We completely classify the possible extensions between semistable vector bundles on the Fargues–Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder–Narasimhan (HN) polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze’s language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the Euclidean geometry of HN polygons.

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

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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.

We introduce a derived enhancement of local Galois deformation rings that we call the "spectral Hecke algebra", in analogy to a construction in the Geometric Langlands program. This is a Hecke algebra that acts on the spectral side of the Langlands correspondence, i.e. on moduli spaces of Galois representations. We verify the simplest form of derived local-global compatibility between the action of the spectral Hecke algebra on the derived Galois deformation ring of Galatius-Venkatesh, and the action of Venkatesh's derived Hecke algebra on the cohomology of arithmetic groups.