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Shimura Varieties, Galois Representations and Motives Shimura varieties, Galois representations and motives Gregorio Baldi A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy of University College London. Department of Mathematics University College London September 12, 2020 2 I, Gregorio Baldi, confirm that the work presented in this thesis is my own, except the contents of Chapter5, which is based on a collaboration with Giada Grossi, and Chapter7, which is based on a collaboration with Emmanuel Ullmo. Where information has been derived from other sources, I confirm that this has been indicated in the work. Abstract This thesis is about Arithmetic Geometry, a field of Mathematics in which tech- niques from Algebraic Geometry are applied to study Diophantine equations. More precisely, my research revolves around the theory of Shimura varieties, a special class of varieties including modular curves and, more generally, moduli spaces parametrising principally polarised abelian varieties of given dimension (possibly with additional prescribed structures). Originally introduced by Shimura in the ‘60s in his study of the theory of complex multiplication, Shimura varieties are com- plex analytic varieties of great arithmetic interest. For example, to an algebraic point of a Shimura variety there are naturally attached a Galois representation and a Hodge structure, two objects that, according to Grothendieck’s philosophy of mo- tives, should be intimately related. The work presented here is largely motivated by the (recent progress towards the) Zilber–Pink conjecture, a far reaching conjecture generalising the André–Oort and Mordell–Lang conjectures. More precisely, we first prove a conjecture of Buium–Poonen which is an in- stance of the Zilber-Pink conjecture (for a product of a modular curve and an elliptic curve). We then present Galois-theoretical sufficient conditions for the existence of rational points on certain Shimura varieties: the moduli space of K3 surfaces and Hilbert modular varieties (the latter case is joint work with G. Grossi). The main idea underlying such works comes from Langlands programme: to a compatible system of Galois representations one can attach an analytic object (like a classi- cal/Hilbert modular form or a Hodge structure), which in turn determines a motive which eventually gives an algebraic point of a Shimura variety. We then prove a geometrical version of Serre’s Galois open image theorem for arbitrary Shimura Abstract 4 varieties. We finally discuss representation-theoretical conditions for a variation of Hodge structures to admit an integral structure (joint work with E. Ullmo). Impact Statement I expect that the results of this thesis will have impact in various areas of Mathemat- ics. Indeed, I have been using tools from Number Theory, Complex and Algebraic Geometry, and Representation Theory. Thus, I believe my work will influence and generate new research in these areas. To achieve this impact, I have already pub- lished in peer-reviewed journals three papers, and have three more posted on the arXiv. As evidence of interest in my work, I have been invited to present my re- search in various occasions around Europe. For example at the XXI congress of the Italian Mathematical Society (Pavia, IT), at the UMI-PTM-SIMAI conference (Poland), at the Summer school on explicit and computational approaches to Galois representations (Luxemburg), at the university of Amsterdam and at the Workshop on K3 surfaces and Galois representation (Shepperton, UK). More recently I have also been invited to give an online talk at the R.O.W. seminar. Acknowledgements First I would like to thank Andrei Yafaev for being a unique supervisor and friend. You have always believed in me and supported my choices, encouraging me to find my own path in the beautiful theory of Shimura varieties. I am happy to thank Emmanuel Ullmo for countless discussions. Collaborating with you was certainly a pleasure and an honour. I am grateful to Alexei Skorobogatov for discussions about rational points and K3 surfaces, to Martin Orr for comments related to the André–Pink conjecture and to Bruno Klingler for various conversations about Hodge theory and `-adic coho- mology. I would like to thank also Minhyong Kim and Luis Garcia for agreeing to being the referees of this thesis. I thank Giada for our lovely collaboration on the content of Chapter5. Not only these years in London wouldn’t have been the same without you, but all my life. Thank you Giada for always being with me. I have been fortunate enough to meet many other students and, among them, make many friends. Thank you to Andrea, Marco, Petru, Udhav and Joe. For all the crazy discussions about anything, thank you Emilianomari! For even crazier