Abstract: Let $\ell$ be an odd prime, and for a non-zero integer $A$, let $C_A$ be the normalisation of the curve given by the affine equation $y^2 = x^\ell + A$, and let $J_A$ be its Jacobian, which is a $\half(\ell - 1)$--dimensional abelian variety defined over $\Q$. We use a method invented by Ed Schaefer to compute the $(1-\zeta_\ell)$--Selmer group $\Sel^{(1-\zeta_\ell)}(K, J_A)$ of $J_A$ over $K = \Q(\zeta_\ell)$ under suitable hypotheses on $A$. This leads to bounds for the Mordell--Weil ranks of $J_A(K)$ and of $J_A(\Q)$.

If $\ell = 3$, $J_A = E_A$ is an elliptic curve. The family of elliptic curves $E_A : y^2 = x^3 + A$ has been the subject of extensive study. We give an overview of the results on the rank of $E_A(\Q)$ in the (recent) literature and relate them to the results obtained here in this special case.

If $\ell = 5$, we obtain conditions on $A$ under which the Mordell--Weil ranks of $J_A(\Q)$ and of $J_A(K)$ are zero. Together with some computations of $2$--Selmer groups, this result can be used to show the existence of non-trivial 2--torsion in $\Sha(\Q, J_A)$ for certain values of~$A$.

File: cm-selmer-new.dvi.gz, needs to be gunzipped.

Michael Stoll <stoll@math.uni-duesseldorf.de>