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Average analytic ranks of elliptic curves over number fields

Tristan Phillips

TL;DR

The paper establishes a conditional upper bound for the average analytic rank of isomorphism classes of elliptic curves over a number field $K$, ordered by naive height, under the assumptions that all curves are modular and satisfy GRH. The author recasts counting elliptic curves as counting points on the moduli stack $\mathcal{X}_{GL_2(\mathbb{Z})}$, identified with the weighted projective stack $\mathcal{P}(4,6)$, and develops a general theory for counting points of bounded height on weighted projective stacks with prescribed local conditions. A robust weighted geometry-of-numbers framework over number fields is established, including o-minimal definability and a Box Lemma, to obtain asymptotics with explicit constants for local-condition counts. These counting results feed into the explicit formula for $L$-functions of elliptic curves, enabling bounds on the average analytic rank that culminate in the unconditional bound $(9\deg(K)+1)/2$ under the stated hypotheses. The work extends prior Q-based results to arbitrary number fields and provides tools for counting elliptic curves with local data, which may have independent interest in arithmetic geometry and Diophantine geometry.

Abstract

A conditional bound is given for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field $K$ are modular and have $L$-functions which satisfy the Generalized Riemann Hypothesis, it is shown that the average analytic rank of isomorphism classes of elliptic curves over $K$ is bounded above by $(9\ \text{deg}(K)+1)/2$, when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.

Average analytic ranks of elliptic curves over number fields

TL;DR

The paper establishes a conditional upper bound for the average analytic rank of isomorphism classes of elliptic curves over a number field , ordered by naive height, under the assumptions that all curves are modular and satisfy GRH. The author recasts counting elliptic curves as counting points on the moduli stack , identified with the weighted projective stack , and develops a general theory for counting points of bounded height on weighted projective stacks with prescribed local conditions. A robust weighted geometry-of-numbers framework over number fields is established, including o-minimal definability and a Box Lemma, to obtain asymptotics with explicit constants for local-condition counts. These counting results feed into the explicit formula for -functions of elliptic curves, enabling bounds on the average analytic rank that culminate in the unconditional bound under the stated hypotheses. The work extends prior Q-based results to arbitrary number fields and provides tools for counting elliptic curves with local data, which may have independent interest in arithmetic geometry and Diophantine geometry.

Abstract

A conditional bound is given for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field are modular and have -functions which satisfy the Generalized Riemann Hypothesis, it is shown that the average analytic rank of isomorphism classes of elliptic curves over is bounded above by , when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.
Paper Structure (19 sections, 25 theorems, 208 equations, 2 tables)

This paper contains 19 sections, 25 theorems, 208 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Average analytic ranks of elliptic curves
  3. Counting points on weighted projective stacks
  4. Asymptotic notation
  5. Organization
  6. Acknowledgements
  7. Preliminaries on weighted projective stacks
  8. Heights on weighted projective stacks
  9. Weighted geometry of numbers over number fields
  10. O-minimal geometry
  11. Weighted geometry-of-numbers
  12. Counting points on weighted projective stacks
  13. Counting elliptic curves
  14. Counting elliptic curves with a prescribed local condition
  15. Average analytic ranks
  16. ...and 4 more sections

Key Result

Theorem 1.1.1

Let $K$ be a number field of degree $d$. Assume that all elliptic curves over $K$ are modular and that their $L$-functions satisfy the Generalized Riemann Hypothesis. Then the average analytic rank of isomorphism classes of elliptic curves over $K$, when ordered by naive height, is bounded above by

Theorems & Definitions (55)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Theorem 1.1.3
  • Remark 1.2.1
  • Definition 2.1.1: Height
  • Definition 2.1.2: Scaling ideal
  • Remark 2.1.3
  • Definition 3.1.1: Semi-algebraic set
  • Definition 3.1.2: Structure
  • Definition 3.1.3: O-minimal structure
  • ...and 45 more