Around 1987, K. Rubin envisioned a signed Iwasawa theory for CM elliptic curves at supersingular primes p over the anticyclotomic Z_p-extension of the CM field, conjectural on a fundamental sign decomposition of the local Iwasawa cohomology. This conjecture was resolved in 2021 (joint with A. Burungale and K. Ota). The essential features of the setting are conjugate self-duality and local root numbers. On the other hand, with progress towards the parity conjecture, the (arithmetic) Gan-Gross-Prasad conjectures and their p-adic variants, the importance of symplectic self-dual p-adic Galois representations is increasing. In this talk, I will report on a recent development on the local sign decomposition for certain symplectic self-dual p-adic Galois representations, generalizing Rubin's decomposition (joint with A. Burungale, K. Nakamura and K. Ota).

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