Second moments and the bias conjecture for the family of cubic pencils (2012.11306v2)

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, in parallel to the first moment case, that the largest lower order term that does not average to 0 is on the average negative (i.e. has a negative bias). Miller verified the Bias Conjecture for the number of pencils of cubics $E_k:y^2=P(x)k+Q(x)$, where $\textrm{deg} P(x), \textrm{deg} Q(x) \le 3$. In this paper, assuming Sato-Tate conjecture for genus two curves, we prove that the Bias Conjecture holds for the pencil of the cubics E_k with the generic choice of polynomials $P(x)$ and $Q(x)$ with $\textrm{deg} P(x), \textrm{deg} Q(x)\le 3$.