attachment:demo.sage of days22/dokchitser

   1 def problems(E):
   2     for d in E.local_data():
   3         problems_local(E, d.prime())
   4 
   5 def problems_local(E, P):
   6     print "at", P, "we feel",
   7     K = E.base_field()
   8     d = E.local_data(P)
   9     if d.has_good_reduction():
  10         print "good :)"
  11     elif d.has_nonsplit_multiplicative_reduction():
  12         print "nonsplit multiplicative :)"
  13     elif d.has_split_multiplicative_reduction():
  14         print "split multiplicative :)"
  15     else: # additive reduction
  16         j = E.j_invariant()
  17         jv = j.valuation(P) if not K is QQ else valuation(j, P.gen())
  18         if jv < 0:
  19             print "bad but potential multiplicative"
  20             # potential multiplicative reduction
  21             return
  22         else:
  23             # potential good reduction
  24             p = P if K is QQ else P.smallest_integer()
  25             if p==2: # prime | 2
  26                 t = polygen(K)
  27                 assert E.a1() == 0
  28                 assert E.a3() == 0
  29                 f = t^3+E.a2()*t^2+E.a4()*t+E.a6()
  30 
  31                 d = 1
  32                 roots = []
  33                 for g,_ in f.factor():
  34                     if g.degree() == 1:
  35                         roots += g.roots(multiplicities=False)
  36                     if g.degree() == 2:
  37                         d = 2
  38                         K2.<a2> = K.extension(g)
  39                         # roots += g.change_ring(K2).roots(multiplicities=False)
  40                     if g.degree() == 3:
  41                         d = 3
  42                         K3.<a3> = K.extension(g)
  43                         for h,_ in g.change_ring(K3).factor():
  44                             if h.degree() == 1:
  45                                 # roots += h.roots(multiplicities=False)
  46                                 pass
  47                             if h.degree() == 2:
  48                                 d = 6
  49                                 K6.<a6> = K3.extension(h)
  50                 print "bad and potential good -- at 2, d=", d, " :("
  51                 return
  52             if p==3: # prime | 3
  53                 print "bad and potential good -- at 3 :("
  54                 return
  55 
  56             print "bad and potential good"
  57 
  58 
  59 # def _root_number_local_3(E, P):
  60 #     def _val(a, P):
  61 #         if P.base_ring() is ZZ:
  62 #             return valuation(a, P.gen())
  63 #         return a.valuation(P)
  64 #     def quadr_residue_symbol(a):
  65 #         return Kr(a).is_square() and 1 or -1
  66 #     def quadr_symbol(a):
  67 #         av =  _val(a, P)
  68 #         if av%2==1:
  69 #             return -1
  70 #         if P.base_ring() is ZZ:
  71 #             pi = P.gen()
  72 #         else:
  73 #             pi = P.number_field().uniformizer(P)
  74 #         # print "piv = ", pi.valuation(P)
  75 #         b = a / pi^av
  76 #         return quadr_residue_symbol(b)
  77 #     # assume potential good reduction
  78 #     Es = E.local_data(P).minimal_model(tidy=False)
  79 #     Es = Es.short_weierstrass_model(complete_cube=False)
  80 
  81 #     disc = Es.discriminant()
  82 #     vD = _val(disc, P)
  83 #     Kr = P.residue_field()
  84 
  85 #     ks = str(Es.kodaira_symbol(P))
  86 #     if ks=='I0' or ks=='I0*':
  87 #         assert vD%2==0, "error -- valuation of discriminant should be even"
  88 #         return quadr_residue_symbol(-1)^(vD/2)
  89 #     elif ks=='III' or ks=='III*':
