Descent on elliptic surfaces and arithmetic bounds for the Mordell-Weil rank (Q2133388)

Let \(S\) be a smooth, projective and geometrically integral curve over a characteristic not equal to \(2\) or \(3\) field \(k\). Let \(E\) be a nonconstant elliptic curve over \(k(S)\), the function field of \(S\). In the article under review, the authors' main result gives a bound for the Mordell-Weil rank of \(E\). The bound depends on \(k\) and is obtained via the authors' function field \(p\)-descent techniques. These \(p\)-descent techniques are analogous to the number field case. To describe the authors' main result, in further detail, let \(p \not = \operatorname{char}(k)\) be an odd prime number and let \(E[p]\) be the group of \(p\)-torsion points of \(E\). Let \(\mathcal{E} \rightarrow S\) be the Néron model of \(E\) over \(S\) and \(\mathcal{E}[p]\rightarrow S\) its group scheme of \(p\)-torsion points. Let \(\Sigma \subset S\) be the set of places of bad reduction of \(\mathcal{E}\). Furthermore, assume that the Galois group \(\operatorname{Gal}\left(\overline{k(S)} / \overline{k}(S)\right)\) acts transitively on \(E[p] \setminus \{0\}\), let \(C\) be the smooth compactification of \(\mathcal{E}[p] \setminus \{0\}\) and let \(C^+\) be the quotient of \(C\) by the involution \(P \mapsto - P\). Then, with this notation and hypothesis, the authors' prove that \[ \operatorname{rk}_{\mathbb{Z}} E(k(S)) \leq \operatorname{dim}_{\mathbb{F}_p} \operatorname{Pic}(C)[p] - \operatorname{dim}_{\mathbb{F}_p} \operatorname{Pic}(C^+)[p] + \#\{v \in \Sigma : p \mid c_v \} \text{.} \] Here \(c_v\) is the Tamagawa number of \(\mathcal{E}\) at \(v\). The authors obtain a similar bound for the case that \(p = 2\). To place their main result into context, the authors explain how it relates to the well known geometric rank bound, in the sense of \textit{J. H. Silverman} [J. Reine Angew. Math. 577, 153--169 (2004; Zbl 1105.11016)]. They also present a number of examples. Finally, the authors describe several situations in which their main result takes a more refined form. In doing so, they make progress towards questions that were raised in [\textit{D. Ulmer}, Math. Sci. Res. Inst. Publ. 49, 285--315 (2004; Zbl 1062.11033)].