The Lang-Trotter conjecture on average for genus-2 curves with Klein-4 reduced automorphism group
Year: 2025
Type: article
Abstract: For an elliptic curve E over $$\mathbb {Q}$$ Q without complex multiplication, the Lang–Trotter conjecture is that $$ \# \{ p # { p < X ∣ E has a supersingular reduction at p } ∼ c X log X as $$X \rightarrow \infty $$ X → ∞ , where $$c>0$$ c > 0 is a constant depending only on E . Fouvry and Murty obtained an average estimation related to the Lang–Trotter conjecture, called the Lang–Trotter conjecture on average. We consider the Lang–Trotter conjecture for curves of genus 2 and obtain a similar result to the Lang–Trotter conjecture on average for the family of curves $$C_{\lambda }:y^2=x(x-1)(x+1)(x-{\lambda })(x-1/ \lambda )$$ C λ : y 2 = x ( x - 1 ) ( x + 1 ) ( x - λ ) ( x - 1 / λ ) . These curves are characterized as curves of genus two with a reduced automorphism group containing the Klein 4-group.
Source: Research in Number Theory
Author: Chihiro Ando
Institution: Yokohama National University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,590
OpenAlex ID: W4414120016