# HG changeset patch # User David Loeffler # Date 1331057340 0 # Node ID 02300303e9243260c6e638effe87e2eb20a8c5d7 # Parent 58790db46f55ee4d8d78096f465bc7b76079ee3a #11709: Farey symbol reviewer changes diff --git a/.hgignore b/.hgignore --- a/.hgignore +++ b/.hgignore @@ -53,6 +53,7 @@ sage/rings/complex_double_api.h sage/misc/allocator.h sage/symbolic/pynac.h +sage/modular/arithgroup/farey_symbol.h doc/output/ doc/en/reference/sage/ doc/en/reference/sagenb/ diff --git a/doc/en/bordeaux_2008/modular_forms_and_hecke_operators.rst b/doc/en/bordeaux_2008/modular_forms_and_hecke_operators.rst --- a/doc/en/bordeaux_2008/modular_forms_and_hecke_operators.rst +++ b/doc/en/bordeaux_2008/modular_forms_and_hecke_operators.rst @@ -55,10 +55,24 @@ enumerate) to obtain a list of generators [[ref my modular forms book]]. +The list of generators Sage computes is unfortunately large. Improving this +would be an excellent Sage development project, which would involve much +beautiful mathematics. + +UPDATE (March 2012): The project referred to above has been carried out (by +several people, notably Hartmut Monien, building on earlier work of Chris +Kurth). Sage now uses a much more advanced algorithm based on Farey symbols +which calculates a *minimal* set of generators. + :: sage: Gamma0(2).gens() ([1 1] + [0 1], + [ 1 -1] + [ 2 -1]) + sage: Gamma0(2).gens(algorithm="todd-coxeter") # the old implementation + ([1 1] [0 1], [-1 0] [ 0 -1], @@ -69,12 +83,8 @@ [-1 1] [-2 1]) sage: len(Gamma1(13).gens()) - 260 + 15 -As you can see above, the list of generators Sage computes is -unfortunately large. Improving this would be an excellent Sage -development project, which would involve much beautiful -mathematics. Modular Forms ------------- diff --git a/sage/modular/arithgroup/arithgroup_element.py b/sage/modular/arithgroup/arithgroup_element.py --- a/sage/modular/arithgroup/arithgroup_element.py +++ b/sage/modular/arithgroup/arithgroup_element.py @@ -133,7 +133,7 @@ This once caused a segfault (see trac #5443):: - sage: r,s,t,u,v = Gamma0(2).gens() + sage: r,s,t,u,v = map(SL2Z, [[1, 1, 0, 1], [-1, 0, 0, -1], [1, -1, 0, 1], [1, -1, 2, -1], [-1, 1, -2, 1]]) sage: v == s*u True sage: s*u == v diff --git a/sage/modular/arithgroup/arithgroup_generic.py b/sage/modular/arithgroup/arithgroup_generic.py --- a/sage/modular/arithgroup/arithgroup_generic.py +++ b/sage/modular/arithgroup/arithgroup_generic.py @@ -853,20 +853,48 @@ return ZZ(1 + (self.projective_index()) / ZZ(12) - (self.nu2())/ZZ(4) - (self.nu3())/ZZ(3) - self.ncusps()/ZZ(2)) - def generators(self): + def farey_symbol(self): r""" - Return generators for this congruence subgroup. - - This is carried out using an "inverse Todd-Coxeter" algorithm. - A Cython version of this for the special case of `\Gamma_0` and `\Gamma_1` is - implemented in the function ``congroup_pyx.generators_helper``. See the documentation - there for further details. Here we are assuming far less about the - group, so the computation is exceptionally slow! + Return the Farey symbol associated to this subgroup. See the + :mod:`~sage.modular.arithgroup.farey_symbol` module for more + information. EXAMPLE:: + sage: Gamma1(4).farey_symbol() + FareySymbol(Congruence Subgroup Gamma1(4)) + """ + from farey_symbol import Farey + return Farey(self) + + @cached_method + def generators(self, algorithm="farey"): + r""" + Return a list of