Higher Siegel--Weil formula for unitary groups: the non-singular terms

Abstract

We construct special cycles on the moduli stack of unitary shtukas. We prove an identity between (1) the r 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 unitary shtukas with 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. Part 2. The geometric side 28 6. Moduli of unitary shtukas 28 7. Special cycles: basic properties 31 8. Hitchin-type moduli spaces 37 9. Special cycles of corank one 41 10. Comparison of two cycle classes 46 11. Local intersection number and trace formula 54 Part 3. The comparison 59 12. Matching of sheaves 59 References 62

Special cycles for Shtukas are closed

“…The main observation is that the forgetful map Sht r G (W) • → Sht r G is proper. This argument was used in [7] to show that Kudla-Rapoport special cycles are closed.…”

“…Let G = U n using a double cover as above, and let E be a vector bundle of rank m ≤ n over X . In [7,8] we define a special cycle Z r E on Sht r G parametrizing unitary Shtukas with a Frobenius invariant map from E. These are function field analogues of the special cycles defined by Kudla and Rapoport [11,12]. In the case m = n, they are 0-cycles and their degrees are given by the Fourier coefficients of the rth central derivative of Siegel-Eisenstein series [7].…”

Special cycles for Shtukas are closed

“…In [7,8] we define a special cycle Z r E on Sht r G parametrizing unitary Shtukas with a Frobenius invariant map from E. These are function field analogues of the special cycles defined by Kudla and Rapoport [11,12]. In the case m = n, they are 0-cycles and their degrees are given by the Fourier coefficients of the rth central derivative of Siegel-Eisenstein series [7]. A basic question, before one can even call special cycles "cycles", is to show that their image in Sht r G is closed.…”

On the arithmetic Siegel--Weil formula for GSpin Shimura varieties

“…[LL20,LL21] by Liu and one of us in the unitary case). It is also worth mentioning several other recent advances in arithmetic Siegel-Weil formula in the unitary case, including cases of the singular term formula [BH21] by Bruinier-Howard and the higher derivative formula over function fields [FYZ21] by Feng, Yun and one of us, and it would be interesting to study the (arguably more difficult) orthogonal analogues.…”