The Hodge conjecture and arithmetic quotients of complex balls

Preprint, Other literature type English OPEN
Bergeron, Nicolas ; Millson, John ; Moeglin, Colette (2016)
  • Publisher: Institut Mittag-Leffler
  • Journal: (issn: 0001-5962)
  • Related identifiers: doi: 10.1007/s11511-016-0136-2
  • Subject: Mathematics - Algebraic Geometry | Mathematics - Number Theory

Let $S$ be a closed Shimura variety uniformized by the complex $n$-ball. The Hodge conjecture predicts that every Hodge class in $H^{2k} (S, \Q)$, $k=0, \ldots, n$, is algebraic. We show that this holds for all degree $k$ away from the neighborhood $]n/3, 2n/3[$ of the ... View more
  • References (79)
    79 references, page 1 of 8

    (1) Ψ factors through LL, that is Ψ : WR × SL2(C) Ψ→L LL → LU(m) where the last map is the canonical extension [65, Proposition 1.3.5] of the injection L∨ ⊂ U(m)∨, and (2) ϕΨL is the L-parameter of the trivial representation of L. r L = Y U(pj, qj) j=1 with Pj pj = p and Pj qj = q. Moreover: if pjqj = 0 then either pj or qj is equal to 1. We let mj = pj + qj (j = 0, . . . , r) and set 7→ ψ(trace(bβ))

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