On the Shafarevich-Tate group of the Jacobian of a quotient of the Fermat curve
Let \(F_{a,b,c}\) be a complete non-singular model over \({\mathbb{Q}}\) of the plane curve defined by the equation \(y^ p=x^ a(1-x)^ b\), where p is an odd prime integer and a,b,c are integers with \(0<a,b<p\) and \(a+b+c=0\). The author shows that the Shafarevich-Tate group {Russian{Sh}} of the Jacobian variety over \({\mathbb{Q}}(\mu_ p)\) of \(F_{a,b,c}\) contains a subgroup isomorphic to (\({\mathbb{Z}}/p{\mathbb{Z}})\oplus ({\mathbb{Z}}/p{\mathbb{Z}})\), provided that \(p\equiv 1\quad (\pmod 4),\) \(p\nmid B_{(p-1)/2}\cdot B_{(p+3)/2}\) and \(-2abc\{(a^ ab^ bc^ c)^{p-1}-1\}/p\in {\mathbb{F}}_ p^{*2}\), where \(B_ i\) is the i-th Bernoulli number. He gets the result by showing that the Cassels-Tate pairing is non-trivial in the case of \(F_{a,b,c}\) over \({\mathbb{Q}}_ p(\mu_ p)\) having a wild split reduction.
scientific article; zbMATH DE number 4079614
scientific article