Manin symbols¶

This module defines the class ManinSymbol. A Manin Symbol of weight $k$ , level $N$ has the form $[P(X,Y),(u:v)]$ where $P(X,Y)\in\mathbb{Z}[X,Y]$ is homogeneous of weight $k-2$ and $(u:v)\in\mathbb{P}^1(\mathbb{Z}/N\mathbb{Z}).$ The ManinSymbol class holds a “monomial Manin Symbol” of the simpler form $[X^iY^{k-2-i},(u:v)]$ , which is stored as a triple $(i,u,v)$ ; the weight and level are obtained from the parent structure, which is a ManinSymbolList.

Integer matrices $[a,b;c,d]$ act on Manin Symbols on the right, sending $[P(X,Y),(u,v)]$ to $[P(aX+bY,cX+dY),(u,v)g]$ . Diagonal matrices (with $b=c=0$ , such as $I=[-1,0;0,1]$ and $J=[-1,0;0,-1]$ ) and anti-diagonal matrices (with $a=d=0$ , such as $S=[0,-1;1,0]$ ) map monomial Manin Symbols to monomial Manin Symbols, up to a scalar factor. For general matrices (such as $T=[0,1,-1,-1]$ and $T^2=[-1,-1;0,1]$ ) the image of a monomial Manin Symbol is expressed as a formal sum of monomial Manin Symbols, with integer coefficients.

There are various different classes holding lists of Manin symbols of different types. The hierarchy is as follows:

class ManinSymbolList(SageObject)

class ManinSymbolList_group(ManinSymbolList)
    class ManinSymbolList_gamma0(ManinSymbolList_group)
    class ManinSymbolList_gamma1(ManinSymbolList_group)
    class ManinSymbolList_gamma_h(ManinSymbolList_group)

class ManinSymbolList_character(ManinSymbolList)

class sage.modular.modsym.manin_symbols.ManinSymbol(parent, t)¶

Bases: sage.structure.sage_object.SageObject

A Manin symbol $[X^i\cdot Y^{k-2-i},(u,v)]$ .

INPUT:

parent - ManinSymbolList.
t - a 3-tuple $(i,u,v)$ of integers.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,2)
sage: s = ManinSymbol(m,(2,2,3)); s
(2,3)
sage: s == loads(dumps(s))
True

::

sage: m = ManinSymbolList_gamma0(5,8)
sage: s = ManinSymbol(m,(2,2,3)); s
[X^2*Y^4,(2,3)]

apply(a, b, c, d)¶

Return the image of self under the matrix $[a,b;c,d]$ .

Not implemented for raw ManinSymbol objects, only for members of ManinSymbolLists.

EXAMPLE:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,2)
sage: m.apply(10,[1,0,0,1]) # not implemented for base class

endpoints(N=None)¶

Returns cusps $alpha$ , $beta$ such that this Manin symbol, viewed as a symbol for level $N$ , is $X^i*Y^{k-2-i} \{alpha, beta\}$ .

EXAMPLES:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: s = ManinSymbol(m,(2,2,3)); s
[X^2*Y^4,(2,3)]
sage: s.endpoints()
(1/3, 1/2)

i¶

Return the $i$ field of this ManinSymbol $(i,u,v)$ .

EXAMPLES:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: s = ManinSymbol(m,(2,2,3))
sage: s._ManinSymbol__get_i()
2
sage: s.i
2

level()¶

Return the level of this ManinSymbol.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: s = ManinSymbol(m,(2,2,3))
sage: s.level()
5

lift_to_sl2z(N=None)¶

Returns a lift of this Manin Symbol to $SL_2(\mathbb{Z})$ .

If this Manin symbol is $(c,d)$ and $N$ is its level, this function returns a list $[a,b, c',d']$ that defines a 2x2 matrix with determinant 1 and integer entries, such that $c=c'$ (mod $N$ ) and $d=d'$ (mod $N$ ).

EXAMPLES:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: s = ManinSymbol(m,(2,2,3))
sage: s
[X^2*Y^4,(2,3)]
sage: s.lift_to_sl2z()
[1, 1, 2, 3]

modular_symbol_rep()¶

Returns a representation of self as a formal sum of modular symbols.

The result is not cached.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: s = ManinSymbol(m,(2,2,3))
sage: s.modular_symbol_rep()
 144*X^6*{1/3, 1/2} - 384*X^5*Y*{1/3, 1/2} + 424*X^4*Y^2*{1/3, 1/2} - 248*X^3*Y^3*{1/3, 1/2} + 81*X^2*Y^4*{1/3, 1/2} - 14*X*Y^5*{1/3, 1/2} + Y^6*{1/3, 1/2}

parent()¶

Return the parent of this ManinSymbol.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: s = ManinSymbol(m,(2,2,3))
sage: s.parent()
Manin Symbol List of weight 8 for Gamma0(5)

tuple()¶

Return the 3-tuple $(i,u,v)$ of this ManinSymbol.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: s = ManinSymbol(m,(2,2,3))
sage: s.tuple()
(2, 2, 3)

u¶

Return the $u$ field of this ManinSymbol $(i,u,v)$ .

EXAMPLES:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: s = ManinSymbol(m,(2,2,3))
sage: s.u # indirect doctest
2

v¶

Return the $v$ field of this ManinSymbol $(i,u,v)$ .

EXAMPLES:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: s = ManinSymbol(m,(2,2,3))
sage: s.v # indirect doctest
3

weight()¶

Return the weight of this ManinSymbol.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbols import ManinSymbol, ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: s = ManinSymbol(m,(2,2,3))
sage: s.weight()
8

class sage.modular.modsym.manin_symbols.ManinSymbolList(weight, list)¶

Bases: sage.structure.sage_object.SageObject

Base class for lists of all Manin symbols for a given weight, group or character.

apply(j, X)¶

Apply the matrix $X=[a,b;c,d]$ to the $j$ -th Manin symbol.