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Let $E/\mathbf{Q}$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. The goal of this paper is two-fold: (1) we formulate a $p$-adic BSD conjecture for the $p$-adic $L$-function $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$introduced by Bertolini–Darmon–Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148]; and (2) for an algebraic analogue $F_{\stackrel{¯<\#comment/>}{\mathfrak{p}}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$of $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$, 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 $\left(F_{\stackrel{¯<\#comment/>}{\mathfrak{p}}}^{\mathrm{B}\mathrm{D}\mathrm{P}}\right)=\left(L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}\right)$is known, our results determine the leading coefficient of $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$at $T=0$up to a $p$-adic unit. Moreover, by adapting the approach of Burungale–Castella–Kim [Algebra Number Theory 15 (2021), pp. 1627–1653], we prove the main conjecture for supersingular primes $p$under mild hypotheses. In the $p$-ordinary case, and under some additional hypotheses, similar results were obtained by Agboola–Castella [J. Théor. Nombres Bordeaux 33 (2021), pp 629–658], but our method is new and completely independent from theirs, and apply to all good primes. 

By investigating second coordination sphere effects on the magnetic anisotropy of a Co(ii) single molecule magnet (SMM), we aim to clarify their role in magneto-structural correlations to enhance the performance of transition metal SMMs. 

A new deep-blue emitting and highly fluorescent anthracene (ANTH) derivative containing perfluorobenzyl (Bn_F) groups, 9,10-ANTH(Bn_F)₂, was synthesized in a single step reaction of ANTH or ANTH(Br)₂with Bn_FI, using either a high-temperature Cu-/Na₂S₂O₃-promoted reaction or via a room-temperature photochemical reaction. Its structure was elucidated by NMR spectroscopy and single crystal X-ray diffractometry. The latter revealed no π–π interaction between neighboring ANTH cores. A combination of high photoluminescence quantum yield (PLQY) of 0.85 for 9,10-ANTH(Bn_F)₂, its significantly improved photostability compared to ANTH and 9,10-ANTH derivatives, and a simple synthetic access makes it an attractive material as a deep-blue OLED emitter and an efficient fluorescent probe.