A Note on the Growth of Sha in Dihedral Extensions
Abstract
We provide a formula for the order of the Tate--Shafarevich group of elliptic curves over dihedral extensions of number fields of order 2n, up to 4th powers and primes dividing n. Specifically, for odd n it is equal to the order of the Tate--Shafarevich group over the quadratic subextension. A similar formula holds for even n.
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