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We study algebraic cycles in the moduli space of PGL2-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 (π, s) and Waldspurger's toric period integral. When L(π, 1/2) ≠ 0, this gives a new geometric interpretation for the Taylor series expansion. When L(π, 1/2) 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 + ƒ ∈ sl2 on (ℚℓ²)⊗2d. The comparison of relative trace formulas needed to prove our formula is then a consequence of Schur--Weyl duality. [ABSTRACT FROM AUTHOR]

We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, the central value of the L-function of the relevant motive is non-vanishing and the Ceresa cycle is torsion in the Griffiths group, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients. [ABSTRACT FROM AUTHOR]

Let F be the field of rational functions on a smooth projective curve over a finite field, and let π be an unramified cuspidal automorphic representation for PGL 2 over F. We prove a variant of the formula of Yun and Zhang relating derivatives of the L -function of π to the self-intersections of Heegner-Drinfeld cycles on moduli spaces of shtukas. In our variant, instead of a self-intersection, we compute the intersection pairing of Heegner-Drinfeld cycles coming from two different quadratic extensions of F , and relate the intersection to the r -th derivative of a product of two toric period integrals. [ABSTRACT FROM AUTHOR]