#### Vol. 1, No. 1, 2013

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Explicit descent in the Picard group of a cyclic cover of the projective line

### Brendan Creutz

Vol. 1 (2013), No. 1, 295–315
##### Abstract

Given a curve $X$ of the form ${y}^{p}=h\left(x\right)$ over a number field, one can use descents to obtain explicit bounds on the Mordell-Weil rank of the Jacobian or to prove that the curve has no rational points. We show how, having performed such a descent, one can easily obtain additional information which may rule out the existence of rational divisors on $X$ of degree prime to $p$. This can yield sharper bounds on the Mordell-Weil rank by demonstrating the existence of nontrivial elements in the Shafarevich-Tate group. As an example we compute the Mordell-Weil rank of the Jacobian of a genus $\mathfrak{4}$ curve over $ℚ$ by determining that the $\mathfrak{3}$-primary part of the Shafarevich-Tate group is isomorphic to $ℤ∕\mathfrak{3}×ℤ∕\mathfrak{3}$.

##### Keywords
abelian variety, Mordell-Weil group, explicit descent
Primary: 11G10
Secondary: 11Y50
##### Milestones
Published: 14 November 2013
##### Authors
 Brendan Creutz School of Mathematics and Statistics University of Sydney Sydney, NSW 2006 Australia