Anatomy of torsion in the CM case

Abstract

Let T 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 CM (d) as an "arithmetic function". Whereas a recent result of the last two authors computes the upper order of T CM (d), here we determine the lower order, the typical order and the average order of T CM (d) as well as study the number of isomorphism classes of groups G of order T 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 ) [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 Q.

Torsion Subgroups of CM Elliptic Curves over Odd Degree Number Fields:

“…From earlier work of Breuer [6], this lim sup is positive. Hence, T CM (d) has upper order d log log d. Several other statistics concerning T CM (d) are investigated in [4]; e.g., it is shown there that the average of T CM (d) for d ≤ x is x/(log x) 1+o (1) , as x → ∞.…”

“…Call d an Olson degree if G CM (d) = G CM (1). The following complete classification of Olson degrees was proved in [4]. To state the result, we need one more piece of notation.…”

Numbers Divisible by a Large Shifted Prime and Large Torsion Subgroups of CM Elliptic Curves

“…its normal order? Such statistical questions were investigated in [2]. Here we content ourselves with recalling only the normal order result.…”

“…Here we content ourselves with recalling only the normal order result. Using the Erdős-Wagstaff theorem, it was shown in [2] that T CM (d) is typically bounded. By this, we mean that the set of d with T CM (d) > y has upper density that tends to 0 as y → ∞.…”

On the minimal degree of definition of $p$-primary torsion subgroups of elliptic curves

“…In the CM case, there are known divisibility bounds for the field of definition of a point of order N (and for p-primary torsion structures when the field of definition does not contain the quadratic field of complex multiplication) given by Silverberg [19], and Prasad and Yogananda [15]. More generally, Bourdon, Clark, and Pollack [6], have recently shown divisibility bounds for p-primary torsion structures, similar to those of our Theorem 1.6.…”