Chabauty for symmetric powers of curves (Q1025423)

Let \(C\) be a smooth projective absolute irreducible curve of genus \(g \geq 2\) over a number field \(K\) and \(J\) its Jacobian variety. Let \(d\) be an integer \(\geq 1\) and denote by \(C^{(d)}\) the \(d\)-th symmetric power of \(C\). In the paper under review the method of Chabauty-Coleman is refined in order to compute \(C^{(d)}(K)\) in many cases. More precisely, the author assumes that \(\text{rank}\, J(K) \leq g-d\) and for a given nonarchimedean prime \(v\) of \(K\) gives a criterion for an element of \(C^{(d)}(K)\) to be the unique \(K\)-rational element in the corresponding fibre of the reduction map \(C^{(d)}(K_v)\mapsto C^{(d)}(k_v)\). To prove that the known elements of \(C^{(d)}(K)\) are all its elements, he uses this criterion with well-chosen primes \(v_1,\ldots,v_l\). Furthermore, he presents a modification of its method, in case where \(\rho : C \rightarrow C^{\prime}\) is a degree-\(d\) morphism defined over \(K\) and the genus of \(C^{\prime}\) is 0 or 1, in order to compute \(C^{(d)}(K)\setminus \rho^*(C^{\prime}(K))\) provided that \(\text{rank}\, J_C(K)-\text{rank}\,J_{C^{\prime}}(K) \leq g_C-g_{C^{\prime}}-d+1\).