On the anticyclotomic Iwasawa theory of CM forms at supersingular primes

Abstract

In this paper, we study the anticyclotomic Iwasawa theory of a CM form $f$ of even weight $w\ge 2$ 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 (the latter seems linked to the local divisibility of Heegner points). Assuming the truth of these conjectures, we establish a formula for the variation of the sizes of the Selmer groups attached to the central critical twist of $f$ as one climbs up the anticyclotomic tower. We also prove a statement which may be regarded as a form of the anticyclotomic main conjecture (without $p$-adic $L$-functions) for the central critical twist of $f$.