论文标题
二次chabauty和$ p $ -adic Gross-Zagier
Quadratic Chabauty and $p$-adic Gross-Zagier
论文作者
论文摘要
令$ x $为模块化曲线的商$ x_0（n）$，其jacobian $ j_x $是$ j_0（n）^{new} $ a $ \ mathbb {q} $的简单因素。令$ f $为$ n $ and wever 2与$ j_x $相关的新形式；假设$ f $具有分析等级1。我们通过计算两个$ f $ $ f $的二$ p $ addic gross-zagier公式来提供分析方法来确定$ x $的合理点。二次chabauty需要在曲线或其雅各布式上提供理性点；这种新方法消除了这一要求。为了实现这一目标，我们提供了一种算法来计算由Bertolini，Darmon和Prasanna构建的$ f $ $ f $ punction的反风速$ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ prasanna的特殊价值。
Let $X$ be a quotient of the modular curve $X_0(N)$ whose Jacobian $J_X$ is a simple factor of $J_0(N)^{new}$ over $\mathbb{Q}$. Let $f$ be the newform of level $N$ and weight 2 associated with $J_X$; assume $f$ has analytic rank 1. We give analytic methods for determining the rational points of $X$ using quadratic Chabauty by computing two $p$-adic Gross-Zagier formulas for $f$. Quadratic Chabauty requires a supply of rational points on the curve or its Jacobian; this new method eliminates this requirement. To achieve this, we give an algorithm to compute the special value of the anticyclotomic $p$-adic $L$-function of $f$ constructed by Bertolini, Darmon, and Prasanna, which lies outside of the range of interpolation.
