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This is a very naive question. Has there been any study in mod p theta correspondence? I am particularly interested in knowing to what extend the results of Harris - Kudla - Sweet's paper - http://www.ams.org/journals/jams/1996-9-04/S0894-0347-96-00198-1/home.html are (expected to be) true in this setting.

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I believe Sug Woo Shin has done some work on a mod p theta correspondence; he has given a few talks with that title. I don't know if he's written anything up for public distribution though.

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