def problems(E): for d in E.local_data(): problems_local(E, d.prime()) def problems_local(E, P): print "at", P, "we feel", K = E.base_field() d = E.local_data(P) if d.has_good_reduction(): print "good :)" elif d.has_nonsplit_multiplicative_reduction(): print "nonsplit multiplicative :)" elif d.has_split_multiplicative_reduction(): print "split multiplicative :)" else: # additive reduction j = E.j_invariant() jv = j.valuation(P) if not K is QQ else valuation(j, P.gen()) if jv < 0: print "bad but potential multiplicative" # potential multiplicative reduction return else: # potential good reduction p = P if K is QQ else P.smallest_integer() if p==2: # prime | 2 t = polygen(K) assert E.a1() == 0 assert E.a3() == 0 f = t^3+E.a2()*t^2+E.a4()*t+E.a6() d = 1 roots = [] for g,_ in f.factor(): if g.degree() == 1: roots += g.roots(multiplicities=False) if g.degree() == 2: d = 2 K2. = K.extension(g) # roots += g.change_ring(K2).roots(multiplicities=False) if g.degree() == 3: d = 3 K3. = K.extension(g) for h,_ in g.change_ring(K3).factor(): if h.degree() == 1: # roots += h.roots(multiplicities=False) pass if h.degree() == 2: d = 6 K6. = K3.extension(h) print "bad and potential good -- at 2, d=", d, " :(" return if p==3: # prime | 3 print "bad and potential good -- at 3 :(" return print "bad and potential good" # def _root_number_local_3(E, P): # def _val(a, P): # if P.base_ring() is ZZ: # return valuation(a, P.gen()) # return a.valuation(P) # def quadr_residue_symbol(a): # return Kr(a).is_square() and 1 or -1 # def quadr_symbol(a): # av = _val(a, P) # if av%2==1: # return -1 # if P.base_ring() is ZZ: # pi = P.gen() # else: # pi = P.number_field().uniformizer(P) # # print "piv = ", pi.valuation(P) # b = a / pi^av # return quadr_residue_symbol(b) # # assume potential good reduction # Es = E.local_data(P).minimal_model(tidy=False) # Es = Es.short_weierstrass_model(complete_cube=False) # disc = Es.discriminant() # vD = _val(disc, P) # Kr = P.residue_field() # ks = str(Es.kodaira_symbol(P)) # if ks=='I0' or ks=='I0*': # assert vD%2==0, "error -- valuation of discriminant should be even" # return quadr_residue_symbol(-1)^(vD/2) # elif ks=='III' or ks=='III*': # return quadr_residue_symbol(-2) # else: # assert ks=='II' or ks=='II*' or ks=='IV' or ks=='IV*', "error -- unexpected kodeira symbol" # # print "vD = ", vD, disc # d = quadr_symbol(disc) # c = Es.a6() # cv = _val(c, P) # assert cv%3!=0, "error -- 3 should not divide the valuation of the constant term anymore" # # print disc, c, P # hs = generalized_hilbert_symbol(disc, c, P) # # hs = hilbert_symbol_magma(disc, c, P) # qr1 = quadr_residue_symbol(cv)^vD # qr2 = quadr_residue_symbol(-1)^(vD*(vD-1)/2) # # print d, hs, qr1, qr2 # return d*hs*qr1*qr2 # def transl_ell_curve_w_2_isog(E, rt): # Echange = E.change_weierstrass_model(1,rt,0,0) # E2 = EllipticCurve(E.base_field(), [0,-2*Echange.a2(),0,Echange.a2()^2-4*Echange.a4(),0]) # # print E.is_isogenous(E2) # return E2 # def _root_number_local_2(E, P): # K = E.base_field() # if K is QQ: # rn = -1 # else: # rn = (-1)^(P.residue_class_degree() * P.ramification_index()) # t = polygen(K) # assert E.a1() == 0 # assert E.a3() == 0 # f = t^3+E.a2()*t^2+E.a4()*t+E.a6() # d = 1 # roots = [] # for g,_ in f.factor(): # if g.degree() == 1: # roots += g.roots(multiplicities=False) # if g.degree() == 2: # d = 2 # K2. = K.extension(g) # # roots += g.change_ring(K2).roots(multiplicities=False) # if g.degree() == 3: # d = 3 # K3. = K.extension(g) # for h,_ in g.change_ring(K3).factor(): # if h.degree() == 1: # # roots += h.roots(multiplicities=False) # pass # if h.degree() == 2: # d = 6 # K6. = K3.extension(h) # # roots += h.change_ring(K6).roots(multiplicities=False) # print "d = ", d # def _compute_H(E, P, ord2=True, ord3=False): # t = E.tamagawa_number(P) # if ord2: # cv = valuation(t,2) + _u_of_E(E, P) # H2 = 1 if cv%2==0 else -1 # if ord3: # cv = valuation(t,3) # H3 = 1 if cv%2==0 else -1 # if ord2 and ord3: return (H2, H3) # if ord2: return H2 # return H3 # def _val(a, P): # if P.base_ring() is ZZ: # return valuation(a, P.gen()) # return a.valuation(P) # def _u_of_E(E, P): # disc = E.discriminant() # Emin = E.local_data(P).minimal_model(tidy=False) # discmin = Emin.discriminant() # assert _val(discmin, P) == E.local_data(P)._val_disc # # print "quot = ", disc/discmin # # return (disc/discmin).norm()^(1/12) # # print "quot = ", (disc/discmin).valuation(P) / 12 # # print "rd = ", P.residue_class_degree() # # if not E.base_field() is QQ: # # print P.ramification_index() # if E.base_field() is QQ: # return Integer(_val(disc/discmin, P) / 12) # else: # return Integer(P.residue_class_degree() * _val(disc/discmin, P) / 12) # if d==1: # # print roots # H = _compute_H(E, P) # E1 = transl_ell_curve_w_2_isog(E, roots[0]) # H1 = _compute_H(E1, P) # E2 = transl_ell_curve_w_2_isog(E, roots[1]) # H2 = _compute_H(E2, P) # E3 = transl_ell_curve_w_2_isog(E, roots[2]) # H3 = _compute_H(E3, P) # return rn * H * H1 * H2 * H3 # elif d==2: # H = _compute_H(E, P) # # 2-isogenous curve over base field # # print roots[0] # E1 = transl_ell_curve_w_2_isog(E, roots[0]) # root[0] is the root over the base field # # print E1 # H1 = _compute_H(E1, P) # # a2 in K2 is root not in base field # K2b. = K2.absolute_field() # from2, to2 = K2b.structure() # # Unfairly picking one prime factor # Pfactors = K2b.ideal([ to2(pgen) for pgen in P.gens() ]).factor() # print Pfactors # P2b = K2b.ideal([ to2(pgen) for pgen in P.gens() ]).factor()[0][0] # E2b = E.change_ring(K2b) # # print E2b # H2 = _compute_H(E2b, P2b) # # print "H2 = ", H2 # E3b = transl_ell_curve_w_2_isog(E2b, to2(a2)) # assert E3b.a6() == 0 # H3 = _compute_H(E3b, P2b) # return rn * H * H1 * H2 * H3 # elif d==3: # # a3 in K3 # K3b. = K3.absolute_field() # from3, to3 = K3b.structure() # # Unfairly picking one prime factor # # print K3b.ideal([ to3(pgen) for pgen in P.gens() ]).factor() # P3b = K3b.ideal([ to3(pgen) for pgen in P.gens() ]).factor()[0][0] # E3b = E.change_ring(K3b) # H = _compute_H(E3b, P3b) # # print "H = ", H # E3bt = transl_ell_curve_w_2_isog(E3b, to3(a3)) # H2 = _compute_H(E3bt, P3b) # # print "H2 = ", H2 # return rn * H * H2 # elif d==6: # # a3 in K3 # K3b. = K3.absolute_field() # from3, to3 = K3b.structure() # # Unfairly picking one prime factor # P3b = K3b.ideal([ to3(pgen) for pgen in P.gens() ]).factor()[0][0] # E3b = E.change_ring(K3b) # H = _compute_H(E3b, P3b) # E3bt = transl_ell_curve_w_2_isog(E3b, to3(a3)) # H2 = _compute_H(E3bt, P3b) # # a6 in K6 # K6b. = K6.absolute_field() # from6, to6 = K6b.structure() # # Unfairly picking one prime factor # P6b = K6b.ideal([ to6(pgen) for pgen in P.gens() ]).factor()[0][0] # E6b = E.change_ring(K6b) # H3, H33 = _compute_H(E6b, P6b, ord2=True, ord3=True) # E6bt = transl_ell_curve_w_2_isog(E6b, to6(a6)) # H4 = _compute_H(E6bt, P6b) # x = polygen(K) # Kd. = K.extension(x^2 - E.discriminant()) # KDb. = Kd.absolute_field() # fromD, toD = KDb.structure() # # Unfairly picking one prime factor # PDb = KDb.ideal([ toD(pgen) for pgen in P.gens() ]).factor()[0][0] # EDb = E.change_ring(KDb) # HD = _compute_H(EDb, PDb, ord2=False, ord3=True) # return rn * H * H2 * H3 * H33 * H4 * HD # raise NotImplementedError("additive reduction over a prime dividing 2")