论文标题
$ l $功能的Manin-Drinfeld周期和衍生物
Manin-Drinfeld cycles and derivatives of $L$-functions
论文作者
论文摘要
我们在$ \ mathrm {pgl} _2 $ -shtukas的模量空间中研究代数周期，由对角线圆环产生。我们的主要结果表明，他们与Heegner-Drinfeld周期配对是$ r $ th-th中央衍生物的产物，该衍生物是汽车$ l $ l $ -function $ l（π，s）$和Waldspurger的复合周期的积分。当$ l（π，\ frac12）\ neq 0 $时，这为泰勒系列扩展提供了新的几何解释。当$ l（π，\ frac12）= 0 $时，配对消失了，表明模块化雅各比式中尖牙消失的高阶类似物以及其他新现象。 我们的证明阐明了Yun和Zhang介绍的代数对应关系的新启示，这是``区分$ l $ function''的几何化身。我们将其视为$ e+f \ e+f \ in \ mathfrak in \ mathfrak {sl} _2 $ on $ e+f \ in $ e+f \ on $ e+f \ on $ e+f \ on $ e+f \ in $ e+f \ in $ e+nimie 2d} $。
We study algebraic cycles in the moduli space of $\mathrm{PGL}_2$-shtukas, arising from the diagonal torus. Our main result shows that their intersection pairing with the Heegner-Drinfeld cycle is the product of the $r$-th central derivative of an automorphic $L$-function $L(π,s)$ and Waldspurger's toric period integral. When $L(π,\frac12) \neq 0$, this gives a new geometric interpretation for the Taylor series expansion. When $L(π,\frac12) = 0$, the pairing vanishes, suggesting higher order analogues of the vanishing of cusps in the modular Jacobian, as well as other new phenomena. Our proof sheds new light on the algebraic correspondence introduced by Yun and Zhang, which is the geometric incarnation of ``differentiating the $L$-function". We realize it as the Lie algebra action of $e+f \in \mathfrak{sl}_2$ on $(\mathbb{Q}_\ell^2)^{\otimes 2d}$. The comparison of relative trace formulas needed to prove our formula is then a consequence of Schur-Weyl duality.
