Abstract.
Nagao has recently given a conjectural limit formula for the rank of an elliptic surface 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(E,s) essentially implies Nagao's formula; in particular, we prove Nagao's formula for rational elliptic surfaces. In the case that 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.
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Oblatum 2-V-1997 & 15-VII-1997
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Rosen, M., Silverman, J. On the rank of an elliptic surface. Invent math 133, 43–67 (1998). https://doi.org/10.1007/s002220050238
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DOI: https://doi.org/10.1007/s002220050238