On Shalika Periods and a Theorem of Jacquet-Martin

Details On Shalika Periods and a Theorem of Jacquet-Martin On Shalika Periods and a Theorem of Jacquet-Martin

2007-05-11

arxiv.org/abs/0705.1576v3

Mathematics

Number Theory

Scientific paper

Let \pi be a cuspidal automorphic representation of GL_4 with central character \mu^2. It is known that \pi has Shalika period with respect to \mu if and only if the L-function L^S(s, \pi, \bigwedge^2\otimes\mu^{-1}) has a pole at s=1. Recentlt, Jacquet and Martin considered the analogous question for cuspidal representations \pi_D of the inner form GL_2(D)(\A), and obtained a partial result via the relative trace formula. In this paper, we provide a complete solution to this problem via the method of theta correspondence, and give necessary and sufficient conditions for the existence of Shalika period for \pi_D. We also resolve the analogous question in the local setting.

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