[Paper Review] Selmer Ranks of Quadratic Twists of Elliptic Curves

Zev Klagsbrun|arXiv (Cornell University)|Nov 9, 2011

Analytic Number Theory Research10 references3 citations

TL;DR

This paper establishes the existence and computes the stable limit of 2-Selmer ranks in families of quadratic twists of any elliptic curve over a number field, showing the distribution is an explicit product of local factors. Under Galois group condition Gal(K(E[2])/K) = S₃, it proves the density of twists with Selmer rank r exists and follows an equilibrium distribution governed by a universal Markov process.

ABSTRACT

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We first prove that the of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. Under the assumption that Gal(K(E[2])/K) = S_3 we also show that the density (counted in a non-standard way) of twists with Selmer rank r exists for all positive integers r, and is given via an equilibrium distribution, depending only on the parity fraction alluded to above, of a certain Markov Process that is itself independent of E and K. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolute Galois group of K by characters of order p.

Motivation & Objective

• To determine the distribution of 2-Selmer ranks across quadratic twists of a fixed elliptic curve over a number field.
• To establish the existence of a stable limit for the proportion of twists with even 2-Selmer rank.
• To compute this stable limit as an explicit product of local factors depending on the curve and base field.
• To analyze the density of twists with arbitrary Selmer rank r under the assumption that the Galois group of the 2-torsion is S₃.
• To generalize results to p-Selmer ranks of 2-dimensional self-dual F_p-representations twisted by characters of order p.

Proposed method

• Uses the theory of quadratic twists and Galois cohomology to analyze the 2-Selmer group structure in families.
• Expresses the stable limit of even 2-Selmer rank proportions as a product over local factors at all places of the number field.
• Applies a Markov process model to describe the distribution of Selmer ranks under the S₃ Galois condition.
• Demonstrates that the equilibrium distribution of the Markov process depends only on the parity fraction, not on the specific curve or field.
• Extends the framework to p-Selmer ranks for self-dual F_p-representations of the absolute Galois group twisted by characters of order p.
• Employs representation-theoretic and arithmetic techniques to ensure independence of the Markov process from the underlying curve and field.

Experimental results

Research questions

• RQ1 What is the stable limit of the proportion of quadratic twists with even 2-Selmer rank over a fixed number field?
• RQ2 How can this stable limit be expressed in terms of local invariants of the elliptic curve and base field?
• RQ3 Under what Galois-theoretic conditions does the density of twists with a given Selmer rank r exist?
• RQ4 What is the structure of the distribution of Selmer ranks across twists when Gal(K(E[2])/K) = S₃?
• RQ5 Can the results be generalized beyond elliptic curves to p-Selmer ranks of 2-dimensional self-dual F_p-representations?

Key findings

• The stable limit of the proportion of quadratic twists with even 2-Selmer rank exists and is given by an explicit product of local factors.
• For a fixed elliptic curve E over a number field K, the proportion of twists with even 2-Selmer rank is computable as a product over all places of K.
• There exists an elliptic curve E such that as K varies, the fraction of twists with even 2-Selmer rank is dense in the interval [0, 1].
• Under the assumption that Gal(K(E[2])/K) = S₃, the density of twists with Selmer rank r exists for all positive integers r.
• This density is determined by an equilibrium distribution of a universal Markov process independent of E and K, depending only on the parity fraction.
• The results generalize to p-Selmer ranks of 2-dimensional self-dual F_p-representations of the absolute Galois group of K twisted by characters of order p.