Second moments and the bias conjecture for the family of cubic pencils
Matija Kazalicki, Bartosz Naskręcki
TL;DR
The paper derives an explicit second-moment formula for the one-parameter cubic pencil $\mathcal{E}_U: y^2=P(x)U+Q(x)$ with $\deg P,\deg Q\le 3$ by relating $\tilde{M}_{2,q}(\mathcal{E}_U)$ to point counts on a genus-2 curve built from a conic fibration on a Kummer threefold. A precise identity $\tilde{M}_{2,q}(\mathcal{E}_U)=q(-\#\Delta(\mathbb{F}_q)+\#C(\mathbb{F}_q)+[\sum_{P(x)\equiv0}\phi_q(Q(x))]^2)$ is established for odd $q$, with $\Delta(x_1,x_2)=P(x_1)Q(x_2)-P(x_2)Q(x_1)$ and $C$ the intersection with the discriminant locus. The authors introduce the notions of $K$-generic/$K$-typical pairs $(\tilde{\Delta},\tilde{C})$, show their equivalence, and prove the Bias conjecture for typical curves, giving the bias as $-m-\delta$ where $m$ counts irreducible factors of a certain polynomial $S$ and $\delta\in\{0,1\}$ depends on the fiber at infinity. They also classify non-typical cases and illustrate that the bias phenomenon persists in broad non-generic settings. Overall, the work connects arithmetic statistics of elliptic-fibered surfaces to explicit geometric data on genus-2 curves and Kummer-related threefolds, reinforcing the bias phenomenon in a natural cubic-pencil family.
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 $\mathcal{E}_U$.