Arithmetic and Geometry: Ten Years in Alpbach (AMS-202)
By Gisbert Wüstholz and Clemens Fuchs
()
About this ebook
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.
Related to Arithmetic and Geometry
Titles in the series (100)
An Introduction to Linear Transformations in Hilbert Space. (AM-4), Volume 4 Rating: 0 out of 5 stars0 ratingsIntroduction to Non-Linear Mechanics. (AM-11), Volume 11 Rating: 0 out of 5 stars0 ratingsConsistency of the Continuum Hypothesis. (AM-3), Volume 3 Rating: 0 out of 5 stars0 ratingsThe Calculi of Lambda-Conversion (AM-6), Volume 6 Rating: 4 out of 5 stars4/5Meromorphic Functions and Analytic Curves. (AM-12) Rating: 0 out of 5 stars0 ratingsTopics in Topology. (AM-10), Volume 10 Rating: 0 out of 5 stars0 ratingsContributions to Fourier Analysis. (AM-25) Rating: 0 out of 5 stars0 ratingsMetric Methods of Finsler Spaces and in the Foundations of Geometry. (AM-8) Rating: 0 out of 5 stars0 ratingsTranscendental Numbers. (AM-16) Rating: 0 out of 5 stars0 ratingsConvergence and Uniformity in Topology. (AM-2), Volume 2 Rating: 0 out of 5 stars0 ratingsThe Two-Valued Iterative Systems of Mathematical Logic. (AM-5), Volume 5 Rating: 0 out of 5 stars0 ratingsCohomology Operations (AM-50), Volume 50: Lectures by N. E. Steenrod. (AM-50) Rating: 0 out of 5 stars0 ratingsSurface Area. (AM-35), Volume 35 Rating: 0 out of 5 stars0 ratingsProbleme General de la Stabilite du Mouvement. (AM-17), Volume 17 Rating: 0 out of 5 stars0 ratingsExistence Theorems in Partial Differential Equations. (AM-23), Volume 23 Rating: 0 out of 5 stars0 ratingsAlgebraic Theory of Numbers. (AM-1), Volume 1 Rating: 0 out of 5 stars0 ratingsFourier Transforms. (AM-19), Volume 19 Rating: 0 out of 5 stars0 ratingsIsoperimetric Inequalities in Mathematical Physics. (AM-27), Volume 27 Rating: 0 out of 5 stars0 ratingsLectures on Differential Equations. (AM-14), Volume 14 Rating: 0 out of 5 stars0 ratingsFinite Dimensional Vector Spaces. (AM-7), Volume 7 Rating: 4 out of 5 stars4/5Automata Studies. (AM-34), Volume 34 Rating: 5 out of 5 stars5/5Contributions to the Theory of Games (AM-24), Volume I Rating: 0 out of 5 stars0 ratingsDegree of Approximation by Polynomials in the Complex Domain. (AM-9), Volume 9 Rating: 0 out of 5 stars0 ratingsContributions to the Theory of Games (AM-28), Volume II Rating: 0 out of 5 stars0 ratingsFunctional Operators (AM-21), Volume 1: Measures and Integrals. (AM-21) Rating: 0 out of 5 stars0 ratingsA Theory of Cross-Spaces. (AM-26), Volume 26 Rating: 0 out of 5 stars0 ratingsLinear Inequalities and Related Systems. (AM-38), Volume 38 Rating: 0 out of 5 stars0 ratingsContributions to the Theory of Nonlinear Oscillations (AM-20), Volume I Rating: 0 out of 5 stars0 ratingsCurvature and Betti Numbers. (AM-32), Volume 32 Rating: 0 out of 5 stars0 ratingsThe Neumann Problem for the Cauchy-Riemann Complex. (AM-75), Volume 75 Rating: 4 out of 5 stars4/5
Related ebooks
Complex Analysis Rating: 3 out of 5 stars3/5Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188) Rating: 0 out of 5 stars0 ratingsTopos Theory Rating: 0 out of 5 stars0 ratingsFunctional Analysis: Introduction to Further Topics in Analysis Rating: 5 out of 5 stars5/5Homotopical Topology Rating: 0 out of 5 stars0 ratingsTopological Methods in Euclidean Spaces Rating: 0 out of 5 stars0 ratingsThe Arithmetic of Elliptic Curves Rating: 5 out of 5 stars5/5Real Analysis: Measure Theory, Integration, and Hilbert Spaces Rating: 4 out of 5 stars4/5Fourier Analysis: An Introduction Rating: 3 out of 5 stars3/5Differential Forms in Algebraic Topology Rating: 5 out of 5 stars5/5Theory of H[superscript