On the rank of an elliptic surface
\textit{K. Nagao} [Manuscr. Math. 92, No. 1, 13-32 (1997; Zbl 0870.11034)] has recently given a conjectural limit formula for the rank of an elliptic surface \({\mathcal E}\) in terms of a weighted average of fibral Frobenius trace values. We show that Tate's conjecture on the order of vanishing of \(L_2 ({\mathcal E}, s)\) essentially implies Nagao's formula; in particular, we prove Nagao's formula for rational elliptic surfaces. In the case that \({\mathcal E}\) is a twist, we reduce Nagao's and Tate's conjectures to the case of products of curves, and we verify the conjectures for many new classes of elliptic surfaces of Kodaira dimension 0 and 1.
scientific article; zbMATH DE number 1186162