Refinements on vertical Sato-Tate

2023/10/13 by Zhao Yu Ma, Ma, Zhao Yu
Mathematics · #11G07 (Primary) 14H52 #11G20 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2310.08791

openalex publication_date 2023/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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→ ∞. 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^ε 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.

Related

• The Sato-Tate Distribution in Thin Parametric Families of Elliptic Curves
• Explicit Sato-Tate type distribution for a family of K3 surfaces
• The Sato--Tate Distribution in Families of Elliptic Curves with a Rational Parameter of Bounded Height
• Pseudorandomness of Sato-Tate Distributions for Elliptic Curves
• Lang-Trotter and Sato-Tate Distributions in Single and Double Parametric Families of Elliptic Curves