Higher Siegel–Weil formula over number fields

Develop a higher Siegel–Weil formula over number fields for unitary groups, relating nonsingular Fourier coefficients of the r-th derivative (for r>1) of Siegel Eisenstein series to intersection numbers of higher-leg special cycles on appropriate moduli spaces or Shimura varieties, extending the known function-field results to the number-field setting.

Background

Feng–Yun–Zhang proved a higher Siegel–Weil formula over function fields in unramified settings, connecting r-th derivatives of Eisenstein series and intersections of special cycles on moduli of Drinfeld shtukas, with parallels to classical (r=0) and arithmetic (r=1) Siegel–Weil. The author notes that no analogous number-field result is known for r>1, highlighting a significant gap in the theory and an important direction for future work.