Higher theta series for unitary groups over function fields

Abstract

In previous work, we defined certain virtual fundamental classes for special cycles on the moduli stack of Hermitian shtukas, and related them to the higher derivatives of non-singular Fourier coefficients of Siegel-Eisenstein series. In the present article, we construct virtual fundamental classes in greater generality, including those expected to relate to the higher derivatives of singular Fourier coefficients. We assemble these classes into "higher" theta series, which we conjecture to be modular. Two types of evidence are presented: structural properties affirming that the cycle classes behave as conjectured under certain natural operations such as intersection products, and verification of modularity in several special situations. One innovation underlying these results is a new approach to special cycles in terms of derived algebraic geometry. Part 2. Properties of the special cycles 25 5. Derived Hitchin stacks 25 6. Fundamental classes of derived special cycles 37 7. Linear Invariance 45 8. Compatibility with the cycle classes of [FYZ21] 50 Part 3. Evidence 54 9. Nonsingular Fourier coefficients for unitary similitude groups 54 10. Modularity: the case of U (1) 60 11. The corank one case: testing against CM cycles 69 References 721.4. Notation. Throughout this paper, k = F q is a finite field of odd characteristic p. Let ℓ = p be a prime. Let ψ 0 : k → Q × ℓ be a nontrivial character. For any space over F q , we denote by Frob = Frob q the q-power Frobenius endomorphism. 1.4.1. Let X denote a smooth, projective, geometrically connected curve over k, of genus g X . Let ω X be the line bundle of 1-forms on X.Let F = k(X) denote the function field of X. Let |X| be the set of closed points of X. For v ∈ |X|, let O v be the completed local ring of X at v with fraction field F v and residue field k v . Let A = A F denote the ring of adèles of F , and. Let X ′ be another smooth curve over k and ν : X ′ → X be a finite map of degree 2 that is generically étale. We denote by σ the non-trivial automorphism of X ′ over X. The case where X ′ is geometrically disconnected is allowed unless stated otherwise; it is usually allowed in Parts 1 and 2 but not in Part 3.

Special cycles for Shtukas 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].…”