Let E/Q be an elliptic curve and let p be an odd prime of good
reduction for E. Let K be an imaginary quadratic field satisfying the
classical Heegner hypothesis and in which p splits. In a previous work,
Agboola--Castella formulated an analogue of the Birch--Swinnerton-Dyer
conjecture for the p-adic L-function LpBDPβ of
Bertolini--Darmon--Prasanna attached to E/K, assuming the prime p to be
ordinary for E. The goal of this paper is two-fold:
(1) We formulate a p-adic BSD conjecture for LpBDPβ
for all odd primes p of good reduction.
(2) For an algebraic analogue FpβBDPβ of
LpBDPβ, we show that the ``leading coefficient'' part of
our conjecture holds, and that the ``order of vanishing'' part follows from the
expected ``maximal non-degeneracy'' of an anticyclotomic p-adic height.
In particular, when the Iwasawa--Greenberg Main Conjecture
(FpβBDPβ)=(LpBDPβ) is
known, our results determine the leading coefficient of LpBDPβ at T=0 up to a p-adic unit. Moreover, by adapting the approach of
Burungale--Castella--Kim in the p-ordinary case, we prove the main conjecture
for supersingular primes p under mild hypotheses.Comment: 34 page