{"product_id":"arithmetic-and-geometry-gisbert-wzstholz-9780691193779","title":"Arithmetic and Geometry: Ten Years in Alpbach","description":"\u003cp\u003e\u003ci\u003eArithmetic and Geometry\u003c\/i\u003e presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures-which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria-provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings height and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach. \u003c\/p\u003e\u003cp\u003e\u003c\/p\u003eThe first course, taught by Peter Scholze, contains his recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces. \u003cp\u003e\u003c\/p\u003eThe second course, taught by Umberto Zannier, addresses the famous Pell equation-not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians. \u003cp\u003e\u003c\/p\u003eThe third course, taught by Shou-Wu Zhang, originates in the Chowla-Selberg formula, which was taken up by Gross and Zagier to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Zhang, X. Yuan, and Wei Zhang prove the Gross-Zagier formula on Shimura curves and verify the Colmez conjecture on average.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eAuthor:\u003c\/b\u003e Gisbert Wüstholz\u003cbr\u003e\u003cb\u003eISBN-10:\u003c\/b\u003e 0691193770\u003cbr\u003e\u003cb\u003eISBN-13:\u003c\/b\u003e 9780691193779\u003cbr\u003e\u003cb\u003ePublisher:\u003c\/b\u003e Princeton University Press\u003cbr\u003e\u003cb\u003eLanguage:\u003c\/b\u003e English\u003cbr\u003e\u003cb\u003ePublished:\u003c\/b\u003e 10\/08\/2019\u003cbr\u003e\u003cb\u003ePages:\u003c\/b\u003e 186\u003cbr\u003e\u003cb\u003eFormat:\u003c\/b\u003e Paperback\u003cbr\u003e\u003cb\u003eWeight:\u003c\/b\u003e 0.60lbs\u003cbr\u003e\u003cb\u003eSize:\u003c\/b\u003e 9.10h x 6.10w x 0.40d","brand":"Gisbert Wüstholz","offers":[{"title":"Paperback","offer_id":48443611021567,"sku":"9780691193779","price":87.0,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0662\/2982\/9887\/files\/img_5a0201a3-9862-4e4c-9ec5-0adec34936db.jpg?v=1777206613","url":"https:\/\/www.whiterainbookhouse.com\/products\/arithmetic-and-geometry-gisbert-wzstholz-9780691193779","provider":"WR Book House","version":"1.0","type":"link"}