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In this paper, we study the anticyclotomic Iwasawa theory of a CM form f of even weight w ≥ 2 at a supersingular prime, generalizing the results in weight 2, due to Agboola and Howard. In due course, we are naturally lead to a conjecture on universal norms that generalizes a theorem of Perrin-Riou and Berger and another that generalizes a conjecture of Rubin (which seems ultimately linked to the local divisibility of Heegner points). Assuming the truth of these conjectures, we establish adoi:10.4171/rmi/828 fatcat:tdouik6qybhmnneg4x5a7d3tvu