p] spaces Rating: 0 out of 5 stars0 ratingsComplex Ball Quotients and Line Arrangements in the Projective Plane (MN-51) Rating: 0 out of 5 stars0 ratingsTopological Methods in Galois Representation Theory Rating: 0 out of 5 stars0 ratingsNumber Theory Rating: 4 out of 5 stars4/5States of Matter Rating: 4 out of 5 stars4/5Numerical Methods for Roots of Polynomials - Part I Rating: 0 out of 5 stars0 ratingsComplex Numbers: Lattice Simulation and Zeta Function Applications Rating: 0 out of 5 stars0 ratingsThe Story of Algebraic Numbers in the First Half of the 20th Century: From Hilbert to Tate Rating: 0 out of 5 stars0 ratingsPopular Lectures on Mathematical Logic Rating: 0 out of 5 stars0 ratingsExploring University Mathematics: Lectures Given at Bedford College, London Rating: 0 out of 5 stars0 ratingsSieve Methods Rating: 0 out of 5 stars0 ratingsDifferential Geometry of Curves and Surfaces: Revised and Updated Second Edition Rating: 4 out of 5 stars4/5Topoi: The Categorial Analysis of Logic Rating: 5 out of 5 stars5/5Uniform Distribution of Sequences Rating: 4 out of 5 stars4/5Rings of Continuous Functions Rating: 0 out of 5 stars0 ratingsThe Origins of Cauchy's Rigorous Calculus Rating: 5 out of 5 stars5/5General Topology Rating: 4 out of 5 stars4/5An Introduction to Fourier Series and Integrals Rating: 5 out of 5 stars5/5Mathematics of Classical and Quantum Physics Rating: 3 out of 5 stars3/5Algebraic and Analytic Methods in Representation Theory Rating: 0 out of 5 stars0 ratings
Mathematics For You
Quantum Physics for Beginners Rating: 4 out of 5 stars4/5Algebra - The Very Basics Rating: 5 out of 5 stars5/5My Best Mathematical and Logic Puzzles Rating: 5 out of 5 stars5/5Basic Math & Pre-Algebra For Dummies Rating: 4 out of 5 stars4/5The Thirteen Books of the Elements, Vol. 1 Rating: 0 out of 5 stars0 ratingsThe Everything Guide to Algebra: A Step-by-Step Guide to the Basics of Algebra - in Plain English! Rating: 4 out of 5 stars4/5Calculus Made Easy Rating: 4 out of 5 stars4/5The Little Book of Mathematical Principles, Theories & Things Rating: 3 out of 5 stars3/5The Everything Everyday Math Book: From Tipping to Taxes, All the Real-World, Everyday Math Skills You Need Rating: 5 out of 5 stars5/5Algebra I Workbook For Dummies Rating: 3 out of 5 stars3/5The Golden Ratio: The Divine Beauty of Mathematics Rating: 5 out of 5 stars5/5Standard Deviations: Flawed Assumptions, Tortured Data, and Other Ways to Lie with Statistics Rating: 4 out of 5 stars4/5ACT Math & Science Prep: Includes 500+ Practice Questions Rating: 3 out of 5 stars3/5Algebra I For Dummies Rating: 4 out of 5 stars4/5The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics Rating: 3 out of 5 stars3/5Precalculus: A Self-Teaching Guide Rating: 4 out of 5 stars4/5Introducing Game Theory: A Graphic Guide Rating: 4 out of 5 stars4/5Flatland Rating: 4 out of 5 stars4/5Mental Math Secrets - How To Be a Human Calculator Rating: 5 out of 5 stars5/5Relativity: The special and the general theory Rating: 5 out of 5 stars5/5Real Estate by the Numbers: A Complete Reference Guide to Deal Analysis Rating: 0 out of 5 stars0 ratingsLimitless Mind: Learn, Lead, and Live Without Barriers Rating: 4 out of 5 stars4/5The Math of Life and Death: 7 Mathematical Principles That Shape Our Lives Rating: 4 out of 5 stars4/5Mental Math: Tricks To Become A Human Calculator Rating: 5 out of 5 stars5/5Math Review: a QuickStudy Laminated Reference Guide Rating: 5 out of 5 stars5/5Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game - Updated Edition Rating: 4 out of 5 stars4/5
Reviews for Arithmetic and Geometry
0 ratings0 reviews