A Gross-Kohnen-Zagier formula for Heegner-Drinfeld cycles

Abstract

Let $F$ be the field of rational functions on a smooth projective curve over a finite field, and let $\pi$ be an unramified cuspidal automorphic representation for ${\mathrm{PGL}}_2$ over $F$. We prove a variant of the formula of Yun and Zhang relating derivatives of the $L$-function of $\pi$ 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.

arXiv comment66 pages; made minor changes to Theorem B, added some details regarding Theorem C, and added a few simplifications suggested by the referee