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The average rank of elliptic $n$-folds

Remke Kloosterman

Abstract

Let $V/\mathbb{F}_q$ be a variety of dimension at least two. We show that the density of elliptic curves $E/\mathbb{F}_q(V)$ with positive rank is zero if $V$ has dimension at least 3 and is at most $1-ζ_V(3)^{-1}$ if $V$ is a surface.

The average rank of elliptic $n$-folds

Abstract

Let be a variety of dimension at least two. We show that the density of elliptic curves with positive rank is zero if has dimension at least 3 and is at most if is a surface.

Paper Structure

This paper contains 3 sections, 14 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Model in a ${\mathbf{P}}^2$-bundle
  3. Densities

Key Result

Theorem 1.1

Let $R$, $t$ and $V/{\mathbf{F}}_q$ be as above. Let Then Moreover, if $\dim V\geq 3$ then $\mu_0=1$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2: Gysin sequence
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 16 more