = 5
sage: prec = 6
sage: phi = E.pollack_stevens_modular_symbol()
sage: Phi = phi.p_stabilize_and_lift(p=p,M=prec,alpha=None,algorithm='stevens',eigensymbol=True) #long time
sage: L = pAdicLseries(Phi)          # long time
sage: L.symbol() is Phi              # long time
True


Examples using Greenberg’s algorithm:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: Phi = phi.lift(11,8,algorithm='greenberg',eigensymbol=True)
sage: Phi2 = phi.lift(11,8,algorithm='stevens',eigensymbol=True)
sage: Phi == Phi2
True


An example in higher weight:

sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
sage: f = ps_modsym_from_simple_modsym_space(Newforms(7, 4)[0].modular_symbols(1))
sage: fs = f.p_stabilize(5)
sage: FsG = fs.lift(M=6, eigensymbol=True,algorithm='greenberg') # long time
sage: FsG.values()[0]                                            # long time
5^-1 * (2*5 + 5^2 + 3*5^3 + 4*5^4 + O(5^7), O(5^6), 2*5^2 + 3*5^3 + O(5^5), O(5^4), 5^2 + O(5^3), O(5^2))
sage: FsS = fs.lift(M=6, eigensymbol=True,algorithm='stevens')   # long time
sage: FsS == FsG                                                 # long time
True

p_stabilize(p=None, M=20, alpha=None, ap=None, new_base_ring=None, ordinary=True, check=True)

Return the $$p$$-stabilization of self to level $$N p$$ on which $$U_p$$ acts by $$\alpha$$.

Note that since $$\alpha$$ is $$p$$-adic, the resulting symbol is just an approximation to the true $$p$$-stabilization (depending on how well $$\alpha$$ is approximated).

INPUT:

• p – prime not dividing the level of self

• M – (default: 20) precision of $$\QQ_p$$

• alpha$$U_p$$ eigenvalue

• ap – Hecke eigenvalue

• new_base_ring – change of base ring

• ordinary – (default: True) whether to return the ordinary

(at p) eigensymbol.

• check – (default: True) whether to perform extra sanity checks

OUTPUT:

A modular symbol with the same Hecke eigenvalues as self away from $$p$$ and eigenvalue $$\alpha$$ at $$p$$. The eigenvalue $$\alpha$$ depends on the parameter ordinary.

If ordinary == True: the unique modular symbol of level $$N p$$ with the same Hecke eigenvalues as self away from $$p$$ and unit eigenvalue at $$p$$; else the unique modular symbol of level $$N p$$ with the same Hecke eigenvalues as self away from $$p$$ and non-unit eigenvalue at $$p$$.

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: p = 5
sage: prec = 4
sage: phi = E.pollack_stevens_modular_symbol()
sage: phis = phi.p_stabilize(p,M = prec)
sage: phis
Modular symbol of level 55 with values in Sym^0 Q_5^2
sage: phis.hecke(7) == phis*E.ap(7)
True
sage: phis.hecke(5) == phis*E.ap(5)
False
sage: phis.hecke(3) == phis*E.ap(3)
True
sage: phis.Tq_eigenvalue(5)
1 + 4*5 + 3*5^2 + 2*5^3 + O(5^4)
sage: phis.Tq_eigenvalue(5,M = 3)
1 + 4*5 + 3*5^2 + O(5^3)

sage: phis = phi.p_stabilize(p,M = prec,ordinary=False)
sage: phis.Tq_eigenvalue(5)
5 + 5^2 + 2*5^3 + O(5^5)


A complicated example (with nontrivial character):

sage: chi = DirichletGroup(24)([-1, -1, -1])
sage: f = Newforms(chi,names='a')[0]
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
sage: phi = ps_modsym_from_simple_modsym_space(f.modular_symbols(1))
sage: phi11, h11 = phi.completions(11,20)[0]
sage: phi11s = phi11.p_stabilize()
sage: phi11s.is_Tq_eigensymbol(11) # long time
True

p_stabilize_and_lift(p, M, alpha=None, ap=None, new_base_ring=None, ordinary=True, algorithm='greenberg', eigensymbol=False, check=True)

$$p$$-stabilize and lift self

INPUT:

• p – prime, not dividing the level of self

• M – precision

• alpha – (default: None) the $$U_p$$ eigenvalue, if known

• ap – (default: None) the Hecke eigenvalue at p (before stabilizing), if known

• new_base_ring – (default: None) if specified, force the resulting eigensymbol to take values in the given ring

• ordinary – (default: True) whether to return the ordinary

(at p) eigensymbol.

• algorithm – (default: ‘greenberg’) a string, either ‘greenberg’ or ‘stevens’, specifying whether to use the lifting algorithm of M.Greenberg or that of Pollack–Stevens. The latter one solves the difference equation, which is not needed. The option to use Pollack–Stevens’ algorithm here is just for historical reasons.

• eigensymbol – (default: False) if True, return an overconvergent eigensymbol. Otherwise just perform a naive lift

• check – (default: True) whether to perform extra sanity checks

OUTPUT:

$$p$$-stabilized and lifted version of self.

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: f = E.pollack_stevens_modular_symbol()
sage: g = f.p_stabilize_and_lift(3,10)  # long time
sage: g.Tq_eigenvalue(5)                # long time
1 + O(3^10)
sage: g.Tq_eigenvalue(7)                # long time
1 + 2*3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + O(3^10)
sage: g.Tq_eigenvalue(3)                # long time
2 + 3^2 + 2*3^3 + 2*3^4 + 2*3^6 + 3^8 + 2*3^9 + O(3^10)