{"title":"Series: Annals of Mathematics Studies","description":null,"products":[{"product_id":"moduli-stacks-of-etale-ϕ-γ-modules-and-the-existence-of-crystalline-lifts-by-matthew-emerton-9780691241357","title":"Moduli Stacks of Étale (ϕ, Γ)-Modules and the Existence of Crystalline Lifts","description":"\u003cdiv class=\"book-description\"\u003e\n\u003cp\u003e\u003cb\u003eA foundational account of a new construction in the \u003ci\u003ep\u003c\/i\u003e-adic Langlands correspondence\u003c\/b\u003e\u003c\/p\u003e\n\n\u003cp\u003eMotivated by the \u003ci\u003ep\u003c\/i\u003e-adic Langlands program, \u003ci\u003eModuli Stacks of Étale (ϕ, Γ)-Modules and the Existence of Crystalline Lifts\u003c\/i\u003e constructs stacks that algebraize Mazur's formal deformation rings of local Galois representations. More precisely, it constructs Noetherian formal algebraic stacks over Spf \u003cb\u003eZ\u003c\/b\u003e\u003csub\u003ep\u003c\/sub\u003e that parameterise étale (ϕ, Γ)-modules. The formal completions of these stacks at points in their special fibres recover the universal deformation rings of local Galois representations.\u003c\/p\u003e\n\n\u003cp\u003eThese stacks are then used to show that all mod \u003ci\u003ep\u003c\/i\u003e representations of the absolute Galois group of a \u003ci\u003ep\u003c\/i\u003e-adic local field lift to characteristic zero, and indeed admit crystalline lifts. The book explicitly describes the irreducible components of the underlying reduced substacks and discusses the relationship between the geometry of these stacks and the Breuil-Mézard conjecture.\u003c\/p\u003e\n\n\u003cp\u003eAlong the way, it proves a number of foundational results in \u003ci\u003ep\u003c\/i\u003e-adic Hodge theory that may be of independent interest.\u003c\/p\u003e\n\u003c\/div\u003e","brand":"NewSouth Books","offers":[{"title":"Default Title","offer_id":46854682083564,"sku":"9780691241357","price":160.0,"currency_code":"NZD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0705\/7784\/8556\/files\/e38480fa1a874c3fcd55dcb90942656e.jpg?v=1759266384"},{"product_id":"simple-algebras-base-change-and-the-advanced-theory-of-the-trace-formula-by-james-arthur-9780691085180","title":"Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula","description":"\u003cdiv class=\"book-description\"\u003e\n\u003cp\u003eA general principle, discovered by Robert Langlands and named by him the \"functoriality principle,\" predicts relations between automorphic forms on arithmetic subgroups of different reductive groups. Langlands functoriality relates the eigenvalues of Hecke operators acting on the automorphic forms on two groups (or the local factors of the \"automorphic representations\" generated by them). In the few instances where such relations have been probed, they have led to deep arithmetic consequences.\u003c\/p\u003e\n\n\u003cp\u003e\u003cem\u003eSimple Algebras, Base Change, and the Advanced Theory of the Trace Formula\u003c\/em\u003e studies one of the simplest general problems in the theory—that of relating automorphic forms on arithmetic subgroups of GL(n,E) and GL(n,F) when E\/F is a cyclic extension of number fields. This is known as the base change problem for GL(n). The problem is attacked and solved by means of the trace formula.\u003c\/p\u003e\n\n\u003cp\u003eThe book relies on deep and technical results obtained by several authors during the last twenty years. It could not serve as an introduction to them, but, by giving complete references to the published literature, the authors have made the work useful to a reader who does not know all the aspects of the theory of automorphic forms.\u003c\/p\u003e\n\u003c\/div\u003e","brand":"NewSouth Books","offers":[{"title":"Default Title","offer_id":46854830915820,"sku":"9780691085180","price":182.0,"currency_code":"NZD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0705\/7784\/8556\/files\/1a123ccf153afb0340ffae533133dfed.jpg?v=1759264247"}],"url":"https:\/\/bookhero.co.nz\/collections\/series-annals-of-mathematics-studies.oembed","provider":"Book Hero","version":"1.0","type":"link"}