The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1 (1312.7859v1)

Abstract: In this article, we prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by height, is less than $1$ (in fact, less than $.885$). As a consequence of our methods, we also prove that at least four fifths of all elliptic curves over $\mathbb{Q}$ have rank either 0 or 1; furthermore, at least one fifth of all elliptic curves in fact have rank 0. The primary ingredient in the proofs of these theorems is a determination of the average size of the $5$-Selmer group of elliptic curves over $\mathbb{Q}$; we prove that this average size is $6$. Another key ingredient is a new lower bound on the equidistribution of root numbers of elliptic curves; we prove that there is a family of elliptic curves over $\mathbb{Q}$ having density at least $55\%$ for which the root number is equidistributed.

Analyzing the 5-Selmer Group and Rank Distribution in Elliptic Curves

This paper, authored by Manjul Bhargava and Arul Shankar, presents significant advancements in understanding the average rank and distribution of the 5-Selmer group in elliptic curves over the rational numbers $\mathbb{Q}$. The pivotal result obtained is that the average rank of elliptic curves is less than one, specifically less than 0.885. This has profound implications on the distribution of elliptic curve ranks, especially in decreasing the perceived density of curves with non-zero ranks.

Key Results and Theorems

1. Average Size of 5-Selmer Groups:

Theorem 1 establishes that when elliptic curves defined over $\mathbb{Q}$ are ordered by height, the average size of their 5-Selmer group is exactly 6. This validates the conjecture related to the sum of divisors of $n$-Selmer groups, confirming that for $n=5$, the predicted value holds true.

1. Rank Distribution:

Leveraging the calculation of Selmer group sizes, Theorem 3 concludes that the average rank is less than 0.885. In maintaining their rank either zero or one, a remarkable density of 83.75% of all elliptic curves is predicted, as noted in Theorem 4.

1. Equidistribution of Root Numbers:

The authors introduce a mechanism for improving the bounds on average rank by equidistributing the root numbers. A family of elliptic curves that adheres to these principles is aptly constructed, resulting in Theorem 6, where at least 55.01% of curves have equidistributed root numbers.

Methodological Approach

The authors employ innovative algebraic strategies, especially in parametrizing elements of the 5-Selmer group. They utilize a bespoke representation of genus-one curves as intersections of quadrics in projective space. The involved combinatorial count of G-orbits in a skew-symmetric matrix setup is skillfully handled to derive average sizes of the Selmer group. Advanced arguments in geometry-of-numbers, alongside deliberately structured sieve methods, underpin the calculations.

Implications and Insights

• Practical Implications: Knowing the statistical distribution of elliptic curve ranks directly impacts number theory and cryptographic applications where elliptic curves play a foundational role. Calculations of rank determine the computational security level and efficiency of cryptographic systems.
• Theoretical Contributions: Verification of conjectures related to average ranks of elliptic curves substantiates ongoing discourse in analytic number theory. Moreover, the architecture provided around equidistributed root numbers revisits long-standing assumptions, reducing ambiguity in the symmetry or asymmetry of rank parity distributions.

Speculations for AI

Considering future research directions, AI's involvement could revolve around automated computations and conjecture verification using extensive datasets in number theory. Machine learning algorithms might assist in detecting non-trivial patterns amongst rank distributions that aren’t feasible analytically. Enhanced computational power and pattern recognition capabilities can result in refinements and extensions of the results depicted herein.

This paper presents numerical results that enrich the corpus of knowledge surrounding elliptic curves, their Selmer group sizes, and rank distributions, charting pathways for future exploration in computational and analytical number theory.