For ordinary automorphic forms on unitary groups U(r,s) over totally real fields, it proves that a rank-zero Selmer group forces the central L-value to be nonzero.
Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of $p$-adic $L$-functions
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abstract
We study the derivative of the standard $p$-adic $L$-function associated with a $P$-ordinary Siegel modular form (for $P$ a parabolic subgroup of $\mathrm{GL}(n)$) when it presents a semi-stable trivial zero. This implies part of Greenberg's conjecture on the order and leading coefficient of $p$-adic $L$-functions at such trivial zero. We use the method of Greenberg-Stevens. For the construction of the improved $p$-adic $L$-function we develop Hida theory for non-cuspidal Siegel modular forms.
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Iwasawa theory for $\mathrm{U}(r,s)$, Bloch-Kato conjecture and Functional Equation
For ordinary automorphic forms on unitary groups U(r,s) over totally real fields, it proves that a rank-zero Selmer group forces the central L-value to be nonzero.