The density of primes dividing a particular non-linear recurrence sequence

Abstract

Define the ECHO sequence {bn} recursively by (b 0 , b 1 , b 2 , b 3 ) = (1, 1, 2, 1) and for n ≥ 4,We relate this sequence {bn} to the coordinates of points on the elliptic curve E : y 2 + y = x 3 − 3x + 4. We use Galois representations attached to E to prove that the density of primes dividing a term in this sequence is equal to 179 336 . Furthermore, we describe an infinite family of elliptic curves whose Galois images match those of E.

Density of odd order reductions for elliptic curves with a rational point of order 2

“…(See for example [9], [8], [6], and [3].) In [1], the authors study the problem of determining the image of τ subject to the constraints that (i) the usual mod 2 k Galois representation ρ E,2 k : Gal(Q/Q) → GL 2 (Z/2 k Z) is surjective for all k, and (ii) α ∈ E(Q) is not equal to 2γ for any γ ∈ E(Q). They show that under these hypotheses, there are two possibilities for the image of τ : AGL 2 (Z/2 k Z), and an index 4 subgroup thereof.…”

“…is a square in Q, but is a fourth power in Q(β 1 , E [8]), and this means that To compute the interpretation for one subgroup in each of our batches, we will use a technique from [4] (see the proof of the Lemma on page 962) and [1] (see the proof of Lemma 9.1). Next, we present tables describing the various images.…”

“…To compute the interpretation for one subgroup in each of our batches, we will use a technique from [4] (see the proof of the Lemma on page 962) and [1] (see the proof of Lemma 9.1). Given a subgroup H ⊆ AGL 2 (Z/2 k Z) and an elliptic curve E (depending on the parameters a, c and k), we wish to compute a polynomial…”

“…We wish to give a brief accounting of the connection between the image of τ E,2 k and the relative density of primes p for which α ∈ E(F p ) has odd order. For a more detailed explanation, see Theorem 5.1 of [1].…”

Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves

“…In [3], the authors show that if F = Q, ℓ = 2, ρ E,2 ∞ is surjective, and α is strongly 2-indivisible, then ω E,α,2 ∞ is either surjective (in which case the density of primes p for which α has odd order is 11/21), or the image of ω E,α,2 ∞ has index 4 in Z × 2 ⋊GL 2 (Z 2 ), and the odd order density is 179/336. This latter case arises when Q(β 1 ) ⊂ Q(E [4]).…”