Algebraic cycles on Hilbert modular fourfolds and poles of L-functions

Abstract

In this paper we give some evidence for the Tate (and Hodge) conjecture(s) for a class of Hilbert modular fourfolds X, whose connected components arise as arithmetic quotients of the fourfold product of the upper half plane by congruence subgroups Γof SL(2, O_K), where O_K denotes the ring of integers of a quartic, Galois, totally real number field K. The expected relationship to the orders of poles of the associated L-functions is verified for abelian extensions of \Q. Also shown is the existence of homologically non-trivial cycles of codimension two which are not intersections of divisors.

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• A conjecture on Whittaker-Fourier coefficients of cusp forms
• A Cuspidality Criterion for the Exterior Square Transfer of Cusp Forms on GL(4)
• Remarks on the symmetric powers of cusp forms on GL(2)
• Tate cycles on some quaternionic Shimura varieties mod p
• On the size of Satake parameters for unitary cuspidal automorphic representations for GL(4)
• Automorphic tensor products and cuspidal cohomology of the ${\mathrm{GL}}_4$