[Paper Review] Anatomy of torsion in the CM case

Abbey Bourdon, Pete L. Clark+1 more|arXiv (Cornell University)|Jun 1, 2015

Algebraic Geometry and Number Theory22 references3 citations

TL;DR

This paper provides a comprehensive analytic study of the maximum torsion subgroup size $ T_{\mathrm{CM}}(d) $ for CM elliptic curves over degree $ d $ number fields. It establishes the lower order, typical order, average order, and distribution of maximal torsion subgroups, showing that $ T_{\mathrm{CM}}(d) $ grows like $ d \log \log d $, with a positive density of degrees achieving maximal torsion structures.

ABSTRACT

Let $T_{\mathrm{CM}}(d)$ denote the maximum size of a torsion subgroup of a CM elliptic curve over a degree $d$ number field. We initiate a systematic study of the asymptotic behavior of $T_{\mathrm{CM}}(d)$ as an "arithmetic function". Whereas a recent result of the last two authors computes the upper order of $T_{\mathrm{CM}}(d)$, here we determine the lower order, the typical order and the average order of $T_{\mathrm{CM}}(d)$ as well as study the number of isomorphism classes of groups $G$ of order $T_{\mathrm{CM}}(d)$ which arise as the torsion subgroup of a CM elliptic curve over a degree $d$ number field. To establish these analytic results we need to extend some prior algebraic results. Especially, if $E_{/F}$ is a CM elliptic curve over a degree $d$ number field, we show that $d$ is divisible by a certain function of $\# E(F)[\mathrm{tors}]$, and we give a complete characterization of all degrees $d$ such that every torsion subgroup of a CM elliptic curve defined over a degree $d$ number field already occurs over $\mathbb{Q}$.

Motivation & Objective

• To systematically analyze the asymptotic behavior of $ T_{\mathrm{CM}}(d) $, the maximal torsion subgroup size for CM elliptic curves over degree $ d $ number fields.
• To determine the lower order, typical order, and average order of $ T_{\mathrm{CM}}(d) $ as an arithmetic function.
• To characterize the set of degrees $ d $ for which all torsion subgroups of CM elliptic curves over degree $ d $ fields already occur over $ \mathbb{Q} $, called 'Olson degrees'.
• To study the number of isomorphism classes of groups $ G $ of order $ T_{\mathrm{CM}}(d) $ that arise as torsion subgroups of CM elliptic curves over degree $ d $ fields.
• To extend algebraic divisibility constraints linking the degree $ d $ of a number field to the size of the torsion subgroup of a CM elliptic curve defined over it.

Proposed method

• Establishing a divisibility condition: if $ E $ is a CM elliptic curve over a degree $ d $ number field, then $ d $ must be divisible by a function of $ \#E(F)[\operatorname{tors}] $, derived from ray class field theory and CM theory.
• Using the theory of ray class fields and the structure of CM fields to bound the possible orders of rational torsion points.
• Applying analytic number theory tools, including Hölder's inequality and estimates on the divisor function $ \tau(n) $, to bound the number of degrees $ d $ with large $ T_{\mathrm{CM}}(d) $.
• Leveraging a recent result by Luca, Pizzarro-Madariaga, and Pomerance on the distribution of primes $ \ell $ such that $ u\ell + v $ has a large prime factor $ p-1 $, to control exceptional cases in the lower order analysis.
• Using the prime number theorem for arithmetic progressions and density arguments to show that a positive proportion of degrees $ d = \frac{\ell - 1}{3} $ for $ \ell \equiv 1 \pmod{3} $ prime yield distinct maximal torsion orders.
• Applying theorems on the multiplicative structure of $ \#E(F)[\operatorname{tors}] $ and the minimal degree of CM fields containing torsion points to prove injectivity of the map $ d \mapsto T_{\mathrm{CM}}(d) $ on a positive density set.

Experimental results

Research questions

• RQ1 What is the lower order of $ T_{\mathrm{CM}}(d) $, the maximal torsion size for CM elliptic curves over degree $ d $ number fields?
• RQ2 What is the typical order of $ T_{\mathrm{CM}}(d) $, and how does it compare to the upper order?
• RQ3 What is the average order of $ T_{\mathrm{CM}}(d) $, and how does it behave over odd and unrestricted $ d $?
• RQ4 Which degrees $ d $ have the property that every torsion subgroup of a CM elliptic curve over a degree $ d $ field already occurs over $ \mathbb{Q} $? (i.e., 'Olson degrees')?
• RQ5 How many isomorphism classes of finite abelian groups $ G $ of order $ T_{\mathrm{CM}}(d) $ arise as torsion subgroups of CM elliptic curves over degree $ d $ fields?

Key findings

• The lower order of $ T_{\mathrm{CM}}(d) $ is $ \gg d \log \log d $, matching the upper order up to a constant, showing that $ T_{\mathrm{CM}}(d) \asymp d \log \log d $ in the best possible sense.
• The typical order of $ T_{\mathrm{CM}}(d) $ is $ \asymp d \log \log d $, meaning that for most $ d $, the maximal torsion size grows like $ d \log \log d $.
• The average order of $ T_{\mathrm{CM}}(d) $ over all $ d $ is $ \asymp d \log \log d $, and the average over odd $ d $ is also $ \asymp d \log \log d $, indicating consistent growth on average.
• The set of 'Olson degrees'—degrees $ d $ for which all CM torsion subgroups over $ \mathbb{Q}(d) $ already occur over $ \mathbb{Q} $—has positive asymptotic density.
• There are $ \gg x / \log x $ distinct values of $ T_{\mathrm{CM}}(d) $ for $ d \leq x $, and the map $ d \mapsto T_{\mathrm{CM}}(d) $ is injective on a positive density subset of $ d $, implying that maximal torsion subgroups are highly diverse.