generators for this congruence subgroup. The result is cached. + + INPUT: + + - ``algorithm`` (string): either ``farey`` or ``todd-coxeter``. + + If ``algorithm`` is set to ``"farey"``, then the generators will be + calculated using Farey symbols, which will always return a *minimal* + generating set. See :mod:`~sage.modular.arithgroup.farey_symbol` for + more information. + + If ``algorithm`` is set to ``"todd-coxeter"``, a simpler algorithm + based on Todd-Coxeter enumeration will be used. This is *exceedingly* + slow for general subgroups, and the list of generators will be far from + minimal (indeed it may contain repetitions). + + EXAMPLE:: + sage: Gamma(2).generators() [[1 2] + [0 1], [ 3 -2] + [ 2 -1], [-1 0] + [ 0 -1]] + sage: Gamma(2).generators(algorithm="todd-coxeter") + [[1 2] [0 1], [-1 0] [ 0 -1], [ 1 0] [-2 1], [-1 0] @@ -875,13 +903,18 @@ [ 2 -1], [1 0] [2 1]] """ - return self.todd_coxeter()[1] + if algorithm=="farey": + return self.farey_symbol().generators() + elif algorithm == "todd-coxeter": + return self.todd_coxeter()[1] + else: + raise ValueError, "Unknown algorithm '%s' (should be either 'farey' or 'todd-coxeter')" % algorithm - def gens(self): + def gens(self, *args, **kwds): r""" Return a tuple of generators for this congruence subgroup. - The generators need not be minimal. + The generators need not be minimal. For arguments, see :meth:`~generators`. EXAMPLES:: @@ -890,7 +923,7 @@ [ 1 0], [1 1] [0 1]) """ - return tuple(self.generators()) + return tuple(self.generators(*args, **kwds)) def gen(self, i): r""" @@ -907,18 +940,17 @@ def ngens(self): r""" - Return the number of generators for this arithmetic subgroup. - - This need not be the minimal number of generators of self. + Return the size of the minimal generating set of self returned by + :meth:`generators`. EXAMPLES:: sage: Gamma0(22).ngens() - 42 + 8 sage: Gamma1(14).ngens() - 219 + 13 sage: GammaH(11, [3]).ngens() - 31 + 3 sage: SL2Z.ngens() 2 """ @@ -1129,9 +1161,9 @@ Return self as an arithmetic subgroup defined in terms of the permutation action of `SL(2,\ZZ)` on its right cosets. - This method uses Todd-coxeter enumeration (via the method todd_coxeter) which - can be extremly slow for arithmetic subgroup with relatively large index in - `SL(2,\ZZ)`. + This method uses Todd-coxeter enumeration (via the method + :meth:`~todd_coxeter`) which can be extremely slow for arithmetic + subgroups with relatively large index in `SL(2,\ZZ)`. EXAMPLES:: diff --git a/sage/modular/arithgroup/congroup_gamma0.py b/sage/modular/arithgroup/congroup_gamma0.py --- a/sage/modular/arithgroup/congroup_gamma0.py +++ b/sage/modular/arithgroup/congroup_gamma0.py @@ -375,36 +375,60 @@ yield SL2Z(lift_to_sl2z(z[0], z[1], N)) @cached_method - def generators(self): + def generators(self, algorithm="farey"): r""" Return generators for this congruence subgroup. - The result is cached. + INPUT: + + - ``algorithm`` (string): either ``farey`` (default) or + ``todd-coxeter``. + + If ``algorithm`` is set to ``"farey"``, then the generators will be + calculated using Farey symbols, which will always return a *minimal* + generating set. See :mod:`~sage.modular.arithgroup.farey_symbol` for + more information. + + If ``algorithm`` is set to ``"todd-coxeter"``, a simpler algorithm + based on Todd-Coxeter enumeration will be used. This tends to return + far larger sets of generators. EXAMPLE:: - sage: