UCLA UCLA Electronic Theses and Dissertations Title On the non-triviality of arithmetic invariants modulo p Permalink https://escholarship.org/uc/item/82p4z5rt Author Burungale, Ashay A. Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California University of California Los Angeles On the Non-Triviality of Arithmetic Invariants Modulo p A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Ashay Burungale 2015 (cid:13)c Copyright by Ashay Burungale 2015 Abstract of the Dissertation On the Non-Triviality of Arithmetic Invariants Modulo p by Ashay Burungale Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Haruzo Hida, Chair Arithmetic invariants are often naturally associated to motives over number (cid:28)elds. One of the basic questions is the non-triviality of the invariants. One typically expects generic non- triviality of the invariants as the motive varies in a family. For a prime p, the invariants can often be normalised to be p-integral. One can thus further ask for the generic non-triviality of the invariants modulo p. The invariants can often be expressed in terms of modular forms. Accordingly, one can try to recast the non-triviality as a modular phenomenon. If the phe- nomena can be proven, the non-triviality typically follows in turn. This principle can be found in the work of Hida and Vatsal among a few others. We have been trying to explore a strategy initiated by Hida in the case of central criti- cial Hecke L-values over the Z -anticyclotomic extension of a CM-(cid:28)eld. The strategy crucially p relies on a linear indepedence of mod p Hilbert modular forms. Several arithmetic invariants seem to admit modular expression analogous to the case of Hecke L-values. This includes the case of Katz p-adic L-function, its cyclotomic derivative and p-adic Abel-Jacobi image of generalised Heegner cycles. We approach the non-triviality of these invariants based on the independence. An analysis of the zero set of the invariants suggests (cid:28)ner versions of the independence. We approach the versions based on Chai’s theory of Hecke stable subvarieties of a mod p Shimura variety. We formulate a conjecture regarding the analogue of the inde- pendence for mod p modular forms on other Shimura varieties. We prove the analogue in the case of quaternionic Shimura varieties over a totally real (cid:28)eld. ii The dissertation of Ashay Burungale is approved. Don Blasius Chandrashekhar Khare Weng Kee Wong Haruzo Hida, Committee Chair University of California, Los Angeles 2015 iii To my teachers ... iv Table of Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Iwasawa µ-invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 p-rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Generalised Heegner cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Conjectural Linear Independence of Mod p Modular Forms . . . . . . . . . 14 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 PEL Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.3 Local algebraic stabiliser . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.4 Distinguished points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.5 Igusa tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.6 Relative Igusa tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.7 Mod p modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.8 Relative mod p modular forms . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.9 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.10 Relative linear independence . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 Locally stable subvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.1 De(cid:28)nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.2 Relative de(cid:28)nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 v 2.3.3 Global structure of locally stable subvarieties . . . . . . . . . . . . . . . 38 2.3.4 Global structure of relative locally stable subvarieties . . . . . . . . . . 41 2.3.5 Chai’s global rigidity conjecture . . . . . . . . . . . . . . . . . . . . . . 43 3 Linear Independence of Mod p Modular Forms on Shimura Curves . . . . 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.2 p-integral model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.3 Idempotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.4 CM points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.5 Igusa tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.6 Mod p modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Serre-Tate deformation space . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.1 Serre-Tate deformation theory . . . . . . . . . . . . . . . . . . . . . . . 56 3.3.2 Reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.3 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.2 Locally stable subvarieties . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.3 Local structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4.4 Global structure I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4.5 Global structure II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.4.6 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4 Non-Triviality of Generalised Heegner Cycles Modulo p . . . . . . . . . . . 78 vi 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 µ-invariant of Anticyclotomic Rankin-Selberg p-adic L-functions . . . . . . . . 87 4.2.1 p-depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.2.2 Anticyclotomic Rankin-Selberg p-adic L-functions . . . . . . . . . . . . 90 4.2.3 µ-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.3 Non-triviality of generalised Heegner cycles modulo p . . . . . . . . . . . . . . 96 4.3.1 p-adic Waldspurger formula . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3.2 Non-triviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 vii Acknowledgments I am grateful to Don Blasius, Chandrashekhar Khare, Haruzo Hida and Weng Kee Wong for being part of the thesis committee. I am also grateful to late Carol Newton, an ex-committee member. Weng Kee Wong kindly agreed to be her replacement. I am indebted to my advisor Haruzo Hida for continuous guidance, encouragement and sup- port. I feel fortunate enough for his suggestion of the thesis topic. The thesis builds on his ideas on the topic. My naive perception of the mathematical world is rather heavily in(cid:29)uenced by his lectures, books and research articles. He has been very generous in sharing his ideas and intuition. His mathematical work will continue to be an inspiration. I am also indebted to my other advisor Chandrashekhar Khare for kind advice and encouragement. He has introduced me to new worlds, mathematical and otherwise, generously sharing his ideas and view points. My interest in Galois representations and literature owes heavily to him. In the near future, I hope to closely take up his mathematical suggestions. For the time to come, conversations with my advisors will continue to be a source of inspiration. I hope that opportunity comes again and again to work with them. I am grateful to my coauthor Ming-Lun Hsieh. At the early stages, he patiently answered questions regarding his work on the non-triviality and his assistance was critical in provid- ing an entry point to the topic. I am grateful to Ching-Li Chai for a number of instructive comments. The thesis crucially relies on his theory of Hecke stable subvarieties of a mod p Shimura variety. I am also grateful to Christopher Skinner for suggestions and encourage- ment. The questions treated in chapter 4 was essentially suggested by him. I thank Mahesh Kakde and Jacques Tilouine for suggestive conversations and encouragement over the years. I thank Francesc Castella and Marc-Hubert Nicole for helpful conversations. I also thank Miljan Brakocevic, Ernest Hunter Brooks, Henri Darmon, Ben Howard, Dimitar Jetchev, Jan NekovÆ(cid:176), Kartik Prasanna, Romyar Shari(cid:28), Richard Taylor, Burt Totaro, Vinayak Vatsal and XinWanfortheirinterestandsuggestions. Ihavebene(cid:28)tedtremendouslyfromvariouscourses viii