Modular symbols¶

To an elliptic curves $E$ over the rational numbers one can associate a space - or better two spaces - of modular symbols of level $N$ , equal to the conductor of $E$ ; because $E$ is known to be modular.

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

The normalisation of our modular symbols attached to $E$ can be chosen, too. For instance one can make it depended on $E$ rather than on its isogeny class. This is useful for $p$ -adic L-functions.

For more details on modular symbols consult the following

REFERENCES:

[MTT] 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.
[Cre] John Cremona, Algorithms for modular elliptic curves, Cambridge University Press, 1997.
[SW] William Stein and Christian Wuthrich, Computations About Tate-Shafarevich Groups using Iwasawa theory, preprint 2009.

AUTHORS:

William Stein (2007): first version
Chris Wuthrich (2008): add scaling and reference to eclib

class sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbol¶

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 from $r$ to $\infty$ .

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

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

REFERENCES:

[MTT] 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.
[SW] William Stein and Christian Wuthrich, Computations About Tate-Shafarevich Groups using Iwasawa theory, preprint 2009.

_ModularSymbol__lalg(D)¶: For positive $D$ , this function evaluates the quotient $L(E_D,1)\cdot \sqrt(D)/\Omega_E$ where $E_D$ is the twist of $E$ by $D$ , $\Omega_E$ is the least positive period of $E$ . For negative $E$ , it is the quotient $L(E_D,1)\cdot \sqrt(-D)/\Omega^{-}_E$ where $\Omega^{-}_E$ is the least positive imaginary part of a non-real period of $E$ .

_ModularSymbol__scale_by_periods_only()¶: If we fail to scale with _find_scaling_L_ratio, we drop here to try and find the scaling by the quotient of the periods to the $X_0$ -optimal curve. The resulting _scaling is not guaranteed to be correct, but could well be.

__weakref__¶: list of weak references to the object (if defined)

_find_scaling_L_ratio()¶

This function is use to set _scaling, the factor used to adjust the scalar multiple of the modular symbol. If $[0]$ , the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rational number with small denominator. Otherwise we try to scale using quadratic twists.

_scaling will be set to a rational non-zero multiple if we succeed and to 1 otherwise. Even if we fail we scale at least to make up the difference between the periods of the $X_0$ -optimal curve and our given curve $E$ in the isogeny class.

EXAMPLES:

sage : m = EllipticCurve('11a1').modular_symbol(use_eclib=True)
sage : m._scaling
1
sage: m = EllipticCurve('11a2').modular_symbol(use_eclib=True)
sage: m._scaling
5
sage: m = EllipticCurve('11a3').modular_symbol(use_eclib=True)
sage: m._scaling
1/5
sage: m = EllipticCurve('11a1').modular_symbol(use_eclib=False)
sage: m._scaling
1/5
sage: m = EllipticCurve('11a2').modular_symbol(use_eclib=False)
sage: m._scaling
1
sage: m = EllipticCurve('11a3').modular_symbol(use_eclib=False)
sage: m._scaling
1/25
sage: m = EllipticCurve('37a1').modular_symbol(use_eclib=False)
sage: m._scaling
1
sage: m = EllipticCurve('37a1').modular_symbol(use_eclib=True)
sage: m._scaling
-1
sage: m = EllipticCurve('389a1').modular_symbol(use_eclib=True)
sage: m._scaling
-1/2
sage: m = EllipticCurve('389a1').modular_symbol(use_eclib=False)
sage: m._scaling
2

Some cases that check on the negative twists:

sage: m = EllipticCurve('121b1').modular_symbol(use_eclib=False)
sage: m._scaling  
-2           
sage: m = EllipticCurve('196a1').modular_symbol(use_eclib=False)
sage: m._scaling  
1/2  

Some checks of consistency:

sage: rk0 = ['11a1', '11a2', '15a1', '27a1', '37b1']
sage: for la in rk0:            
...          E = EllipticCurve(la)
...          me = E.modular_symbol(use_eclib = True)
...          ms = E.modular_symbol(use_eclib = False)
...          print E.lseries().L_ratio()*E.real_components(), me(0), ms(0)                      
1/5 1/5 1/5
1 1 1
1/4 1/4 1/4
1/3 1/3 1/3
2/3 2/3 2/3

sage: rk1 = ['37a1','43a1','53a1', '91b1','91b2','91b3']
sage: [EllipticCurve(la).modular_symbol(use_eclib=True)(0) for la in rk1]
[0, 0, 0, 0, 0, 0]
sage: for la in rk1:            
...       E = EllipticCurve(la)
...       m = E.modular_symbol(use_eclib = True)
...       lp = E.padic_lseries(5)
...       for D in [5,17,12,8]:
...           ED = E.quadratic_twist(D)
...           md = sum([kronecker(D,u)*m(ZZ(u)/D) for u in range(D)])
...           etaa = lp._quotient_of_periods_to_twist(D)
...           if ED.lseries().L_ratio()*ED.real_components()*etaa != md:
...               print 'oyoyoy a bug !!!'                      

