# Quadratic non-residues in elliptic divisibility sequences

Let $$E: y^2 = x^3 + ax + b$$ be an elliptic curve over $$\mathbb{Q}$$ with $$a,b \in \mathbb{Z}$$. Recall that any rational point $$P = (x,y)$$ can be written uniquely as $$P = (u/d^2, v/d^3)$$ with $$u,v,d \in \mathbb{Z}, d > 0, \gcd(u,d) = \gcd(v,d) = 1$$. We write $$d(P)$$ for the number appearing in the denominator.

The sequence $$d(nP)$$ for $$n \in \mathbb{N}$$ is the elliptic divisibility sequence associated to $$P$$. My question is about existence of quadratic non-residues in these sequences modulo all but finitely many primes $$p$$.

Let $$P$$ be a point of infinite order. Does there exists a finite set of primes $$S$$ such that for all $$p \notin S$$, there is $$n \in \mathbb{N}$$ such that $$\left( \frac{d(nP)}{p}\right) = -1 \quad ? \quad (*)$$

Remarks:

1. It is known that the sequence $$d(nP)$$ contains only finitely many squares, by Theorem 1.1 of

Everest, Reynolds, Stevens - On the denominators of rational points on elliptic curves.

In particular, for all but finitely many $$n$$, one finds that $$d(nP)$$ is a quadratic non-residue modulo half of all primes. I want to know that as one varies $$n$$ and takes all these primes together, one only excludes finitely many primes.

2. There is a slightly different definition of elliptic divisibility sequences in terms of division polynomials and recurrence sequences, which can differ from the above sequence by a sign and some small primes (this is the original definition of Ward). I'm also interested in the analogous problem for such sequences.

• Is this known for Lucas sequences? – Dror Speiser Jul 27 at 20:15
• Good question. I don't know. – Daniel Loughran Jul 28 at 8:22
• Is it even known that the primes excluded have density zero? – Stanley Yao Xiao Jul 30 at 12:03
• @Stanley Yao Xiao up to some work reconciling the definition difference, I think so: combine the main result in Reductions of Points on Elliptic Curves by Akbary et al, with the main result in Character Sums with Division Polynomials by Shparlinski and Stange. – Dror Speiser Jul 30 at 17:08
• It appears that for a fixed $E$ and $P$, the primes for which there is no $n$ so that $\left(\frac{d(nP)}{p}\right) = -1$ can be reasonably large. I just looked at the example of $E : y^{2} = x^{3} - 2x$ and $P = (2,2)$, and found that there is no such $n$ for $p = 17$, $257$, $1009$, $1361$, $26881$, and $141041$. – Jeremy Rouse Aug 5 at 2:06