  90 #         return quadr_residue_symbol(-2)
  91 #     else: 
  92 #         assert ks=='II' or ks=='II*' or ks=='IV' or ks=='IV*', "error -- unexpected kodeira symbol"
  93 #         # print "vD = ", vD, disc
  94 #         d = quadr_symbol(disc)
  95 #         c = Es.a6()
  96 #         cv = _val(c, P)
  97 #         assert cv%3!=0, "error -- 3 should not divide the valuation of the constant term anymore"
  98 #         # print disc, c, P
  99 #         hs = generalized_hilbert_symbol(disc, c, P)
 100 #         # hs = hilbert_symbol_magma(disc, c, P)
 101 #         qr1 = quadr_residue_symbol(cv)^vD
 102 #         qr2 = quadr_residue_symbol(-1)^(vD*(vD-1)/2)
 103 #         # print d, hs, qr1, qr2
 104 #         return d*hs*qr1*qr2
 105 
 106 
 107 # def transl_ell_curve_w_2_isog(E, rt):
 108 #     Echange = E.change_weierstrass_model(1,rt,0,0)
 109 #     E2 = EllipticCurve(E.base_field(), [0,-2*Echange.a2(),0,Echange.a2()^2-4*Echange.a4(),0])
 110 #     # print E.is_isogenous(E2)
 111 #     return E2
 112 
 113 # def _root_number_local_2(E, P):
 114 #     K = E.base_field()
 115 
 116 #     if K is QQ:
 117 #         rn = -1
 118 #     else:
 119 #         rn = (-1)^(P.residue_class_degree() * P.ramification_index())
 120 
 121 #     t = polygen(K)
 122 #     assert E.a1() == 0
 123 #     assert E.a3() == 0
 124 #     f = t^3+E.a2()*t^2+E.a4()*t+E.a6()
 125 
 126 #     d = 1
 127 #     roots = []
 128 #     for g,_ in f.factor():
 129 #         if g.degree() == 1:
 130 #             roots += g.roots(multiplicities=False)
 131 #         if g.degree() == 2:
 132 #             d = 2
 133 #             K2.<a2> = K.extension(g)
 134 #             # roots += g.change_ring(K2).roots(multiplicities=False)
 135 #         if g.degree() == 3:
 136 #             d = 3
 137 #             K3.<a3> = K.extension(g)
 138 #             for h,_ in g.change_ring(K3).factor():
 139 #                 if h.degree() == 1:
 140 #                     # roots += h.roots(multiplicities=False)
 141 #                     pass
 142 #                 if h.degree() == 2:
 143 #                     d = 6
 144 #                     K6.<a6> = K3.extension(h)
 145 #                     # roots += h.change_ring(K6).roots(multiplicities=False)
 146 
 147 #     print "d = ", d
 148 
 149 #     def _compute_H(E, P, ord2=True, ord3=False):
 150 #         t = E.tamagawa_number(P)
 151 #         if ord2:
 152 #             cv = valuation(t,2) + _u_of_E(E, P)
 153 #             H2 = 1 if cv%2==0 else -1
 154 #         if ord3:
 155 #             cv = valuation(t,3)
 156 #             H3 = 1 if cv%2==0 else -1
 157 #         if ord2 and ord3: return (H2, H3)
 158 #         if ord2: return H2
 159 #         return H3
 160 
 161 #     def _val(a, P):
 162 #         if P.base_ring() is ZZ:
 163 #             return valuation(a, P.gen())
 164 #         return a.valuation(P)
 165 
 166 #     def _u_of_E(E, P):
 167 #         disc = E.discriminant()
 168 #         Emin = E.local_data(P).minimal_model(tidy=False)
 169 #         discmin = Emin.discriminant()
 170 #         assert _val(discmin, P) == E.local_data(P)._val_disc
 171 #         # print "quot = ", disc/discmin
 172 #         # return (disc/discmin).norm()^(1/12)
 173 #         # print "quot = ", (disc/discmin).valuation(P) / 12
 174 #         # print "rd = ", P.residue_class_degree()
 175 #         # if not E.base_field() is QQ:
 176 #             # print P.ramification_index()
 177 #         if E.base_field() is QQ:
 178 #             return Integer(_val(disc/discmin, P) / 12)
 179 #         else:
 180 #             return Integer(P.residue_class_degree() * _val(disc/discmin, P) / 12)
