Rational points on Picard groups of some genus-changing curves of genus at least 2 (Q1597949)

Let \(K\) be an algebraic function field in one variable with a finite constant field of characteristic \(p\geq 5\), and let \(C\) be an algebraic curve over \(K\) defined by the equation \(y^2=x^p+a\) (\(a\not \in K^p\)). It is a singular curve of absolute genus \(0\) computed over a separable closure \(K^{\text{sep}}\), and relative genus \(g=(p-1)/2\) computed over \(K\). By the result of \textit{J. F. Voloch} [Bull. Soc. Math. Fr. 119, 121-126 (1991; Zbl 0735.14018)] \(C\) has a finite number of points defined over \(K\). Let now \(\text{Pic}^0(C)\) be the Picard group of divisors of degree zero on \(C\), and let \(\text{Pic}^0_K(C)\) be a subgroup in \(\text{Pic}^0(C)\) consisting of those divisor classes which are invariant under the action of the absolute Galois group \(\text{Gal}(K^{\text{sep}}/K)\). The author constructs a group variety \(X\) over \(K\) of dimension \(g\), such that \(X\) has a finite number of \(K\)-rational points, and then embeds \(\text{Pic}^0_K(C)\) into \(X(K)\). Applying the Albanese map, this result also gives a proof of the finiteness of \(K\)-rational points on the curve \(C\).