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Bounds on the Mordell-Weil rank of elliptic fibrations

Antonella Grassi, Rick Miranda, Kapil Paranjape, Vasudevan Srinivas, Timo Weigand

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.

Bounds on the Mordell-Weil rank of elliptic fibrations

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.

Paper Structure

This paper contains 19 sections, 51 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Bounds from physics
  3. Outline of our paper
  4. Background on elliptic fibrations and their Mordell-Weil group
  5. Movable curves and effective divisors
  6. Elliptic fibrations
  7. Noether's Formula
  8. The Shioda map and the Shioda pairing
  9. Bounds for the Mordell-Weil rank with Functions Fields
  10. Proof of Theorem \ref{['main2']}
  11. Bounds for the Mordell-Weil rank via birational geometry
  12. Explicit bounds for the Mordell-Weil rank of elliptic Calabi-Yau threefolds
  13. Explicit bounds for the Mordell-Weil rank of elliptic Calabi-Yau fourfolds
  14. Case (i): $\rho (B)=1$, $i(B)=3,4$.
  15. Case (ii): $\operatorname{rk} \mathrm{Pic} (B)=1$, $i(B)=2$.
  16. ...and 4 more sections

Key Result

Theorem 1

Let $X \to B$ be an elliptic Calabi-Yau threefold with sections and singular fibers. Then the Mordell-Weil group of $X$ satisfies ${\rm rank }\, \mathrm{MW}(X/B)\leq 28$.

Theorems & Definitions (112)

  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8
  • ...and 102 more