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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.

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    $\begingroup$ Is this known for Lucas sequences? $\endgroup$ – Dror Speiser Jul 27 at 20:15
  • $\begingroup$ Good question. I don't know. $\endgroup$ – Daniel Loughran Jul 28 at 8:22
  • $\begingroup$ Is it even known that the primes excluded have density zero? $\endgroup$ – Stanley Yao Xiao Jul 30 at 12:03
  • $\begingroup$ @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. $\endgroup$ – Dror Speiser Jul 30 at 17:08
  • $\begingroup$ We already have some ideas how to prove this "density zero" result, so are really looking for a proof that there are only finitely many exceptions. $\endgroup$ – Daniel Loughran yesterday

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