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On the indivisibility of derived Kato’s Euler systems and the main conjecture for modular forms


We provide a simple and efficient numerical criterion to verify the Iwasawa main conjecture and the indivisibility of derived Kato’s Euler systems for modular forms of weight two at any good prime under mild assumptions. In the ordinary case, the criterion works for all members of a Hida family once and for all. The key ingredient is the explicit computation of the integral image of the derived Kato’s Euler systems under the dual exponential map. We provide explicit new examples at the end. This work does not appeal to the Eisenstein congruence method at all.

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This project grew out from C.K.’s year-long discussion with Karl Rubin when he was at UC Irvine. C.K. heartily thanks Liang Xiao and Kâzim Büyükboduk for guiding him to study Kato’s Euler systems and for extremely helpful suggestions and strong encouragement, respectively. C.K. learned many details of Kato’s Euler systems from Kentaro Nakamura and Shanwen Wang during “an explicit week” at KIAS. C.K. also greatly appreciates Masato Kurihara’s constant encouragement and thanks him for pointing out the relation of this work with [27] and valuable comments. C.K. thanks Ashay Burungale pointing out the analogy with Heegner points; Keunyoung Jeong for figuring out some computation in Sect. 5 together; Kazuto Ota for pointing out the non-ordinary generalization; Olivier Fouquet, Minhyong Kim, Robert Pollack, Tadashi Ochiai, and Haining Wang for the helpful discussion and encouragement. C.K. appreciates the generous hospitality of Ulsan National Institute of Science and Technology (UNIST), Keio University, and Shanghai Center for Mathematical Sciences during visits. C.K. was partially supported by a KIAS Individual Grant (SP054102) via the Center for Mathematical Challenges at Korea Institute for Advanced Study, by “Overseas Research Program for Young Scientists” through Korea Institute for Advanced Study, by “the 10th MSJ-Seasonal Institute 2017” through Mathematical Society of Japan, and by Basic Science Research Program through the National Research Foundation of Korea (NRF-2018R1C1B6007009). M.K. thanks to Kentaro Nakamura and Shanwen Wang for giving nice lectures about Euler Systems at KIAS. With their lectures, M.K. got a better picture of the subject. M.K. appreciates Robert Pollack for the useful discussion and encouragement. M.K. also thanks to Byungheup Jun, Jungyun Lee, and Peter J. Cho for general support and constant encouragement. H.S. thanks to Ashay Burungale for helpful conversations and comments about modular symbols. H.S. is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF-2017R1A2B4012408). All we deeply thank the organizers of Iwasawa 2017 for providing us with the intensive atmosphere, which makes it possible for us to finish the first draft during the conference. All we deeply thank the referee for his or her extremely careful reading and comments. A number of inaccuracies are corrected and the exposition is improved a lot due to the comments.

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Kim, CH., Kim, M. & Sun, HS. On the indivisibility of derived Kato’s Euler systems and the main conjecture for modular forms. Sel. Math. New Ser. 26, 31 (2020).

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  • Iwasawa theory
  • Iwasawa main conjectures
  • Kato’s Euler systems
  • Euler systems
  • Kolyvagin systems
  • Modular symbols
  • Hida families

Mathematics Subject Classification

  • Primary 11R23
  • Secondary 11F67