论文标题
第二瞬间和分立铅笔家族的偏见
Second moments and the bias conjecture for the family of cubic pencils
论文作者
论文摘要
对于$ \ mathbb {q} $和prime $ p $的1-参数系列$ e_k $ e_k $的椭圆曲线和p $，请考虑第二米刻sum $ $ m_ {2，p}（e_k）= \ sum_ {k \ in \ mathbb in \ mathbb {f} $ a_ {k，p} = p+1- \＃e_k（\ mathbb {f} _p）$。受罗森（Rosen）和西尔弗曼（Silverman）的纳戈（Nagao）猜想证明的启发，这将有理椭圆表面的第一时刻与相应椭圆形曲线的mordell-weil组等级联系起来，S。J。Miller启动了$ m_ {2，p}（e_k）= p^2+o（p^2+o（p^2+o（p^p^p^p^p^p^{3/2 {3/2），共同解释）。他猜想，与第一个时刻案例相似，最大的下阶术语平均为0的偏差为负偏差。 在本文中，我们提供了第二次的明确公式 \ Mathcal {e} _ {u}：y^2 = p（x）u+q（x），$$其中$ \ textrm {deg} p（x），\ textrm {deg} q（x）\ leq 3 $。对于多项式的通用选择，$ p（x）$和$ q（x）$以某个属两曲线的点计数表示。作为一个应用程序，我们证明了偏见的猜想是对立方体的铅笔$ \ mathcal {e} _U $。
For a 1-parametric family $E_k$ of elliptic curves over $\mathbb{Q}$ and a prime $p$, consider the second moment sum $M_{2,p}(E_k)=\sum_{k \in \mathbb{F}_p} a_{k,p}^2$, where $a_{k,p}=p+1-\#E_k(\mathbb{F}_p)$. Inspired by Rosen and Silverman's proof of Nagao conjecture which relates the first moment of a rational elliptic surface to the rank of Mordell-Weil group of the corresponding elliptic curve, S. J. Miller initiated the study of the asymptotic expansion of $M_{2,p}(E_k)=p^2+O(p^{3/2})$ (which by the work of Deligne and Michel has cohomological interpretation). He conjectured that, similar to the first moment case, the largest lower-order term that does not average to 0 has a negative bias. In this paper, we provide an explicit formula for the second moment $M_{2,p}(\mathcal{E}_{U})$ of $$ \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 $\mathcal{E}_U$.
