{"title":"Toby Gee","description":"\u003cp\u003eDiscover the intricate world of mathematics with the works of Toby Gee, a distinguished figure in the realm of number theory. Gee has made significant contributions to the field through his extensive research and publications.\u003c\/p\u003e\n\n\u003cp\u003eA notable author in the Science \u0026amp; Nature section, Toby Gee specialises in areas such as arithmetic algebraic geometry and modular forms, offering valuable insights into complex mathematical concepts. His publications, including \u003cem\u003eModuli Stacks of Étale (ϕ, Γ)-Modules\u003c\/em\u003e and \u003cem\u003eThe Existence of Crystalline Lifts\u003c\/em\u003e, are true reflections of his expertise and commitment to advancing mathematical knowledge.\u003c\/p\u003e\n\n\u003cp\u003ePerfect for maths enthusiasts and scholars alike, Gee's works provide a deep dive into theoretical constructs and their applications. Whether you're expanding your academic understanding or exploring new mathematical horizons, Toby Gee's books are an essential addition to your collection.\u003c\/p\u003e","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"}],"url":"https:\/\/bookhero.co.nz\/collections\/toby-gee.oembed","provider":"Book Hero","version":"1.0","type":"link"}