Modular symbols attached to elliptic curves over \(\QQ\)

To an elliptic curve \(E\) over the rational numbers with conductor \(N\), one can associate a space of modular symbols of level \(N\), because \(E\) is known to be modular. The space is two-dimensional and contains a subspace on which complex conjugation acts as multiplication by \(+1\) and one on which it acts by \(-1\).

There are two implementations of modular symbols, one within Sage and the other in Cremona’s eclib library. One can choose here which one is used.

Associated to \(E\) there is a canonical generator in each space. They are maps \([.]^+\) and \([.]^{-}\), both \(\QQ \to\QQ\). They are normalized such that

\[[r]^{+} \Omega^{+} + [r]^{-}\Omega^{-} = \int_{\infty}^r 2\pi i f(z) dz\]

where \(f\) is the newform associated to the isogeny class of \(E\) and \(\Omega^{+}\) is the smallest positive period of the Néron differential of \(E\) and \(\Omega^{-}\) is the smallest positive purely imaginary period. Note that it depends on \(E\) rather than on its isogeny class.

From eclib version v20161230, both plus and minus symbols are available and are correctly normalized. In the Sage implementation, the computation of the space provides initial generators which are not necessarily correctly normalized; here we implement two methods that try to find the correct scaling factor.

Modular symbols are used to compute \(p\)-adic \(L\)-functions.

EXAMPLES:

sage: E = EllipticCurve("19a1")
sage: m = E.modular_symbol()
sage: m(0)
1/3
sage: m(1/17)
-2/3
sage: m2 = E.modular_symbol(-1, implementation="sage")
sage: m2(0)
0
sage: m2(1/5)
1/2

sage: V = E.modular_symbol_space()
sage: V
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 for Gamma_0(19) of weight 2 with sign 1 over Rational Field
sage: V.q_eigenform(30)
q - 2*q^3 - 2*q^4 + 3*q^5 - q^7 + q^9 + 3*q^11 + 4*q^12 - 4*q^13 - 6*q^15 + 4*q^16 - 3*q^17 + q^19 - 6*q^20 + 2*q^21 + 4*q^25 + 4*q^27 + 2*q^28 + 6*q^29 + O(q^30)

For more details on modular symbols consult the following

REFERENCES:

[MaTaTe]B. Mazur, J. Tate, and J. Teitelbaum, On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones mathematicae 84, (1986), 1-48.
[Crem97]John Cremona, Algorithms for modular elliptic curves, Cambridge University Press, 1997.
[StWu]William Stein and Christian Wuthrich, Algorithms for the Arithmetic of Elliptic Curves using Iwasawa Theory, Mathematics of Computation 82 (2013), 1757-1792.

AUTHORS:

  • William Stein (2007): first version
  • Chris Wuthrich (2008): add scaling and reference to eclib
  • John Cremona (2016): reworked eclib interface
class sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbol

Bases: sage.structure.sage_object.SageObject

A modular symbol attached to an elliptic curve, which is the map \(\QQ\to \QQ\) obtained by sending \(r\) to the normalized symmetrized (or anti-symmetrized) integral \(\infty\) to \(r\).

This is as defined in [MaTaTe], but normalized to depend on the curve and not only its isogeny class as in [StWu].

See the documentation of E.modular_symbol() in elliptic curves over the rational numbers for help.

base_ring()

Return the base ring for this modular symbol.

EXAMPLES:

sage: m = EllipticCurve('11a1').modular_symbol()
sage: m.base_ring()
Rational Field
elliptic_curve()

Return the elliptic curve of this modular symbol.

EXAMPLES:

sage: m = EllipticCurve('11a1').modular_symbol()
sage: m.elliptic_curve()
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sign()

Return the sign of this elliptic curve modular symbol.

EXAMPLES:

sage: m = EllipticCurve('11a1').modular_symbol()
sage: m.sign()
1
sage: m = EllipticCurve('11a1').modular_symbol(sign=-1, implementation="sage")
sage: m.sign()
-1
class sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolECLIB(E, sign)

Bases: sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbol

Modular symbols attached to \(E\) using eclib.

Note that the normalization used within eclib differs from the normalization chosen here by a factor of 2 in the case of elliptic curves with negative discriminant (with one real component) since the convention there is to write the above integral as \([r]^{+}x+[r]^{-}yi\), where the lattice is \(\left<2x,x+yi\right>\), so that \(\Omega^{+}=2x\) and \(\Omega^{-}=2yi\). We allow for this below.

