On the Mazur--Tate refined conjecture for the anticyclotomic tower at inert primes

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with supersingular reduction at $p\ge 5$, and $K$ be an imaginary quadratic field such that $p$ is inert in $K\mathrm{/}\mathbb{Q}$. In this paper, we prove the analogous of the ``weak’’ Mazur–Tate refined conjecture for an anticyclotomic tower over $K$ using the result by A. Burungale–K. Büyükboduk–A. Lei.