Buyukboduk-Sakamoto conjectured a formula for the derivatives of Katz p-adic L-functions for CM fields at s=0 and they showed that the existence of Rubin-Stark elements implies their conjecture.
In this talk, we introduce a proof of the conjecture for imaginary quadratic fields using the congruences between CM modular forms and non-CM modular forms. This method can be viewed as an analogue of the proof of Gross-Stark conjecture by Dasgupta-Kakde-Ventullo. Moreover, we will explain that a certain invariant appearing in the derivative formula coincides with the L-invariant defined by Benois. In particular, the formula can be interpreted as a special case of Perrin-Riou's p-adic Beilinson conjecture.
This is a joint work with Ming-Lun Hsieh.

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