# Elliptic analogue of primes of the form $x^2 + 1$

I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but please let me know if you know of related published articles, if these sort of primes/problems have a name I should search for, etc.

Now the idea...

A well-known conjecture is that there are infinitely many prime numbers $$p$$ of the form $$x^2 + 1$$. (Landau problem number 4, in fact). Or equivalently, letting $$G_m$$ denote the multiplicative group, the conjecture states that there are infinitely many primes $$p$$ such that $$\# G_m({\mathbb F}_p)$$ is a square.

The obvious elliptic analogue is this. Take an elliptic curve $$E$$ over $${\mathbb Q}$$, non-CM let's say. Are there infinitely many primes $$p$$ such that $$\# E({\mathbb F}_p)$$ is a square? (Take a model over $${\mathbb Z}$$, reduce mod $$p$$, blah, etc.)

I can't be the first person to ask this question. Any references would be greatly appreciated, if you've seen this question posed! There's lots more that can be said, heuristics in the spirit of Lang-Trotter, relevant work on Frobenius distributions, questions on average as $$E$$ varies, etc. That's all for my student to investigate, I hope. But has anyone really focused on the simple question above?

References much appreciated!

• Perhaps this seems like a worthwhile starting point? arxiv.org/abs/2003.09951 Feb 5 at 19:51
• Thanks! But that one I found already, and I think it's a different vertical direction about powers of a single prime p rather than allowing p to vary over all primes. Feb 6 at 6:19