Manin-Drinfeld cycles and derivatives of $L$-functions

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 $L(\pi,s)$ and Waldspurger's toric period integral. When $L(\pi,\frac12) \neq 0$, this gives a new geometric interpretation for the Taylor series expansion. When $L(\pi,\frac12) = 0$, 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.