#### DOCUMENTA MATHEMATICA, Vol. 19 (2014), 709-767

M.-L. Hsieh

Special Values of Anticyclotomic Rankin-Selberg L-Functions

In this article, we construct a class of anticyclotomic \padic Rankin-Selberg $L$-functions for Hilbert modular forms, generalizing the construction of Brako\u{c}evic, Bertolini, Darmon and Prasanna in the elliptic case. Moreover, building on works of Hida, we give a necessary and sufficient criterion for the vanishing of the Iwasawa $\mu$-invariant of this \padic $L$-function vanishes and prove a result on the non-vanishing modulo $p$ of central Rankin-Selberg $L$-values with anticyclotomic twists. These results have future applications to Iwasawa main conjecture for Rankin-Selberg convolution and Iwasawa theory for Heegner cycles.

2010 Mathematics Subject Classification: 11F67 11G15

Keywords and Phrases: Iwasawa theory, p-adic L-functions, mu-invariant.

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