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.
p-adic L-functions for unitary groups
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abstract
This paper completes the construction of $p$-adic $L$-functions for unitary groups. More precisely, in 2006, the last three named authors proposed an approach to constructing such $p$-adic $L$-functions (Part I). Building on more recent results, including the first named author's construction of Eisenstein measures and $p$-adic differential operators, Part II of the present paper provides the calculations of local $\zeta$-integrals occurring in the Euler product (including at $p$). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.
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2019 1verdicts
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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.