Refinements on vertical Sato-Tate

Abstract: Vertical Sato-Tate states that the Frobenius trace of a randomly chosen elliptic curve over $\mathbb F_p$ tends to a semicircular distribution as $p\rightarrow \infty$. We go beyond this statement by considering the number of elliptic curves $N_{t,p}'$ with a given trace $t$ over $\mathbb F_p$ and characterizing the 2-dimensional distribution of $(t,N_{t,p}')$. In particular, this gives the distribution of the size of isogeny classes of elliptic curves over $\mathbb F_p$. Furthermore, we show a notion of stronger convergence for vertical Sato-Tate which states that the number of elliptic curves with Frobenius trace in an interval of length $p^\epsilon$ converges to the expected amount. The key step in the proof is to truncate Gekeler's infinite product formula, which relies crucially on an effective Chebotarev's density theorem that was recently developed by Pierce, Turnage-Butterbaugh and Wood.