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In this paper we develop a new method to study Iwasawa theory and Eisensteinfamilies for unitary groups $\mathrm{U}(r,s)$ of general signature over atotally real field $F$. As a consequence we prove that for a motivecorresponding to a regular algebraic cuspidal automorphic representation $\pi$on $\mathrm{U}(r,s)_{/F}$ which is ordinary at $p$, twisted by a Heckecharacter, if its Selmer group has rank $0$, then the corresponding central$L$-value is nonzero. This generalizes a result of Skinner-Urban in their ICM2006 report in the special case when $F=\mathbb{Q}$ and the motive is conjugateself-dual. Along the way we also obtain $p$-adic functional equations for thecorresponding $p$-adic $L$-functions and $p$-adic families of KlingenEisenstein series. Our method does not involve computing Fourier-Jacobicoefficients (as opposed to previous work which only work in low rank cases,e.g. $\mathrm{U}(1,1)$, $\mathrm{U}(2,0)$ and $\mathrm{U}(1,0)$) whoseautomorphic interpretation is unclear in general.

Iwasawa theory for $\mathrm{U}(r,s)$, Bloch-Kato conjecture and Functional Equation