The Hodge conjecture and arithmetic quotients of complex balls

Abstract

In memory of Raquel Maritza Gilbert beloved wife of the second author.Abstract. Let S be a closed Shimura variety uniformized by the complex n-ball. The Hodge conjecture predicts that every Hodge class in H 2k (S, Q), k = 0, . . . , n, is algebraic. We show that this holds for all degrees k away from the neighborhood ]n/3, 2n/3[ of the middle degree. We also prove the Tate conjecture for the same degrees as the Hodge conjecture and the generalized form of the Hodge conjecture in degrees away from an interval (depending on the codimension c of the subvariety) centered at the middle dimension of S. Finally we extend most of these results to Shimura varieties associated to unitary groups of any signature. The proofs make use of the recent endoscopic classification of automorphic representations of classical groups by [1,58]. As such our results are conditional on the stabilization of the trace formula for the (disconnected) groups GL(N ) ⋊ θ associated to base change. At present the stabilization of the trace formula has been proved only for the case of connected groups. But the extension needed is now announced, see §1.13 for more details.

Superconnections, theta series, and period domains

“…For τ in the Siegel half plane of genus r, we can define a theta series θ(τ ; L r ). We prove similarly that its cohomology class [θ(τ ; L r )] is a H * (X L )-valued holomorphic Siegel modular form of weight 1 2 dim V , with Fourier expansion…”

“…] G and study their properties. These forms (defined in [14] for more general symmetric spaces) have proved to be of fundamental importance in the study of special cycles; see, for example, [12,16] or the recent work of Bergeron, Millson and Moeglin, [1], proving the Hodge conjecture for compact arithmetic quotients of the complex n-ball in certain cohomological degrees. The original motivation for this paper was to understand the construction of the forms ϕ KM .…”

Endoscopy and Cohomology of a Quasi-Split U(4)

“…Suppose ψ ∈ Ψ 2 (G * , η χκ ) S has the property that there is π ∈ Π ψ (G, ξ) with h d (g, K ∞ ; π ∞ ) = 0. Proposition 3.1 implies that π v 0 must be a Langlands quotient of a standard representation with an exponent of the form (z/z) p/2 (zz) (a−1)/2 for some a ≥ N − d. Proposition 13.2 of [2] implies that there is i such that n i ≥ N − d, and we assume that this is n 1 . Note that [2,Prop 13.2] implicitly assumes that the other archimedean components of π have regular infinitesimal character, but this is satisfied in our case.…”

“…In this case, dim π Kn ≤ dim I Kn ≤ |K : K n (B ∩ K)| ≤ q 3n (1 + 1/q) 3 . • π is a submodule of a representation I induced from the standard parabolic of type (2,1) or (1,2). Let P be one of these parabolics, and let the representation of the Levi GL 2 × GL 1 that we induce be π ′ ⊗ χ, where π ′ is supercuspidal.…”

“…The main part of the proof involves using the structure of the packets Π ψ (G) to bound the right hand side of (1) in terms of global multiplicities on smaller quasi-split unitary groups, which we then bound using a theorem of Savin. The key fact that allows us to control the power of Nn we obtain is due to Bergeron, Millson, and Moeglin [2,Prop 13.2], and essentially states that if there exists π ∈ Π ψ (G) with H d (g, K ∞ ; π ∞ ) = 0, then ψ must contain a representation ν(n) with n ≥ N − d. We define a shape to be a list of pairs (n 1 , m 1 ), . .…”