5 problems

Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication, and let $\pi_{E,2}(X)$ denote the number of amicable pairs of $E$ up to $X$. Jones's refined conjecture. The…

Let $E/\mathbb{Q}$ be an elliptic curve, and let … count amicable pairs of $E$ up to $X$. Assume that there are infinitely many primes $p$ such that $\#E_p(\mathbb{F}_p)$ is prime.…

Let $k$ be an integer for which the curve $E:y^2=x^3+k$ is defined, and let $\mathcal N_k$ be the set of primes $p\ge5$ of good reduction such that…

The CM amicable-pair ratio conjecture. Either $\mathcal N_E(X)$ is bounded, or

Let $E/\bb{Q}$ be an elliptic curve. Define the amicable-pair counting function … Assume that there are infinitely many primes $p$ such that $\#\widetilde E_p(\mathbb{F}_p)$ is pri…