Comparing anticyclotomic Selmer groups of positive co-ranks for congruent modular forms

Abstract

We study the variation of Iwasawa invariants of the anticyclotomic Selmer groups of congruent modular forms under the Heegner hypothesis. In particular, we show that even if the Selmer groups we study may have positive coranks, the mu-invariant vanishes for one modular form if and only if it vanishes for the other, and that their lambda-invariants are related by an explicit formula. This generalizes results of Greenberg-Vatsal for the cyclotomic extension, as well as results of Pollack-Weston and Castella-Kim-Longo for the anticyclotomic extension when the Selmer groups in question are cotorsion.2010 Mathematics Subject Classification. 11R18, 11F11, 11F33, 11R23 (primary); 11F85 (secondary).

Fine Selmer groups of congruent p-adic Galois representations

“…Moreover, over Z -extensions one can then often prove equality of -invariants (we refer to Section 2 for the definition of the Iwasawa invariants). Analogous results have been obtained for Selmer groups of Galois representations over the anticyclotomic Z -extension of an imaginary quadratic base field (see [HL19]) and for signed Selmer groups of Galois representations over the cyclotomic Z -extension of a number field in the non-ordinary setting (see for example [Pon20, Section 3]). Moreover, there exist vast generalisations to Selmer groups attached to families of modular forms (see for example [EPW06], [Sha09] and [Bar13]).…”