 181 
 182 #     if d==1:
 183 #         # print roots
 184 #         H = _compute_H(E, P)
 185 #         E1 = transl_ell_curve_w_2_isog(E, roots[0])
 186 #         H1 = _compute_H(E1, P)
 187 #         E2 = transl_ell_curve_w_2_isog(E, roots[1])
 188 #         H2 = _compute_H(E2, P)
 189 #         E3 = transl_ell_curve_w_2_isog(E, roots[2])
 190 #         H3 = _compute_H(E3, P)
 191 #         return rn * H * H1 * H2 * H3
 192 #     elif d==2:
 193 #         H = _compute_H(E, P)
 194 #         # 2-isogenous curve over base field
 195 #         # print roots[0]
 196 #         E1 = transl_ell_curve_w_2_isog(E, roots[0]) # root[0] is the root over the base field
 197 #         # print E1
 198 #         H1 = _compute_H(E1, P)
 199 
 200 #         # a2 in K2 is root not in base field
 201 #         K2b.<b2> = K2.absolute_field()
 202 #         from2, to2 = K2b.structure()
 203 #         # Unfairly picking one prime factor
 204 #         Pfactors = K2b.ideal([ to2(pgen) for pgen in P.gens() ]).factor()
 205 #         print Pfactors
 206 #         P2b = K2b.ideal([ to2(pgen) for pgen in P.gens() ]).factor()[0][0]
 207 #         E2b = E.change_ring(K2b)
 208 #         # print E2b
 209 #         H2 = _compute_H(E2b, P2b)
 210 #         # print "H2 = ", H2
 211 
 212 #         E3b = transl_ell_curve_w_2_isog(E2b, to2(a2))
 213 #         assert E3b.a6() == 0
 214 #         H3 = _compute_H(E3b, P2b)
 215 
 216 #         return rn * H * H1 * H2 * H3
 217 #     elif d==3:
 218 #         # a3 in K3
 219 #         K3b.<b3> = K3.absolute_field()
 220 #         from3, to3 = K3b.structure()
 221 #         # Unfairly picking one prime factor
 222 #         # print K3b.ideal([ to3(pgen) for pgen in P.gens() ]).factor()
 223 #         P3b = K3b.ideal([ to3(pgen) for pgen in P.gens() ]).factor()[0][0]
 224 #         E3b = E.change_ring(K3b)
 225 #         H = _compute_H(E3b, P3b)
 226 #         # print "H = ", H
 227 #         E3bt = transl_ell_curve_w_2_isog(E3b, to3(a3))
 228 #         H2 = _compute_H(E3bt, P3b)
 229 #         # print "H2 = ", H2
 230 #         return rn * H * H2
 231 #     elif d==6:
 232 #         # a3 in K3
 233 #         K3b.<b3> = K3.absolute_field()
 234 #         from3, to3 = K3b.structure()
 235 #         # Unfairly picking one prime factor
 236 #         P3b = K3b.ideal([ to3(pgen) for pgen in P.gens() ]).factor()[0][0]
 237 #         E3b = E.change_ring(K3b)
 238 #         H = _compute_H(E3b, P3b)
 239 
 240 #         E3bt = transl_ell_curve_w_2_isog(E3b, to3(a3))
 241 #         H2 = _compute_H(E3bt, P3b)
 242 
 243 #         # a6 in K6
 244 #         K6b.<b6> = K6.absolute_field()
 245 #         from6, to6 = K6b.structure()
 246 #         # Unfairly picking one prime factor
 247 #         P6b = K6b.ideal([ to6(pgen) for pgen in P.gens() ]).factor()[0][0]
 248 #         E6b = E.change_ring(K6b)
 249 #         H3, H33 = _compute_H(E6b, P6b, ord2=True, ord3=True)
 250 
 251 #         E6bt = transl_ell_curve_w_2_isog(E6b, to6(a6))
 252 #         H4 = _compute_H(E6bt, P6b)
 253 
 254 #         x = polygen(K)
 255 #         Kd.<d> = K.extension(x^2 - E.discriminant())
 256 #         KDb.<bd> = Kd.absolute_field()
 257 #         fromD, toD = KDb.structure()
 258 #         # Unfairly picking one prime factor
 259 #         PDb = KDb.ideal([ toD(pgen) for pgen in P.gens() ]).factor()[0][0]
 260 #         EDb = E.change_ring(KDb)
 261 #         HD = _compute_H(EDb, PDb, ord2=False, ord3=True)
 262 
 263 #         return rn * H * H2 * H3 * H33 * H4 * HD
 264 
 265 #     raise NotImplementedError("additive reduction over a prime dividing 2")