Degeneracy maps#
- class sage.modular.hecke.degenmap.DegeneracyMap(matrix, domain, codomain, t)[source]#
Bases:
HeckeModuleMorphism_matrixA degeneracy map between Hecke modules of different levels.
EXAMPLES:
We construct a number of degeneracy maps:
sage: M = ModularSymbols(33) sage: d = M.degeneracy_map(11) sage: d Hecke module morphism degeneracy map corresponding to f(q) |--> f(q) defined by the matrix [ 1 0 0] [ 0 0 1] [ 0 0 -1] [ 0 1 -1] [ 0 0 1] [ 0 -1 1] [-1 0 0] [-1 0 0] [-1 0 0] Domain: Modular Symbols space of dimension 9 for Gamma_0(33) of weight ... Codomain: Modular Symbols space of dimension 3 for Gamma_0(11) of weight ... sage: d.t() 1 sage: d = M.degeneracy_map(11,3) sage: d.t() 3
>>> from sage.all import * >>> M = ModularSymbols(Integer(33)) >>> d = M.degeneracy_map(Integer(11)) >>> d Hecke module morphism degeneracy map corresponding to f(q) |--> f(q) defined by the matrix [ 1 0 0] [ 0 0 1] [ 0 0 -1] [ 0 1 -1] [ 0 0 1] [ 0 -1 1] [-1 0 0] [-1 0 0] [-1 0 0] Domain: Modular Symbols space of dimension 9 for Gamma_0(33) of weight ... Codomain: Modular Symbols space of dimension 3 for Gamma_0(11) of weight ... >>> d.t() 1 >>> d = M.degeneracy_map(Integer(11),Integer(3)) >>> d.t() 3
The parameter d must be a divisor of the quotient of the two levels:
sage: d = M.degeneracy_map(11,2) Traceback (most recent call last): ... ValueError: the level of self (=33) must be a divisor or multiple of level (=11) and t (=2) must be a divisor of the quotient
>>> from sage.all import * >>> d = M.degeneracy_map(Integer(11),Integer(2)) Traceback (most recent call last): ... ValueError: the level of self (=33) must be a divisor or multiple of level (=11) and t (=2) must be a divisor of the quotient
Degeneracy maps can also go from lower level to higher level:
sage: M.degeneracy_map(66,2) Hecke module morphism degeneracy map corresponding to f(q) |--> f(q^2) defined by the matrix [ 2 0 0 0 0 0 1 0 0 0 1 -1 0 0 0 -1 1 0 0 0 0 0 0 0 -1] [ 0 0 1 -1 0 -1 1 0 -1 2 0 0 0 -1 0 0 -1 1 2 -2 0 0 0 -1 1] [ 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 -1 1 0 0 -1 1 0 0 0] [ 0 0 0 0 0 0 0 0 0 2 -1 0 0 1 0 0 -1 1 0 0 1 0 -1 -1 1] [ 0 -1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 -1 0 0 -1 0 0 0 0 0] [ 0 0 0 0 0 0 0 1 -1 0 0 2 -1 0 0 1 0 0 0 -1 0 -1 1 -1 1] [ 0 0 0 0 1 -1 0 1 -1 0 0 0 0 0 -1 2 0 0 0 0 1 0 1 0 0] [ 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0] [ 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 1 0 0 0] Domain: Modular Symbols space of dimension 9 for Gamma_0(33) of weight ... Codomain: Modular Symbols space of dimension 25 for Gamma_0(66) of weight ...
>>> from sage.all import * >>> M.degeneracy_map(Integer(66),Integer(2)) Hecke module morphism degeneracy map corresponding to f(q) |--> f(q^2) defined by the matrix [ 2 0 0 0 0 0 1 0 0 0 1 -1 0 0 0 -1 1 0 0 0 0 0 0 0 -1] [ 0 0 1 -1 0 -1 1 0 -1 2 0 0 0 -1 0 0 -1 1 2 -2 0 0 0 -1 1] [ 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 -1 1 0 0 -1 1 0 0 0] [ 0 0 0 0 0 0 0 0 0 2 -1 0 0 1 0 0 -1 1 0 0 1 0 -1 -1 1] [ 0 -1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 -1 0 0 -1 0 0 0 0 0] [ 0 0 0 0 0 0 0 1 -1 0 0 2 -1 0 0 1 0 0 0 -1 0 -1 1 -1 1] [ 0 0 0 0 1 -1 0 1 -1 0 0 0 0 0 -1 2 0 0 0 0 1 0 1 0 0] [ 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0] [ 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 1 0 0 0] Domain: Modular Symbols space of dimension 9 for Gamma_0(33) of weight ... Codomain: Modular Symbols space of dimension 25 for Gamma_0(66) of weight ...
- t()[source]#
Return the divisor of the quotient of the two levels associated to the degeneracy map.
EXAMPLES:
sage: M = ModularSymbols(33) sage: d = M.degeneracy_map(11,3) sage: d.t() 3 sage: d = M.degeneracy_map(11,1) sage: d.t() 1
>>> from sage.all import * >>> M = ModularSymbols(Integer(33)) >>> d = M.degeneracy_map(Integer(11),Integer(3)) >>> d.t() 3 >>> d = M.degeneracy_map(Integer(11),Integer(1)) >>> d.t() 1