- Prove there are infinitely many pairs (E,p) such that: (a) E(Q) has rank >= 2, and (b) Sha(E/Q)(p) is finite.
- Prove there are infinitely many pairs (E,p) such that: (a) E(Q) has rank >= 2, and (b) Sha(E/Q)(p) is nonzero.
- Prove there are infinitely many pairs (E,p) such that: (a) E(Q) has rank >= 2, and (b) Sha(E/Q)(p) is nonzero and finite.
- Prove 2 assuming the full BSD conjecture. (1 is trivially implied by BSD and the existence of families of curves with 2 marked points.)
- Prove 1 assuming the BSD rank conjecture, but not the BSD formula.
- Same questions but over a number field K.
- Same questions but for triples (E,p,K), where E is over K and we vary over all K.

## Monday, January 31, 2011

### Higher rank curves with nontrivial Sha

I wonder if all of the problems are open. Here we consider pairs (E,p) where E is an elliptic curve over Q and p is a prime.

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## 1 comment:

(1) follows from "the parity conjecture", by which I mean that the rank of E(Q) has the parity predicted by the functional equation.

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