| Résume | I will explain how a conjectural nonvanishing property for central values of GL_n-automorphic L-functions (consistent with all of the folklore conjectures) would follow from a certain equidistribution property for toric orbits in GL_n locally symmetric spaces, and moreover how we expect this
property should follow from the theorems of Ratner and Margulis-Tomanov on p-adic unipotent flows. In the special case where the base field is a CM field, and the cuspidal automorphic representation of GL_n is conjugate self-dual, this problem can also be phrased in terms of automorphic periods on unitary groups using recent progress on the Ichino-Ikeda Gan-Gross-Prasad conjecture, which I will also explain.
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