Summary of Results and The Goal

Theorem 1 (Stein and Grigorov, Jorza, Patrikis, Tarnita)

1. The proof involves combining refinements of the theorems of Kolyvagin and Kato with explicit 2 and 3-descents.
   
   
2. Much work goes into just making this computation practical.
   
   
3. One can completely carry it out using SAGE, except for the 3-descents, which rely on code that Michael Stoll wrote for MAGMA, and three 4-Selmer group computations, which are also available nowhere but in MAGMA. These are the only curves in Cremona's book that conjecturally have nontrivial Sha:

      571a    681b   960d   960n
       4       9      4      4
   

Theorem 2 (Stein and Lum)

1. The proof in the rank 0 case is basically an application of a theorem of Rubin.
   
   
2. When E  has rank 1, a theorem of Mazur and Swinnerton-Dyer gives BSD(E;p)  at split primes, and for inert primes we do an explicit Heegner point calculation and use Kolyvagin's bound.
   
   

Anticipated Theorem 3 (Stein and Wuthrich)

1. Wuthrich implemented a program that should do this in Magma, and I am close to finishing one that does it in SAGE.
   
   
2. My goal is that the entire calculation can be completely done in SAGE (bounding the ranks of three 4 -Selmer groups seems worrisome...)

1. E  is a CM elliptic curve of rank 1  with conductor divisible by p , or
2. E  is a non-CM curve with additive reduction at p  such that either
   1. p  divides a Tamagawa number of E , or
   2. the representation �E;p  is reducible.

cremona = list(cremona_optimal_curves(range(1,1001)))

len(cremona)

2463 2463

cm = [e for e in cremona if e.has_cm()] len(cm)

44 44

cm1 = [e for e in cm if e.rank() == 1]; len(cm1)

19 19

bad_cm = [] for e in cm1: for p in prime_divisors(e.conductor()): if p >= 5: bad_cm.append((e.cremona_label(), p)) print len(bad_cm) print Sequence(bad_cm, cr=True)

10 [ ('121b1', 11), ('225a1', 5), ('361a1', 19), ('441b1', 7), ('441d1', 7), ('675a1', 5), ('784h1', 7), ('800h1', 5), ('800a1', 5), ('900c1', 5) ] 10 [ ('121b1', 11), ('225a1', 5), ('361a1', 19), ('441b1', 7), ('441d1', 7), ('675a1', 5), ('784h1', 7), ('800h1', 5), ('800a1', 5), ('900c1', 5) ]

bad_noncm = [] for e in cremona: if not e.has_cm(): for p, f in factor(e.conductor()): if p >= 5 and f >= 2: # additive prime >= 5. c = e.tamagawa_product() if c % p == 0: z = (e.cremona_label(), p, 'tamagawa'); print z bad_noncm.append(z) else: if e.is_reducible(p): z = (e.cremona_label(), p); print z bad_noncm.append(z) print len(bad_noncm)

('50a1', 5) ('50b1', 5, 'tamagawa') ('75a1', 5) ('75c1', 5, 'tamagawa') ('121a1', 11) ('121c1', 11) ('150a1', 5) ('150b1', 5) ('175a1', 5) ('175c1', 5) ('225e1', 5) ('225d1', 5) ('275b1', 5) ('294a1', 7) ('294b1', 7) ('325e1', 5, 'tamagawa') ('325d1', 5) ('400b1', 5) ('400c1', 5) ('450a1', 5) ('450b1', 5) ('450c1', 5) ('450d1', 5) ('490f1', 7) ('490k1', 7) ('550f1', 5) ('550k1', 5) ('550b1', 5) ('637a1', 7) ('637c1', 7) ('700d1', 5, 'tamagawa') ('735f1', 7, 'tamagawa') ('775c1', 5) ('882c1', 7) ('882d1', 7) ('950a1', 5) ('975j1', 5, 'tamagawa') 37 ('50a1', 5) ('50b1', 5, 'tamagawa') ('75a1', 5) ('75c1', 5, 'tamagawa') ('121a1', 11) ('121c1', 11) ('150a1', 5) ('150b1', 5) ('175a1', 5) ('175c1', 5) ('225e1', 5) ('225d1', 5) ('275b1', 5) ('294a1', 7) ('294b1', 7) ('325e1', 5, 'tamagawa') ('325d1', 5) ('400b1', 5) ('400c1', 5) ('450a1', 5) ('450b1', 5) ('450c1', 5) ('450d1', 5) ('490f1', 7) ('490k1', 7) ('550f1', 5) ('550k1', 5) ('550b1', 5) ('637a1', 7) ('637c1', 7) ('700d1', 5, 'tamagawa') ('735f1', 7, 'tamagawa') ('775c1', 5) ('882c1', 7) ('882d1', 7) ('950a1', 5) ('975j1', 5, 'tamagawa') 37