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title
William Stein's Number Theory Research Blog
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2026-02-27 10:34:34
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William Stein's Number Theory Research Blog skip to main | skip to sidebar William Stein's Number Theory Research Blog 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. 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 v...
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