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# Summary of Results and The Goal

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

 Theorem 1: Suppose that E  is a non-CM elliptic curve of rank Ô1 , conductor Ô1000  and that p  is a prime. If p  is odd, assume further that the mod p  representation ÚE;p  is irreducible and p  does not divide any Tamagawa number of E . Then BSD(E;p)  is true.

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)

 Theorem 2: Suppose that E  is a CM elliptic curve of rank Ô1  (59 rank 0 and 56 rank 1), conductor Ô5000  and that pÕ5  is a prime of good reduction for E . Then BSD(E;p)  is true.

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)

 Anticipaed Theorem 3: Suppose that E  is a non-CM elliptic curve of rank Ô1 , conductor Ô1000  and that pÕ5  is a prime that divides a Tamagawa number of E  or such that the mod p  representation ÚE;p  is reducible. Assume further that E  has good ordinary or bad multiplicative reduction at p . Then BSD(E;p)  is true.

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...)

Once we finish the proof of Theorem 3, we will almost have:

 Eventual Goal: Suppose that E  is an elliptic curve of rank Ô1  in Cremona's book. Then the full Birch and Swinnerton-Dyer conjecture is true for E .

In theory, all that remains is to deal with pairs (E;p)  with pÕ5  such that:
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 ÚE;p  is reducible.

## What is Left?

We finish the talk by computing all pairs (E;p)  in Cremona's book so that none of the above results (or their known refiniments) prove BSD(E;p) . This is a concrete challenge. We compute a list of all the (optimal) curves in Cremona's book:
There are 2463 of them.
 `2463` `2463`
Only 44 have CM.
 `44` `44`
Of these, 19 have rank 1:
 `19` `19`
 ```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) ]```
 ```('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```

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