{"product_id":"arithmetic-and-geometry-ten-years-in-alpbach-9780691193786","title":"Arithmetic and Geometry: Ten Years in Alpbach","description":"\u003cp\u003e • Author(s): Gisbert Wüstholz | Clemens Fuchs\u003cbr\u003e • Publisher: Princeton University Press\u003cbr\u003e • Publisher Imprint: Princeton University Press\u003cbr\u003e • BISAC: Arithmetic\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\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.","brand":"Atlantic Books","offers":[{"title":"Hardcover","offer_id":46419349078167,"sku":"9780691193786","price":21017.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9780691193786.webp?v=1769074147","url":"https:\/\/atlanticbooks.com\/products\/arithmetic-and-geometry-ten-years-in-alpbach-9780691193786","provider":"Atlantic Books","version":"1.0","type":"link"}