Bounds on the Mordell-Weil rank of elliptic fibrations
Antonella Grassi, Rick Miranda, Kapil Paranjape, Vasudevan Srinivas, Timo Weigand
TL;DR
This paper establishes explicit bounds on the Mordell-Weil rank of elliptic fibrations, especially for Calabi-Yau varieties, with implications for string theory and conjectures for higher dimensions.
Contribution
It provides two proofs for rank bounds on elliptic fibrations, including new bounds for Calabi-Yau threefolds and fourfolds, supporting a broader conjecture.
Findings
Explicit bounds for Calabi-Yau threefolds
New bounds for fourfolds under mild assumptions
Motivates conjecture for bounds in all dimensions
Abstract
We present two proofs for a bound on the rank of the Mordell-Weil group of some elliptic fibrations. The bounds apply to Calabi-Yau varieties, which are also of interest to the physics of string theory. We prove explicit bounds for Calabi-Yau threefolds, as predicted by physics, and give new explicit bounds for fourfolds under mild assumptions. These results motivate a conjecture for bounds in any dimensions.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometry and complex manifolds · Geometric and Algebraic Topology