Second moments and the bias conjecture for the family of cubic pencils

Abstract: For a 1-parametric family $E_k$ of elliptic curves over $\mathbb{Q}$ and a prime $p$, consider the second moment sum $M_{2,p}(E_k)=\sum_{k \in \mathbb{F}p} a{k,p}^2$, where $a_{k,p}=p+1-#E_k(\mathbb{F}p)$. Inspired by Rosen and Silverman's proof of Nagao conjecture which relates the first moment of a rational elliptic surface to the rank of Mordell-Weil group of the corresponding elliptic curve, S. J. Miller initiated the study of the asymptotic expansion of $M{2,p}(E_k)=p^2+O(p^{3/2})$ (which by the work of Deligne and Michel has cohomological interpretation). He conjectured that, similar to the first moment case, the largest lower-order term that does not average to 0 has a negative bias. In this paper, we provide an explicit formula for the second moment $M_{2,p}(\mathcal{E}{U})$ of $$ \mathcal{E}{U}:y^2=P(x)U+Q(x), $$ where $\textrm{deg } P(x), \textrm{deg } Q(x)\leq 3$. For a generic choice of polynomials $P(x)$ and $Q(x)$ this formula is expressed in terms of the point count of a certain genus two curve. As an application, we prove that the Bias conjecture holds for the pencil of the cubics $\mathcal{E}_U$.