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Arithmetic and Geometry

Ten Years in Alpbach (AMS-202)

Gisbert Wüstholz (Hrsg.), Clemens Fuchs (Hrsg.)

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Sachbuch / Naturwissenschaft

Beschreibung

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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Schlagwörter

Moduli space, Abelian variety, Computation, Double coset, Weil group, L-function, Complex multiplication, Homomorphism, Automorphic form, Szpiro's conjecture, Tate conjecture, Perverse sheaf, Local system, Polylogarithm, Geometric Langlands correspondence, Series expansion, Diagram (category theory), Geometry, Class number formula, Existential quantification, Integer, Shafarevich conjecture, Intersection number, Taylor series, Dimension, Laurent series, Heegner point, Shimura variety, P-adic number, Base change, Summation, Vector bundle, Polynomial, Formal power series, Eisenstein series, Roth's theorem, Topological ring, Presheaf (category theory), Pell's equation, Diophantine equation, Reductive group, Upper and lower bounds, Modular form, Rigid analytic space, Conjugacy class, Stack (mathematics), Big O notation, Mathematics, Monotonic function, Quaternion algebra, Langlands dual group, Isogeny, Siegel zero, Cohomology, Theorem, Hodge theory, Algebraic space, Formal group, Conjecture, Continued fraction, Automorphism, Isomorphism class, Diophantine geometry, Endomorphism, Logarithmic derivative, Special case, Real number, Logarithmic form, Floor and ceiling functions, Cusp form