On the Iwasawa invariants of BDP Selmer groups and BDP p-adic L-functions

Antonio Lei; Katharina Müller; Jiacheng Xia

doi:10.1515/forum-2023-0049

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On the Iwasawa invariants of BDP Selmer groups and BDP p-adic L-functions

Antonio Lei , Katharina Müller und Jiacheng Xia

Veröffentlicht/Copyright: 4. Oktober 2023

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Abstract

Let p be an odd prime. Let f 1 and f 2 be weight 2 cuspidal Hecke eigenforms with isomorphic residual Galois representations at p. Greenberg–Vatsal and Emerton–Pollack–Weston showed that if p is a good ordinary prime for the two forms, the Iwasawa invariants of their p-primary Selmer groups and p-adic L-functions over the cyclotomic ℤ p -extension of ℚ are closely related. The goal of this article is to generalize these results to the anticyclotomic setting. More precisely, let K be an imaginary quadratic field where p splits. Suppose that the generalized Heegner hypothesis holds with respect to both ( f 1 , K ) and ( f 2 , K ) . We study relations between the Iwasawa invariants of the BDP Selmer groups and the BDP p-adic L-functions of f 1 and f 2 .

Keywords: Anticyclotomic Iwasawa theory; congruences of modular forms; Heegner points

MSC 2020: 11R23; 11G40

Communicated by Jan Bruinier

Funding statement: All three authors’ research is supported by the NSERC Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096.

Acknowledgements

Parts of these works were carried out during Antonio Lei’s visit at University College Dublin in fall 2022 supported by a Distinguished Visiting Professorship and the Seed Funding Scheme. He thanks UCD for the financial support and the warm hospitality. He also thanks Kazim Buyukboduk and Daniele Casazza for interesting discussions on subjects related to topics studied in this paper during his visit. The authors would like to thank Tobias Berger, Francesc Castella, Antonio Cauchi, Daniel Delbourgo, Neil Dummigan, Jeffrey Hatley, Ernest Hunter Brooks, David Loeffler, Kimball Martin, Ariel Pacetti and Jan Vonk for answering their questions during the preparation of the article. The authors would also like to thank Chan-Ho Kim, Chao Li and Luochen Zhao for their helpful suggestions and comments. Finally, we thank the anonymous referee for carefully reading an earlier version of the article as well as constructive feedback.

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Received: 2023-02-16

Revised: 2023-06-14

Published Online: 2023-10-04

Published in Print: 2024-01-01

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https://doi.org/10.1515/forum-2023-0049

Schlagwörter für diesen Artikel

Anticyclotomic Iwasawa theory; congruences of modular forms; Heegner points