Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series

Abstract

Let $V$ be a rational quadratic space of signature $(m;2)$. A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with $\mathrm{SO}(V)$ to the coefficients of the central derivative of an incoherent Siegel Eisenstein series of genus $m+1$. We prove this conjecture for the coefficients of non-singular index $T$ when $T$ is not positive definite. We also prove it when $T$ is positive definite and the corresponding special cycle has dimension 0. To obtain these results, we establish new local arithmetic Siegel–Weil formulas at the archimedian and non-archimedian places.