Arithmetic on elliptic threefolds

Rania Wazir

Type: Article

Publication Date: 2004-04-14

Citations: 80

DOI: https://doi.org/10.1112/s0010437x03000381

Abstract

In a recent paper, Rosen and Silverman showed that Tate's conjecture on algebraic cycles implies a formula of Nagao, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. In this article, we extend their result to the case of elliptic threefolds. The main ingredients of our argument are a Shioda–Tate-like formula for elliptic threefolds, and a relation between the 'average' number of rational points on singular fibers and the Galois action on those fibers.

Locations

arXiv (Cornell University) - PDF
Compositio Mathematica - View - PDF

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Works Cited by This (18)

Action	Title	Year	Authors
+	Introduction to Intersection Theory in Algebraic Geometry	1984	William Fulton
+	Principles of Algebraic Geometry	1978	Phillip Griffiths Joseph Harris
+	Mordell–Weil lattices for higher genus fibration over a curve	1999	Tetsuji Shioda
+	Geometric Invariant Theory	1994	David Mumford John E. Fogarty Frances Kirwan
+	Advanced Topics in the Arithmetic of Elliptic Curves	1994	Joseph H. Silverman
+	On the rank of an elliptic surface	1998	Michael Rosen Joseph H. Silverman
+	Algebraic group schemes in characteristic zero are reduced	1966	Frans J. Oort
+ PDF Chat	On elliptic modular surfaces	1972	Tetsuji Shioda
+	Arithmetic on Elliptic Threefolds	2001	Rania Wazir
+	Fundamentals of Diophantine Geometry	1983	Serge Lang