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We construct special cycles on the moduli stack of hermitian shtukas. We prove an identity between (1) the$r^{\mathrm{th}}$$r^{\mathrm{th}}$central derivative of non-singular Fourier coefficients of a normalized Siegel–Eisenstein series, and (2) the degree of special cycles of “virtual dimension 0” on the moduli stack of hermitian shtukas with$r$$r$legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series.

Fix a positive integernand a finite field$${\mathbb {F}}_q$$$F_q$. We study the joint distribution of the rank$${{\,\mathrm{rk}\,}}(E)$$$\mathrm{rk}\left(E\right)$, then-Selmer group$$\text {Sel}_n(E)$$${\text{Sel}}_n\left(E\right)$, and then-torsion in the Tate–Shafarevich group Equation missing<#comment/>asEvaries over elliptic curves of fixed height$$d \ge 2$$$d\ge 2$over$${\mathbb {F}}_q(t)$$$F_q\left(t\right)$. We compute this joint distribution in the largeqlimit. We also show that the “largeq, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains.