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Arithmetic and Geometry : T...
Arithmetic and Geometry 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.The 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.The 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.The 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
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monografia Rebiun32134014 https://catalogo.rebiun.org/rebiun/record/Rebiun32134014 m|||||o||d|||||||| cr cnu|||||||| 200406t20192019nju fo d z eng d 0-691-19754-7 10.1515/9780691197548 doi UPVA 998799926203706 CBUC 991013145221706708 DE-B1597 eng DE-B1597 rda eng nju US-NJ Arithmetic and Geometry Ten Years in Alpbach (AMS-202) Gisbert Wüstholz, Clemens Fuchs Princeton, NJ Princeton University Press [2019] Princeton, NJ Princeton, NJ Princeton University Press 2019 1 online resource (187 pages) 1 online resource (187 pages) Annals of Mathematics Studies 202 Previously issued in print: 2019 Includes bibliographical references Frontmatter -- Contents -- Preface -- 1. Introduction -- 2. Local Shimura Varieties: Minicourse Given by Peter Scholze -- 3. Hyperelliptic Continued Fractions and Generalized Jacobians: Minicourse Given by Umberto Zannier -- 4. Faltings Heights and L-functions: Minicourse Given by Shou-Wu Zhang -- List of Contributors Arithmetic and Geometry 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.The 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.The 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.The 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 Specialized In English Fuchs, Clemens editor. edt. http://id.loc.gov/vocabulary/relators/edt Wüstholz, Gisbert editor. edt. http://id.loc.gov/vocabulary/relators/edt 0-691-19378-9 0-691-19377-0 Annals of mathematics studies number 202 Princeton scholarship online