for g in Gamma0(3).generators(): - ... print g - ... print '---' - [1 1] - [0 1] - --- - [-1 0] - [ 0 -1] - --- - ... - --- - [ 1 0] - [-3 1] - --- + sage: [x.matrix() for x in Gamma0(3).generators()] + [ + [1 1] [-1 1] + [0 1], [-3 2] + ] + sage: [x.matrix() for x in Gamma0(3).generators(algorithm="todd-coxeter")] + [ + [1 1] [-1 0] [ 1 -1] [1 0] [1 1] [-1 0] [ 1 0] + [0 1], [ 0 -1], [ 0 1], [3 1], [0 1], [ 3 -1], [-3 1] + ] + sage: SL2Z.gens() + ([ 0 -1] + [ 1 0], [1 1] + [0 1]) """ - from sage.modular.modsym.p1list import P1List - from congroup_pyx import generators_helper - level = self.level() - if level == 1: # P1List isn't very happy working mod 1 + if self.level() == 1: + # we return a fixed set of generators for SL2Z, for historical + # reasons, which aren't the ones the Farey symbol code gives return [ self([0,-1,1,0]), self([1,1,0,1]) ] - gen_list = generators_helper(P1List(level), level, Mat2Z) - return [self(g, check=False) for g in gen_list] + + elif algorithm=="farey": + return self.farey_symbol().generators() + + elif algorithm=="todd-coxeter": + from sage.modular.modsym.p1list import P1List + from congroup_pyx import generators_helper + level = self.level() + if level == 1: # P1List isn't very happy working mod 1 + return [ self([0,-1,1,0]), self([1,1,0,1]) ] + gen_list = generators_helper(P1List(level), level, Mat2Z) + return [self(g, check=False) for g in gen_list] + + else: + raise ValueError, "Unknown algorithm '%s' (should be either 'farey' or 'todd-coxeter')" % algorithm def gamma_h_subgroups(self): r""" diff --git a/sage/modular/arithgroup/congroup_gamma1.py b/sage/modular/arithgroup/congroup_gamma1.py --- a/sage/modular/arithgroup/congroup_gamma1.py +++ b/sage/modular/arithgroup/congroup_gamma1.py @@ -237,41 +237,49 @@ raise NotImplementedError @cached_method - def generators(self): + def generators(self, algorithm="farey"): r""" - Return generators for this congruence subgroup. + Return generators for this congruence subgroup. The result is cached. + + INPUT: - The result is cached. + - ``algorithm`` (string): either ``farey`` (default) or + ``todd-coxeter``. + If ``algorithm`` is set to ``"farey"``, then the generators will be + calculated using Farey symbols, which will always return a *minimal* + generating set. See :mod:`~sage.modular.arithgroup.farey_symbol` for + more information. + + If ``algorithm`` is set to ``"todd-coxeter"``, a simpler algorithm + based on Todd-Coxeter enumeration will be used. This tends to return + far larger sets of generators. + EXAMPLE:: - sage: for g in Gamma1(3).generators(): - ... print g - ... print '---' - [1 1] - [0 1] - --- - [-20 9] - [ 51 -23] - --- - [ 4 1] - [-9 -2] - --- - ... - --- - [ 4 -1] - [ 9 -2] - --- - [ -5 3] - [-12 7] - --- - + sage: [x.matrix() for x in Gamma1(3).generators()] + [ + [1 1] [ 1 -1] + [0 1], [ 3 -2] + ] + sage: [x.matrix() for x in Gamma1(3).generators(algorithm="todd-coxeter")] + [ + [1 1] [-20 9] [ 4 1] [-5 -2] [ 1 -1] [1 0] [1 1] [-5 2] + [0 1], [ 51 -23], [-9 -2], [ 3 1], [ 0 1], [3 1], [0 1], [12 -5], + [ 1 0] [ 4 -1] [ -5 3] [ 1 -1] [ 7 -3] [ 4 -1] [ -5 3] + [-3 1], [ 9 -2], [-12 7], [ 3 -2], [12 -5], [ 9 -2], [-12 7] + ] """ - from sage.modular.modsym.g1list import G1list - from congroup_pyx import generators_helper - level = self.level() - gen_list = generators_helper(G1list(level), level, Mat2Z) - return [self(g, check=False) for g in gen_list] + if algorithm=="farey": + return self.farey_symbol().generators() + elif algorithm=="todd-coxeter": + from sage.modular.modsym.g1list import G1list + from congroup_pyx import generators_helper + level = self.level() + gen_list = generators_helper(G1list(level), level, Mat2Z) + return [self(g, check=False) for g in gen_list] + else: + raise ValueError, "Unknown algorithm '%s' (should be either 'farey' or 'todd-coxeter')" % algorithm def __call__(self, x, check=True): r""" diff --git a/sage/modular/arithgroup/congroup_gammaH.py b/sage/modular/arithgroup/congroup_gammaH.py --- a/sage/modular/arithgroup/congroup_gammaH.py +++ b/sage/modular/arithgroup/congroup_gammaH.py @@ -408,44 +408,53 @@ return int(self.level() - 1) in v @cached_method - def generators(self): + def generators(self, algorithm="farey"): r""" - Return generators for this congruence subgroup. + Return generators for this congruence subgroup. The result is cached. + + INPUT: - The result is cached. + - ``algorithm`` (string): either ``farey`` (default) or + ``todd-coxeter``. + + If ``algorithm`` is set to ``"farey"``, then the generators will be + calculated using Farey symbols, which will always return a *minimal* + generating set. See :mod:`~sage.modular.arithgroup.farey_symbol` for + more information. + + If ``algorithm`` is set to ``"todd-coxeter"``, a simpler algorithm + based on Todd-Coxeter enumeration will be used. This tends to return + far larger sets of generators. EXAMPLE:: - sage: for g in GammaH(3, [2]).generators(): - ... print g - ... print '---' - [1 1] - [0 1] - --- - [-1 0] - [ 0 -1] - --- - [ 1 -1] - [ 0 1] - --- - [1 0] - [3 1] - --- - [1 1] - [0 1] - --- - [-1 0] - [ 3 -1] - --- - [ 1 0] - [-3 1] - --- + sage: [x.matrix() for x in GammaH(7, [2]).generators()] + [ + [1 1] [ 2 -1] [ 4 -3] + [0 1], [ 7 -3], [ 7 -5] + ] + sage: [x.matrix() for x in GammaH(7, [2]).generators(algorithm="todd-coxeter")] + [ + [1 1] [-90 29] [ 15 4] [-10 -3] [ 1 -1] [1 0] [1 1] [-3 -1] + [0 1], [301 -97], [-49 -13], [ 7 2], [ 0 1], [7 1], [0 1], [ 7 2], + [-13 4] [-5 -1] [-5 -2] [-10 3] [ 1 0] [ 9 -1] [-20 7] + [ 42 -13], [21 4], [28 11], [ 63 -19], [-7 1], [28 -3], [-63 22], + [1 0] [-3 -1] [ 15 -4] [ 2 -1] [ 22 -7] [-5 1] [ 8 -3] + [7 1], [ 7 2], [ 49 -13], [ 7 -3], [ 63 -20], [14 -3], [-21 8], + [11 5] [-13 -4] + [35 16], [-42 -13] + ] """ - from sage.modular.modsym.ghlist import GHlist - from congroup_pyx import generators_helper - level = self.level() - gen_list = generators_helper(GHlist(self), level, Mat2Z) - return [self(g, check=False) for g in gen_list] + if algorithm=="farey": + return self.farey_symbol().generators() + elif algorithm=="todd-coxeter": + from sage.modular.modsym.ghlist import GHlist + from congroup_pyx import generators_helper + level = self.level() + gen_list = generators_helper(GHlist(self), level, Mat2Z) + return [self(g, check=False) for g in gen_list] + else: + raise ValueError, "Unknown algorithm '%s' (should be either 'farey' or 'todd-coxeter')" % algorithm def _coset_reduction_data_first_coord(G): """ diff --git a/sage/modular/arithgroup/congroup_pyx.pyx b/sage/modular/arithgroup/congroup_pyx.pyx --- a/sage/modular/arithgroup/congroup_pyx.pyx +++ b/sage/modular/arithgroup/congroup_pyx.pyx @@ -283,7 +283,7 @@ EXAMPLES:: - sage: Gamma0(7).generators() # indirect doctest + sage: