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On the rank of an elliptic surface. (English) Zbl 0905.14019

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.

MSC:

14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Citations:

Zbl 0870.11034
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