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Title: Shimura varieties, Galois representations and motives
Author: Baldi, Gregorio
ISNI:       0000 0004 9359 7650
Awarding Body: UCL (University College London)
Current Institution: University College London (University of London)
Date of Award: 2020
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This thesis is about Arithmetic Geometry, a field of Mathematics in which techniques 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 complex 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 motives, 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 instance 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 varieties. We finally discuss representation-theoretical conditions for a variation of Hodge structures to admit an integral structure (joint work with E. Ullmo).
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available