def get_twodim_cocycle(self,sign = 1,use_magma = True,bound = 5, hecke_data = None, return_all = False, outfile = None): F = self.group().base_ring() if F == QQ: F = NumberField(PolynomialRing(QQ,'x').gen(),names='r') component_list = [] good_components = [] if F.signature()[1] == 0 or (F.signature() == (0,1) and 'G' not in self.group()._grouptype): Tinf = self.hecke_matrix(oo).transpose() K = (Tinf-sign).kernel().change_ring(QQ) if K.dimension() >= 2: component_list.append((K, [(oo,Tinf)])) fwrite('Too charpoly = %s'%Tinf.charpoly().factor(),outfile) else: K = Matrix(QQ,self.dimension(),self.dimension(),0).kernel() if K.dimension() >= 2: component_list.append((K, [])) disc = self.S_arithgroup().Gpn._O_discriminant discnorm = disc.norm() try: N = ZZ(discnorm.gen()) except AttributeError: N = ZZ(discnorm) if F == QQ: x = QQ['x'].gen() F = NumberField(x,names='a') q = ZZ(1) g0 = None num_hecke_operators = 0 if hecke_data is not None: qq = F.ideal(hecke_data[0]) pol = hecke_data[1] Aq = self.hecke_matrix(qq.gens_reduced()[0], use_magma = use_magma).transpose().change_ring(QQ) fwrite('ell = (%s,%s), T_ell charpoly = %s'%(qq.norm(), qq.gens_reduced()[0], Aq.charpoly().factor()),outfile) U0 = component_list[0][0].intersection(pol.subs(Aq).left_kernel()) if U0.dimension() != 2: raise ValueError('Hecke data does not suffice to cut out space') good_component = (U0.denominator() * U0,component_list[0][1] + [(qq.gens_reduced()[0],Aq)]) row0 = good_component[0].matrix().rows()[0] col0 = [QQ(o) for o in row0.list()] clcm = lcm([o.denominator() for o in col0]) col0 = [ZZ(clcm * o ) for o in col0] # phi1 = sum([a * phi for a,phi in zip(col0,self.gens())],self(0)) # phi2 = self.apply_hecke_operator(phi1,qq.gens_reduced()[0], use_magma = use_magma) # return [phi1, phi2], [(ell, o.restrict(good_component[0])) for ell, o in good_component[1]] flist = [] for row0 in good_component[0].matrix().rows(): col0 = [QQ(o) for o in row0.list()] clcm = lcm([o.denominator() for o in col0]) col0 = [ZZ(clcm * o ) for o in col0] flist.append(sum([a * phi for a,phi in zip(col0,self.gens())],self(0))) return flist,[(ell, o.restrict(good_component[0])) for ell, o in good_component[1]] while len(component_list) > 0 and num_hecke_operators < bound: verbose('num_hecke_ops = %s'%num_hecke_operators) verbose('len(components_list) = %s'%len(component_list)) q = q.next_prime() verbose('q = %s'%q) fact = F.ideal(q).factor() dfact = F.ideal(disc.gens_reduced()[0]).factor() for qq,e in fact: verbose('Trying qq = %s'%qq) if qq in [o for o,_ in dfact]: verbose('Skipping because qq divides D...') continue if ZZ(qq.norm()).is_prime() and not qq.divides(F.ideal(disc.gens_reduced()[0])): try: Aq = self.hecke_matrix(qq.gens_reduced()[0],g0 = g0,use_magma = use_magma).transpose().change_ring(QQ) except (RuntimeError,TypeError) as e: verbose('Skipping qq (=%s) because Hecke matrix could not be computed...'%qq.gens_reduced()[0]) continue except KeyboardInterrupt: verbose('Skipping qq (=%s) by user request...'%qq.gens_reduced()[0]) num_hecke_operators += 1 sleep(1) continue verbose('Computed hecke matrix at qq = %s'%qq) fwrite('ell = (%s,%s), T_ell charpoly = %s'%(qq.norm(), qq.gens_reduced()[0], Aq.charpoly().factor()),outfile) old_component_list = component_list component_list = [] num_hecke_operators += 1 for U,hecke_data in old_component_list: V = Aq.decomposition_of_subspace(U) for U0,is_irred in V: if U0.dimension() == 1: continue if U0.dimension() == 2 and is_irred: good_components.append((U0.denominator() * U0,hecke_data+[(qq.gens_reduced()[0],Aq)])) else: # U0.dimension() > 2 or not is_irred component_list.append((U0.denominator() * U0,hecke_data + [(qq.gens_reduced()[0],Aq)])) if len(good_components) > 0 and not return_all: flist = [] for row0 in good_components[0][0].matrix().rows(): col0 = [QQ(o) for o in row0.list()] clcm = lcm([o.denominator() for o in col0]) col0 = [ZZ(clcm * o ) for o in col0] flist.append(sum([a * phi for a,phi in zip(col0,self.gens())],self(0))) return flist,[(ell, o.restrict(good_components[0][0])) for ell, o in good_components[0][1]] if len(component_list) == 0 or num_hecke_operators >= bound: break if len(good_components) == 0: if not return_all: raise ValueError('Group does not seem to be attached to an abelian surface') else: return [] if return_all: ans = [] for K,hecke_data in good_components: flist = [] for row0 in K.matrix().rows(): col0 = [QQ(o) for o in row0.list()] clcm = lcm([o.denominator() for o in col0]) col0 = [ZZ(clcm * o ) for o in col0] flist.append(sum([a * phi for a,phi in zip(col0,self.gens())],self(0))) ans.append((flist,[(ell, o.restrict(K)) for ell, o in hecke_data])) return ans else: flist = [] for row0 in good_components[0][0].matrix().rows(): col0 = [QQ(o) for o in row0.list()] clcm = lcm([o.denominator() for o in col0]) col0 = [ZZ(clcm * o ) for o in col0] flist.append(sum([a * phi for a,phi in zip(col0,self.gens())],self(0))) return flist,[(ell, o.restrict(good_components[0][0])) for ell, o in good_components[0][1]]