Lower Order Biases in Moment Expansions of One Parameter Families of
Elliptic Curves
Lower Order Biases in Moment Expansions of One Parameter Families of
Elliptic Curves
For a fixed elliptic curve $E$ without complex multiplication, $a_p := p+1 - \#E(\mathbb{F}_p)$ is $O(\sqrt{p})$ and $a_p/2\sqrt{p}$ converges to a semicircular distribution. Michel proved that for a one-parameter family of elliptic curves $y^2 = x^3 + A(T)x + B(T)$ with $A(T), B(T) \in \mathbb{Z}[T]$ and non-constant $j$-invariant, the second …