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.

Abstract. The Hardy-Littlewood prime k-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field F q (t).

We study the distribution of solutions n to the congruence σ(n) ≡ a ( mod n). After excluding obvious families of solutions, we show that the number of these n ≤ x is at most x½+o(1), as x → ∞, uniformly for integers a with ∣a∣ ≤ x¼. As a concrete example, the number of composite solutions n ≤ x to the congruence σ(n) ≡ 1 ( mod n) is at most x½+o(1). These results are analogues of theorems established for the Euler ϕ-function by the third-named author.

No abstract

Abstract. Let T CM (d) be the maximum size of the torsion subgroup of an elliptic curve with complex multiplication defined over a degree d number field. We show that there is an absolute, effective constant C such thatFor a commutative group G, we denote by G[tors] the torsion subgroup of G.