We describe a method of bounding the Mordell–Weil rank of an elliptic curve E over a number field k. The result of this method may improve upon an upper bound from the p-Selmer group for some odd...Show moreWe describe a method of bounding the Mordell–Weil rank of an elliptic curve E over a number field k. The result of this method may improve upon an upper bound from the p-Selmer group for some odd prime number p and involves an expression for the Cassels–Tate pairing on X(E/k) in terms of certain local pairings, one for each place v of k, which we call Tate local pairings. For each odd prime number p we give explicit formulas for the Tate local pairings both in the case where all p-torsion of E is locally defined over the base field and for the more general case. We prove that in the case where all p-torsion is rational the formula for the general case also suffices. This means that the elements in the two formulas differ by the norm of some element. We conjecture which element this should be and prove our conjecture for small primes.Show less
