Abstract. The purpose of this article is to generalize some results of Vatsal on the special values of Rankin-Selberg L-functions in an anticyclotomic Z p -extension. Let g be a cuspidal Hilbert modular newform of parallel weight (2, ..., 2) and level N over a totally real field F , and let K/F be a totally imaginary quadratic extension of relative discriminant D. We study the l-adic valuation of the special values L(g, χ, 1 2 ) as χ varies over the ring class characters of K of P-power conductor, for some fixed prime ideal P. We prove our results under the only assumption that the prime to P part of N is relatively prime to D.

Abstract. In this paper, we prove that a primitive Hilbert cusp form g is uniquely determined by the central values of the Rankin-Selberg L-functions L(f ⊗ g, 1 2 ), where f runs through all primitive Hilbert cusp forms of level q for infinitely many prime ideals q. This result is a generalization of the work of Luo [10] to the setting of totally real number fields.

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The purpose of this paper is to prove that a primitive Hilbert cusp form g is uniquely determined by the central values of the Rankin-Selberg L-functions L(f ⊗ g, 12 ), where f runs through all primitive Hilbert cusp forms of weight k for infinitely many weight vectors k. This result is a generalization of the work of Ganguly, Hoffstein, and Sengupta [3] to the setting of totally real number fields, and it is a weight aspect analogue of the authors own work [6].

Let k be a positive integer such that k ≡ 3 mod 4, and let N be a positive square-free integer.