In this paper we prove the \pm-main conjecture formulated by Kobayashi for elliptic curves with supersingular reduction at p such that a_p = 0. We also prove as a corollary the p-part of the BSD formula at supersingular primes when the analytic rank is 0.

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We study a partially linearized version of the relative trace formula for the arithmetic Gan--Gross--Prasad conjecture for the unitary group U(V). The linear factor in this relative trace formula provides an SL_2-symmetry which allows us to prove by induction the arithmetic fundamental lemma over Q_p when p is odd and p ≥ dim V.