Arithmetic And Geometry: Ten Years In Alpbach

Arithmetic and Geometry presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lecturesùwhich were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austriaùprovide 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 equationùnot 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 ChowlaûSelberg 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 GrossûZagier formula on Shimura curves and verify the Colmez conjecture on average.

• | Author: Gisbert W?stholz, Clemens Fuchs
• | Publisher: Princeton University Press
• | Publication Date: Oct 08, 2019
• | Number of Pages: 186 pages
• | Language: English
• | Binding: Paperback/Mathematics
• | ISBN-10: 0691193770
• | ISBN-13: 9780691193779