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On the rank of an elliptic surface (1998)

Rosen, Michael ; Silverman, Joseph H.

Springer

Inventiones mathematicae 133 (1998), S. 43-67

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ISSN: 1432-1297

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: 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.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/s002220050238

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