Best results regarding the Lang-Trotter conjecture

Question

Let $E$ be an elliptic curve over $\mathbb Q$, which does not have complex multiplication over the algebraic closure of $\mathbb Q$. For $x>0$, let $P(x)$ be the number of primes $p < x$ such that $E$ has good super-singular reduction at $p$. The Lang-Trotter's conjecture states that $$P(x) = O (x^{1/2}/\log x).$$ In Serre's paper Quelques applications du théorème de densité de Chebotarev, Serre proves $P(x)=O(x^{3/4})$ under the generalized Riemann hypothesis (GRH) for Artin $L$-functions.

Has this bound been improved since? If so, what is the better known bound (under GRH) and where can I find it?

Thanks.

Answer 1

A couple of years after my Ph.D. thesis (whose main result is the infinitude of singular primes, i.e. $P(x) \rightarrow \infty$ as $x \rightarrow \infty$), Kaneko published a result[1] that let me obtain the unconditional upper bound[2] $P(x) = O(x^{3/4} \log x)$ using some of the same ideas:

[1] Masanobu Kaneko: Supersingular $j$-invariants as singular moduli mod $p$, Osaka J. Math. 26 (1989), 849–855.

[2] Noam D. Elkies: Distribution of supersingular primes, Astérisque 198-199-200 (1991; proceedings of Journées Arithmétiques 1989), 127–132.

As noted in [2], the factor $\log x$ can be removed with some more care (averaging over the auxiliary discriminants $-D$ improves on the worst-case estimate), thus exactly matching Serre's conditional bound.

As far as I know, no further improvement has been obtained since then, even assuming GRH.

That paper [2] also reports lower bounds from my thesis, conditional on GRH for quadratic characters: $P(x) \gg \log \log x$, and also $P(x_n) \gg \log x_n$ for an infinite sequence $x_n \rightarrow \infty$.