The Lang-Trotter conjecture on average for genus-2 curves with Klein-4 reduced automorphism group

For an elliptic curve E over Q Q without complex multiplication, the Lang–Trotter conjecture is that # \ p<X E has a supersingular reduction at p \ ~ c √Xlog X # { p < X ∣ E has a supersingular reduction at p } ∼ c X log X as X → ∞ 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 Cλ:y²=x(x-1)(x+1)(x-λ )(x-1/ λ ) 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.