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Abstract: Given a Hecke character of a CM field, the automorphic induction of which to the underlying totally real subfield gives a HIlbert modular form with complex multiplication. By the work of Hida and Tilouine the congruences of such forms are governed by the L-value associated with the Hecke character. On the other hand, the Jacquet-Langlands lift of these modular forms to a quaternion algebra can be realized a the theta lift of the character to the corresponding unitary group. We will focus on the case when the unitary group is definite and discuss how one can p-integrally normalize the theta lift and the Rallis inner product formula. As a result we obtain a similar statement for the congruences. We will also discuss further arithmetic applications if time permits.

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