Arithmetic on elliptic threefolds

By Rania Wazir. Compositio Mathematica, 2001

106 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5a55929a0fafa3d93c70435ae9844dace7aa7090

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Denis Klevers
2. M. Cvetič
3. Hernan Piragua
4. D. S. Park
5. D. Morrison
6. Washington Taylor
7. J. Heckman
8. C. Vafa

In Mathematical Physics

1. Denis Klevers
2. M. Cvetič
3. Hernan Piragua
4. D. S. Park
5. D. Morrison
6. J. Heckman
7. Washington Taylor
8. Tom Rudelius

Citations in the corpus

• Anomaly equations and intersection theory (2011, background): “The number VA is the Mordell-Weil rank of the elliptic fibration [9, 56, 57].”
• F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections (2013, background): “Elliptic curves E with a so-called non-trivial Mordell-Weil group of rational points are a classical subject in mathematics [32, 33, 34, 35, 36].”
• F-theory and the Mordell-Weil group of elliptically-fibered Calabi-Yau threefolds (2012, background): “σ(ŝ) is a projection of the homology class of ŝ to H4(X)/T , it can be shown that σ is a homomorphism from the Mordell-Weil group to the homology lattice [45,46,49], i.”