_repr_()¶

String representation of modular symbols.

EXAMPLES:

sage: m = EllipticCurve('11a1').modular_symbol(use_eclib=True)
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 = EllipticCurve('43a1').modular_symbol(sign=-1)
sage: m
Modular symbol with sign -1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 + x^2 over Rational Field

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)
sage: m.sign()
-1

class sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolECLIB(E, sign, normalize='L_ratio')¶

__call__(r)¶

Evaluates the modular symbol at $r$ .

EXAMPLES:

sage: m = EllipticCurve('11a1').modular_symbol(use_eclib=True)
sage: m(0)
1/5       

__init__(E, sign, normalize='L_ratio')¶

Modular symbols attached to $E$ using eclib.

INPUT:

E - an elliptic curve
sign - an integer, -1 or 1
normalize - either ‘L_ratio’ (default) 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. 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.

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
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolECLIB(E,-1)
...          
NotImplementedError: Negative spaces of Modular Symbols using eclib has not yet been implemented.

This is a rank 1 case with vanishing positive twists. The modular symbol can not be adjusted:

sage: E=EllipticCurve('121b1')
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolECLIB(E,+1)
Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by -1, 2 or -2.
sage: M(0)
0
sage: M(1/7)
-2

sage: M = EllipticCurve('121d1').modular_symbol(use_eclib=True)
sage: M(0)
2
sage: M = EllipticCurve('121d1').modular_symbol(use_eclib=True,normalize='none')
sage: M(0)
4

sage: E = EllipticCurve('15a1')
sage: [C.modular_symbol(use_eclib=True,normalize='L_ratio')(0) for C in E.isogeny_class()[0]]
[1/4, 1/8, 1/4, 1/2, 1/8, 1/16, 1/2, 1]
sage: [C.modular_symbol(use_eclib=True,normalize='none')(0) for C in E.isogeny_class()[0]]
[1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8]

_call_with_caching(r)¶

Evaluates the modular symbol at $r$ , caching the computed value.

EXAMPLES:

sage: m = EllipticCurve('11a1').modular_symbol(use_eclib=True)
sage: m._call_with_caching(0)
1/5

class sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E, sign, normalize='L_ratio')¶

__call__(r)¶

Evaluates the modular symbol at $r$ .

EXAMPLES:

sage: m = EllipticCurve('11a1').modular_symbol(use_eclib=False)
sage: m(0)
1/5       

__init__(E, sign, normalize='L_ratio')¶

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

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)
2
sage: M._scaling
-2

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

sage: E = EllipticCurve('15a1')
sage: [C.modular_symbol(use_eclib=False, normalize='L_ratio')(0) for C in E.isogeny_class()[0]]
[1/4, 1/8, 1/4, 1/2, 1/8, 1/16, 1/2, 1]
sage: [C.modular_symbol(use_eclib=False, normalize='period')(0) for C in E.isogeny_class()[0]]
[1/8, 1/16, 1/8, 1/4, 1/16, 1/32, 1/4, 1/2]
sage: [C.modular_symbol(use_eclib=False, normalize='none')(0) for C in E.isogeny_class()[0]]
[1, 1, 1, 1, 1, 1, 1, 1]

_call_with_caching(r)¶

Evaluates the modular symbol at $r$ , caching the computed value.

EXAMPLES:

sage: m = EllipticCurve('11a1').modular_symbol(use_eclib=False)
sage: m._call_with_caching(0)
1/5

_find_scaling_period()¶

Uses the integral period map of the modular symbol implementation in sage in order to determine the scaling. The resulting modular symbol is correct only for the $X_0$ -optimal curve, at least up to a possible factor +-1 or +-2.

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(1/5, 1)
sage: E = EllipticCurve('11a2')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(1, 5)
sage: E = EllipticCurve('121b2')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(0, 11/2, 0, 11/2, 11/2, 0, 0, -3, 2, 1/2, -1, 3/2)

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 elliptic curves.

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

Modular symbols¶

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