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Compactification of the Drinfeld Modular Surfaces

Compactification of the Drinfeld Modular Surfaces
Author: Thomas Lehmkuhl
Publisher: American Mathematical Soc.
Total Pages: 113
Release: 2009-01-21
Genre: Science
ISBN: 0821842447
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In this article the author describes in detail a compactification of the moduli schemes representing Drinfeld modules of rank 2 endowed with some level structure. The boundary is a union of copies of moduli schemes for Drinfeld modules of rank 1, and its points are interpreted as Tate data. The author also studies infinitesimal deformations of Drinfeld modules with level structure.

Drinfeld Modules

Drinfeld Modules
Author: Mihran Papikian
Publisher: Springer Nature
Total Pages: 541
Release: 2023-03-31
Genre: Mathematics
ISBN: 3031197070
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This textbook offers an introduction to the theory of Drinfeld modules, mathematical objects that are fundamental to modern number theory. After the first two chapters conveniently recalling prerequisites from abstract algebra and non-Archimedean analysis, Chapter 3 introduces Drinfeld modules and the key notions of isogenies and torsion points. Over the next four chapters, Drinfeld modules are studied in settings of various fields of arithmetic importance, culminating in the case of global fields. Throughout, numerous number-theoretic applications are discussed, and the analogies between classical and function field arithmetic are emphasized. Drinfeld Modules guides readers from the basics to research topics in function field arithmetic, assuming only familiarity with graduate-level abstract algebra as prerequisite. With exercises of varying difficulty included in each section, the book is designed to be used as the primary textbook for a graduate course on the topic, and may also provide a supplementary reference for courses in algebraic number theory, elliptic curves, and related fields. Furthermore, researchers in algebra and number theory will appreciate it as a self-contained reference on the topic.

Arithmetic Geometry over Global Function Fields

Arithmetic Geometry over Global Function Fields
Author: Gebhard Böckle
Publisher: Springer
Total Pages: 350
Release: 2014-11-13
Genre: Mathematics
ISBN: 3034808534
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This volume collects the texts of five courses given in the Arithmetic Geometry Research Programme 2009-2010 at the CRM Barcelona. All of them deal with characteristic p global fields; the common theme around which they are centered is the arithmetic of L-functions (and other special functions), investigated in various aspects. Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields: they give respectively a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the construction of Mordell-Weil groups of high rank) and a state of the art survey of Geometric Iwasawa Theory explaining the recent proofs of various versions of the Main Conjecture, in the commutative and non-commutative settings.

Compactifications of Symmetric and Locally Symmetric Spaces

Compactifications of Symmetric and Locally Symmetric Spaces
Author: Armand Borel
Publisher: Springer Science & Business Media
Total Pages: 477
Release: 2006-07-25
Genre: Mathematics
ISBN: 0817644660
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Introduces uniform constructions of most of the known compactifications of symmetric and locally symmetric spaces, with emphasis on their geometric and topological structures Relatively self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to analysis, number theory, algebraic geometry and algebraic topology

Number Fields and Function Fields – Two Parallel Worlds

Number Fields and Function Fields – Two Parallel Worlds
Author: Gerard B. M. van der Geer
Publisher: Springer Science & Business Media
Total Pages: 323
Release: 2006-11-24
Genre: Mathematics
ISBN: 0817644474
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Invited articles by leading researchers explore various aspects of the parallel worlds of function fields and number fields Topics range from Arakelov geometry, the search for a theory of varieties over the field with one element, via Eisenstein series to Drinfeld modules, and t-motives Aimed at graduate students, mathematicians, and researchers interested in geometry and arithmetic and their connections

Yang-Mills Connections on Orientable and Nonorientable Surfaces

Yang-Mills Connections on Orientable and Nonorientable Surfaces
Author: Nan-Kuo Ho
Publisher: American Mathematical Soc.
Total Pages: 113
Release: 2009-10-08
Genre: Mathematics
ISBN: 0821844911
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Download Yang-Mills Connections on Orientable and Nonorientable Surfaces Book in PDF, ePub and Kindle

In ``The Yang-Mills equations over Riemann surfaces'', Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the point of view of Morse theory. In ``Yang-Mills Connections on Nonorientable Surfaces'', the authors study Yang-Mills functional on the space of connections on a principal $G_{\mathbb{R}}$-bundle over a closed, connected, nonorientable surface, where $G_{\mathbb{R}}$ is any compact connected Lie group. In this monograph, the authors generalize the discussion in ``The Yang-Mills equations over Riemann surfaces'' and ``Yang-Mills Connections on Nonorientable Surfaces''. They obtain explicit descriptions of equivariant Morse stratification of Yang-Mills functional on orientable and nonorientable surfaces for non-unitary classical groups $SO(n)$ and $Sp(n)$.

Moduli Spaces of Abelian Surfaces

Moduli Spaces of Abelian Surfaces
Author: Klaus Hulek
Publisher: Walter de Gruyter
Total Pages: 361
Release: 2011-05-03
Genre: Mathematics
ISBN: 3110891921
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The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 6 (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Hypocoercivity

Hypocoercivity
Author: CŽdric Villani
Publisher: American Mathematical Soc.
Total Pages: 154
Release: 2009-10-08
Genre: Mathematics
ISBN: 0821844989
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This memoir attempts at a systematic study of convergence to stationary state for certain classes of degenerate diffusive equations, taking the general form ${\frac{\partial f}{\partial t}}+ L f =0$. The question is whether and how one can overcome the degeneracy by exploiting commutators.