INPUT:

  • E - an elliptic curve
  • sign - an integer, -1 or 1

EXAMPLES:

sage: import sage.schemes.elliptic_curves.ell_modular_symbols
sage: E=EllipticCurve('11a1')
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolECLIB(E,+1)
sage: M
Modular symbol with sign 1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: M(0)
1/5
sage: E=EllipticCurve('11a2')
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolECLIB(E,+1)
sage: M(0)
1

This is a rank 1 case with vanishing positive twists:

sage: E=EllipticCurve('121b1')
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolECLIB(E,+1)
sage: M(0)
0
sage: M(1/7)
1/2

sage: M = EllipticCurve('121d1').modular_symbol(implementation="eclib")
sage: M(0)
2

sage: E = EllipticCurve('15a1')
sage: [C.modular_symbol(implementation="eclib")(0) for C in E.isogeny_class()]
[1/4, 1/8, 1/4, 1/2, 1/8, 1/16, 1/2, 1]

Since trac ticket #10256, the interface for negative modular symbols in eclib is available:

sage: E = EllipticCurve('11a1')
sage: Mplus = E.modular_symbol(+1); Mplus
Modular symbol with sign 1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: [Mplus(1/i) for i in [1..11]]
[1/5, -4/5, -3/10, 7/10, 6/5, 6/5, 7/10, -3/10, -4/5, 1/5, 0]
sage: Mminus = E.modular_symbol(-1); Mminus
Modular symbol with sign -1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: [Mminus(1/i) for i in [1..11]]
[0, 0, 1/2, 1/2, 0, 0, -1/2, -1/2, 0, 0, 0]

The scaling factor relative to eclib’s normalization is 1/2 for curves of negative discriminant:

sage: [E.discriminant() for E in cremona_curves([14])]
[-21952, 941192, -1835008, -28, 25088, 98]
sage: [E.modular_symbol()._scaling for E in cremona_curves([14])]
[1/2, 1, 1/2, 1/2, 1, 1]

TESTS (for trac ticket #10236):

sage: E = EllipticCurve('11a1')
sage: m = E.modular_symbol(implementation="eclib")
sage: m(1/7)
7/10
sage: m(0)
1/5
class sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E, sign, normalize='L_ratio')

Bases: sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbol

Modular symbols attached to \(E\) using sage.

INPUT:

  • E – an elliptic curve
  • sign – an integer, -1 or 1
  • normalize – either ‘L_ratio’ (default), ‘period’, or ‘none’; For ‘L_ratio’, the modular symbol is correctly normalized by comparing it to the quotient of \(L(E,1)\) by the least positive period for the curve and some small twists. The normalization ‘period’ uses the integral_period_map for modular symbols and is known to be equal to the above normalization up to the sign and a possible power of 2. For ‘none’, the modular symbol is almost certainly not correctly normalized, i.e. all values will be a fixed scalar multiple of what they should be. But the initial computation of the modular symbol is much faster, though evaluation of it after computing it won’t be any faster.

EXAMPLES:

sage: E=EllipticCurve('11a1')
sage: import sage.schemes.elliptic_curves.ell_modular_symbols
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1)
sage: M
Modular symbol with sign 1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: M(0)
1/5
sage: E=EllipticCurve('11a2')
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1)
sage: M(0)
1
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,-1)
sage: M(1/3)
1/2

This is a rank 1 case with vanishing positive twists. The modular symbol is adjusted by -2:

sage: E=EllipticCurve('121b1')
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,-1,normalize='L_ratio')
sage: M(1/3)
1
sage: M._scaling
1

sage: M = EllipticCurve('121d1').modular_symbol(implementation="sage")
sage: M(0)
2
sage: M = EllipticCurve('121d1').modular_symbol(implementation="sage", normalize='none')
sage: M(0)
1

sage: E = EllipticCurve('15a1')
sage: [C.modular_symbol(implementation="sage", normalize='L_ratio')(0) for C in E.isogeny_class()]
[1/4, 1/8, 1/4, 1/2, 1/8, 1/16, 1/2, 1]
sage: [C.modular_symbol(implementation="sage", normalize='period')(0) for C in E.isogeny_class()]
[1/8, 1/16, 1/8, 1/4, 1/16, 1/32, 1/4, 1/2]
sage: [C.modular_symbol(implementation="sage", normalize='none')(0) for C in E.isogeny_class()]
[1, 1, 1, 1, 1, 1, 1, 1]
sage.schemes.elliptic_curves.ell_modular_symbols.modular_symbol_space(E, sign, base_ring, bound=None)

Creates the space of modular symbols of a given sign over a give base_ring, attached to the isogeny class of the elliptic curve E.

INPUT:

  • E - an elliptic curve over \(\QQ\)
  • sign - integer, -1, 0, or 1
  • base_ring - ring
  • bound - (default: None) maximum number of Hecke operators to use to cut out modular symbols factor. If None, use enough to provably get the correct answer.

OUTPUT: a space of modular symbols

EXAMPLES:

sage: import sage.schemes.elliptic_curves.ell_modular_symbols
sage: E=EllipticCurve('11a1')
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.modular_symbol_space(E,-1,GF(37))
sage: M
Modular Symbols space of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Finite Field of size 37