discussions, thank you Mimmo. I am grateful to Matteo for being always happy to carefully read any drafts. I have learnt a lot from all of you. I am grateful to the anonymous referees of the published/submitted papers, whose comments have certainly improved the exposition, and for having spotted several inaccuracies. It is hard to publicly acknowledge your work, but I have cer- tainly beneficed from all your comments. I am grateful to the I.H.E.S. for having hosted me, with amazing working con- Acknowledgements 7 ditions, during various visits. In particular most of the work of7 was done during a visit in October 2019. My doctoral studies were funded by the Engineering and Physical Sciences Research Council [EP/L015234/1]. The EPSRC Centre for Doc- toral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London. On the non-mathematical side, thank you to my family for always supporting me. It is hard to describe my gratitude in only a few lines, but I am grateful for your love and support throughout these years. Contents 1 Introduction and main results 12 1.1 What is a rational point?....................... 12 1.2 Rational points and elliptic curves.................. 15 1.3 Rational points of Shimura varieties................. 18 1.4 The geometry of Shimura varieties.................. 20 2 Preliminaries 22 2.1 Notations............................... 22 2.2 Shimura varieties........................... 23 2.2.1 Shimura varieties as moduli spaces of Hodge structures... 25 2.2.2 Canonical models and reflex field.............. 29 2.2.3 Some examples of Shimura varieties............. 29 2.3 Galois representation......................... 33 2.3.1 Weakly compatible systems................. 33 2.3.2 Galois representations attached to algebraic points of Shimura varieties....................... 35 2.4 Motives................................ 37 2.4.1 Tannakian categories..................... 38 2.4.2 Realisations.......................... 39 2.5 Some conjectures in Arithmetic Geometry relating the three worlds. 41 2.5.1 Tate and semisimplicity................... 41 2.5.2 Fontaine–Mazur....................... 42 2.5.3 Hodge............................. 43 Contents 9 2.5.4 Simpson........................... 45 3 On a conjecture of Buium and Poonen 46 3.1 Introduction.............................. 46 3.1.1 Main results.......................... 48 3.1.2 Related work......................... 50 3.2 Preliminaries............................. 51 3.2.1 Hecke operators....................... 52 3.2.2 Equidistribution of Hecke points............... 55 3.3 Proofs of the main results....................... 56 3.3.1 Proof of Theorem 3.3.1.................... 57 3.3.2 Shimura varieties....................... 59 3.3.3 Quaternionic Shimura curves................. 61 3.3.4 A remark on Masser-Wüstholz Isogeny Theorem...... 62 3.4 Mixed Shimura varieties and the Zilber–Pink conjecture....... 63 3.4.1 Zilber–Pink conjecture.................... 64 4 Local to Global principle for the moduli space of K3 surfaces 69 4.1 Introduction.............................. 69 4.1.1 Main results.......................... 71 4.1.2 Comments on the assumptions of Theorem 4.1.3...... 74 4.1.3 How to get rid of the extension L=K ............. 75 4.1.4 Examples and applications.................. 75 4.2 The motive of a surface........................ 78 4.3 Proof of Theorem 4.1.3........................ 79 4.4 Descent to a number field....................... 83 5 Finite descent obstruction for Hilbert modular varieties (joint with G. Grossi) 86 5.1 Introduction.............................. 86 5.1.1 Main results.......................... 87 5.1.2 Serre’s weak conjecture over totally real fields....... 89 Contents 10 5.1.3 Related work......................... 90 5.2 Recap on Hilbert modular varieties and modular forms....... 91 5.2.1 Hilbert modular varieties................... 91 5.2.2 Eichler–Shimura theory................... 94 5.2.3 A remark on polarisations.................. 98 5.3 Producing abelian varieties via Serre’s conjecture.......... 98 5.3.1 Weakly compatible systems with coefficients........ 98 5.3.2 Key proposition........................ 99 5.3.3 Proof of Theorems 5.1.1 and 5.1.2.............. 102 5.3.4 A corollary.......................... 103 5.4 Finite descent obstruction and proof of Theorem 5.1.3........ 104 5.4.1 Recap on the integral finite descent obstruction....... 104 5.4.2 Integral points on Hilbert modular varieties......... 105 5.4.3 Proof of Theorem 5.1.3.................... 108 6 On the geometric Mumford–Tate conjecture for subvarieties of Shimura varieties 110 6.1 Introduction.............................. 110 6.1.1 Geometric and `-adic monodromy.............. 110 6.1.2 Main theorem........................
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