论文标题
椭圆表面整数纤维上的恒定根号
Constant root number on integer fibres of elliptic surfaces
论文作者
论文摘要
里佐（Rizzo）表明，椭圆曲线的家族$ \ Mathcal {w}（t）：y^2 = x^3+tx^2-（t+3）x+1 $，是华盛顿的一个众所周知的例子，具有根号$ w（\ mathcal {w}（w}（t））= - 1 $ 1 $ in \ in \ in \ mathbbbbbbbbbbbbbbbb c} $。在本文中，我们概括了这个示例，并确定了这种现象发生的小程度的家庭。由David，Bettin和Delaunay（Arxiv：1612.03095）和Desjardins（Arxiv：1810.12787）的结果激励，我们详细研究了两个家庭$ \ Mathcal {f} _s _S（t） $ \ MATHCAL {L} _ {W，S，V}（T）：Wy^2 = X^3+3（t^2+V）X^2+3SX+S（T^2+V）$，并描述$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mathcal {f} _s（f} _s（t）的必要条件。我们进一步证明了$ \ Mathcal {L} _ {W，S，V}（T）$上的相似但部分结果。我们的结果给出了整数纤维时等级升高的亚家族的例子。
Rizzo showed that the family of elliptic curves $\mathcal{W}(t) :y^2=x^3+tx^2-(t+3)x+1$, a well-known example of Washington, has root number $W(\mathcal{W}(t))=-1$ for all $t\in\mathbb{Z}$. In this paper we generalize this example and identify the families of small degree on which this phenomenon happens. Motivated by results from David, Bettin and Delaunay (arXiv:1612.03095) and Desjardins (arXiv:1810.12787), we study in detail the two families $\mathcal{F}_s(t):y^2=x^3+3tx^2+3sx+st$ and $\mathcal{L}_{w,s,v}(t): wy^2=x^3+3(t^2+v)x^2+3sx+s(t^2+v)$ and describe necessary and sufficient conditions for which subfamilies of $\mathcal{F}_s(t)$ have constant root number on integer fibres. We further prove similar but partial results on $\mathcal{L}_{w,s,v}(t)$. Our results give examples of subfamilies for which there is rank elevation at integer fibres.
