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 $\text{deg }P(x),\text{deg }Q(x)\le 3$. 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 ${\mathcal{E}}_U$.

28 pages, unconditional proof of the main result, to appear in Mathematische Zeitschrift

Cited by in corpus (1)

• Biases in Moments of the Dirichlet Coefficients in One- and Two-Parameter Families of Elliptic Curves