Gamma0(7).generators(algorithm="todd-coxeter") # indirect doctest [[1 1] [0 1], [-1 0] [ 0 -1], [ 1 -1] diff --git a/sage/modular/arithgroup/farey.cpp b/sage/modular/arithgroup/farey.cpp --- a/sage/modular/arithgroup/farey.cpp +++ b/sage/modular/arithgroup/farey.cpp @@ -248,9 +248,13 @@ a.push_back(0); b.push_back(1); for(size_t i=0; iis_member(SL2Z(0, -1, 1, 1)) ) + // index 2 even subgroup + generators.push_back(SL2Z(0, -1, 1, 1)); + else + // index 4 odd subgroup + generators.push_back(SL2Z(0, 1, -1, -1)); + generators.push_back(SL2Z(-1, 1, -1, 0)); coset.push_back(SL2Z::E); coset.push_back(SL2Z::T); cusp_classes.push_back(1); @@ -446,11 +450,12 @@ if( find(p.begin(), p.end(), pairing[i]) == p.end() ) { SL2Z m = pairing_matrix(i); if( not group->is_member(m) ) m = I*m; + if( pairing[i] == ODD and group->is_member(I) ) m = I*m; gen.push_back(m); if( pairing[i] > NO ) p.push_back(pairing[i]); } } - if( nu2() == 0 and group->is_member(I) ) gen.push_back(I); + if( nu2() == 0 and nu3() == 0 and group->is_member(I) ) gen.push_back(I); return gen; } diff --git a/sage/modular/arithgroup/farey_symbol.pyx b/sage/modular/arithgroup/farey_symbol.pyx --- a/sage/modular/arithgroup/farey_symbol.pyx +++ b/sage/modular/arithgroup/farey_symbol.pyx @@ -104,11 +104,10 @@ index 10 arithmetic subgroup given by Tim Hsu:: sage: from sage.modular.arithgroup.arithgroup_perm import HsuExample10 - sage: [g.matrix() for g in FareySymbol(HsuExample10()).generators()] [ - [1 2] [ 2 -1] [ 4 -3] [-1 0] - [0 1], [ 7 -3], [ 3 -2], [ 0 -1] + [1 2] [-2 1] [ 4 -3] + [0 1], [-7 3], [ 3 -2] ] Calculate the generators of the group `\Gamma' = @@ -166,21 +165,31 @@ ## of the group is called cdef int p if hasattr(group, "level"): p=group.level() - sig_on() if group == SL2Z: + sig_on() self.this_ptr = new cpp_farey() + sig_off() elif is_Gamma0(group): + sig_on() self.this_ptr = new cpp_farey(group, new is_element_Gamma0(p)) + sig_off() elif is_Gamma1(group): + sig_on() self.this_ptr = new cpp_farey(group, new is_element_Gamma1(p)) + sig_off() elif is_Gamma(group): + sig_on() self.this_ptr = new cpp_farey(group, new is_element_Gamma(p)) + sig_off() elif is_GammaH(group): + sig_on() l = group._GammaH_class__H self.this_ptr = new cpp_farey(group, new is_element_GammaH(p, l)) + sig_off() else: + sig_on() self.this_ptr = new cpp_farey(group) - sig_off() + sig_off() def __deallocpp__(self): r""" @@ -351,6 +360,18 @@ [ 1 0], [ 1 -1] ] + The unique index 2 even subgroup and index 4 odd subgroup each get handled correctly:: + + sage: [g.matrix() for g in FareySymbol(ArithmeticSubgroup_Permutation(S2="(1,2)", S3="()")).generators()] + [ + [ 0 -1] [-1 1] + [ 1 1], [-1 0] + ] + sage: [g.matrix() for g in FareySymbol(ArithmeticSubgroup_Permutation(S2="(1,2, 3, 4)", S3="(1,3)(2,4)")).generators()] + [ + [ 0 1] [-1 1] + [-1 -1], [-1 0] + ] """ return self.this_ptr.get_generators() diff --git a/sage/schemes/elliptic_curves/ell_rational_field.py b/sage/schemes/elliptic_curves/ell_rational_field.py --- a/sage/schemes/elliptic_curves/ell_rational_field.py +++ b/sage/schemes/elliptic_curves/ell_rational_field.py @@ -3215,9 +3215,6 @@ This map is actually a map on `X_0(N)`, so equivalent representatives in the upper half plane map to the same point:: - sage: Gamma0(15).gen(5) - [-7 -1] - [15 2] sage: phi((-7*z-1)/(15*z+2)) (8.20822465478524 - 13.1562816054681*I : -8.79855099049... + 69.4006129342...*I : 1.00000000000000)