## Second $$p$$-descents on elliptic curves

### Brendan Creutz

#### Abstract

Let $$p$$ be a prime and let $$C$$ be a genus one curve over a number field $$k$$ representing an element of order dividing $$p$$ in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of $$D$$ in the Shafarevich-Tate group such that $$pD = C$$ and obtains explicit models for these $$D$$ as curves in projective space. This leads to a practical algorithm for performing explicit 9-descents on elliptic curves over $$\mathbb{Q}$$

Keywords: elliptic curves, descent, Shafarevich-Tate group.

: Primary 11G05; secondary 11Y50.

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 Monday, October 29, 2012