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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.

Arithmetical algebraic geometry. --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory --- Abelian variety. --- Algebraic geometry. --- Algebraic independence. --- Algebraic space. --- Analytic number theory. --- Arbitrarily large. --- Automorphic form. --- Automorphism. --- Base change. --- Big O notation. --- Class number formula. --- Cohomology. --- Complex multiplication. --- Computation. --- Conjecture. --- Conjugacy class. --- Continued fraction. --- Cusp form. --- Diagram (category theory). --- Dimension. --- Diophantine equation. --- Diophantine geometry. --- Discriminant. --- Divisible group. --- Double coset. --- Eisenstein series. --- Endomorphism. --- Equation. --- Existential quantification. --- Exponential map (Riemannian geometry). --- Fiber bundle. --- Floor and ceiling functions. --- Formal group. --- Formal power series. --- Formal scheme. --- Fundamental group. --- Geometric Langlands correspondence. --- Geometry. --- Heegner point. --- Hodge structure. --- Hodge theory. --- Homomorphism. --- I0. --- Integer. --- Intersection number. --- Irreducible component. --- Isogeny. --- Isomorphism class. --- Jacobian variety. --- L-function. --- Langlands dual group. --- Laurent series. --- Linear combination. --- Local system. --- Logarithmic derivative. --- Logarithmic form. --- Mathematics. --- Modular form. --- Moduli space. --- Monotonic function. --- Natural topology. --- P-adic analysis. --- P-adic number. --- Pell's equation. --- Perverse sheaf. --- Polylogarithm. --- Polynomial. --- Power series. --- Presheaf (category theory). --- Prime number. --- Projective space. --- Quaternion algebra. --- Rational point. --- Real number. --- Reductive group. --- Rigid analytic space. --- Roth's theorem. --- Series expansion. --- Shafarevich conjecture. --- Sheaf (mathematics). --- Shimura variety. --- Siegel zero. --- Special case. --- Stack (mathematics). --- Subset. --- Summation. --- Szpiro's conjecture. --- Tate conjecture. --- Tate module. --- Taylor series. --- Theorem. --- Theta function. --- Topological ring. --- Topology. --- Torsor (algebraic geometry). --- Upper and lower bounds. --- Vector bundle. --- Weil group. --- Witt vector. --- Zariski topology.

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Das Lösen von Übungsaufgaben gehört im Mathematikstudium zum A und O.  Ihnen fällt die Rolle zu, komplexe Sachverhalte zu illustrieren, und die Beschäftigung mit ihnen fördert und vertieft das Verständnis des Erlernten. Eine Lösung einer Aufgabe gefunden zu haben, manchmal erst nach Stunden, vermittelt das befriedigende Gefühl, die zugrunde liegende Theorie verstanden zu haben und anwenden zu können. Oftmals ist es aber auch einfach schön, sich einer mathematischen Herausforderung zu stellen und diese zu bewältigen. Gerade in der Algebra hat man oft mit nicht unerheblichen Anfangsschwierigkeiten zu kämpfen. Anhand von Übungsaufgaben soll das Buch dazu beitragen, diese faszinierende Welt etwas näher zu bringen. Die Aufgaben in diesem Übungsbuch sind in fünf Gruppen unterteilt. Die Autoren beginnen mit einigen grundlegenden Aufgaben zu Abbildungen und Relationen, anschließend beschäftigen sie sich mit Gruppen, Ringen und Körpern, sowie in einem abschließenden Kapitel mit Darstellungen von endlichen Gruppen. Es werden dann ausführliche Lösungen zu allen Übungsaufgaben gegeben. Zum Schluss wurden noch sechs Probeklausuren mit vollständigen Lösungen angefügt. Sie bieten die Gelegenheit, die Fertigkeit über das Erlernte zu überprüfen.   Der Inhalt: Aufgaben – Lösungen – Probeklausuren – Literaturverzeichnis Die Zielgruppen: - Studierende der Mathematik ab dem 3. Semester - Studierende der Physik und Informatik Die Autoren: Prof. Dr. Clemens Fuchs ist Professor für Mathematik an der Universität Salzburg. Prof. Dr. Gisbert Wüstholz ist Professor für Mathematik an der ETH Zürich.

Algebra.