Mathematical Research Letters

Volume 26 (2019)

Number 4

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

Pages: 1115 – 1144



Jeffrey Hatley (Department of Mathematics, Union College, Schenectady, New York, U.S.A.)

Antonio Lei (Département de Mathématiques et de Statistique, Université Laval, Québec, QC, Canada)


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.

The second-named author’s research is supported by the NSERC Discovery Grants Program 05710.

Received 14 June 2017

Accepted 30 June 2018

Published 25 October 2019