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Elliptic functions. --- Functions, Theta. --- Theta functions --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic

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In its first six chapters this 2006 text seeks to present the basic ideas and properties of the Jacobi elliptic functions as an historical essay, an attempt to answer the fascinating question: 'what would the treatment of elliptic functions have been like if Abel had developed the ideas, rather than Jacobi?' Accordingly, it is based on the idea of inverting integrals which arise in the theory of differential equations and, in particular, the differential equation that describes the motion of a simple pendulum. The later chapters present a more conventional approach to the Weierstrass functions and to elliptic integrals, and then the reader is introduced to the richly varied applications of the elliptic and related functions. Applications spanning arithmetic (solution of the general quintic, the functional equation of the Riemann zeta function), dynamics (orbits, Euler's equations, Green's functions), and also probability and statistics, are discussed.

Elliptic functions --- Elliptic functions. --- Transcendental functions. --- Functions, Transcendental --- Functions, Special --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic

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A comprehensive treatment of elliptic functions is linked by these notes to a study of their application to elliptic curves. This approach provides geometers with the opportunity to acquaint themselves with aspects of their subject virtually ignored by other texts. The exposition is clear and logically carries themes from earlier through to later topics. This enthusiastic work of scholarship is made complete with the inclusion of some interesting historical details and a very comprehensive bibliography.

Elliptic functions. --- Curves, Elliptic. --- Elliptic curves --- Curves, Algebraic --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Curves, Algebraic.

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This book consists of a collection of original, refereed research and expository articles on elliptic aspects of geometric analysis on manifolds, including singular, foliated and non-commutative spaces. The topics covered include the index of operators, torsion invariants, K-theory of operator algebras and L2-invariants. The results presented in this book, which is largely inspired and stimulated by the Atiyah-Singer index theorem, should be of interest to graduates and researchers in mathematical physics, differential topology and differential analysis.

C*-algebras --- Elliptic functions --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Algebras, C star --- Algebras, W star --- C star algebras --- W star algebras --- W*-algebras --- Banach algebras --- Algebra. --- Functional analysis. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Mathematics --- Mathematical analysis

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This book contains a detailed account of the result of the author's recent Annals paper and JAMS paper on arithmetic invariant, including μ-invariant, L-invariant, and similar topics.   This book can be regarded as an introductory text to the author's previous book p-Adic Automorphic Forms on Shimura Varieties.  Written as a down-to-earth introduction to Shimura varieties, this text includes many examples and applications of the theory that provide motivation for the reader.  Since it is limited to modular curves and the corresponding Shimura varieties, this book is not only a great resource for experts in the field, but it is also accessible to advanced graduate students studying number theory.  Key topics include non-triviality of arithmetic invariants and special values of L-functions; elliptic curves over complex and p-adic fields; Hecke algebras; scheme theory; elliptic and modular curves over rings; and Shimura curves.

Curves, Elliptic. --- Invariants. --- Number theory. --- Curves, Elliptic --- Invariants --- Number theory --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Elliptic functions. --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Number study --- Numbers, Theory of --- Mathematics. --- Algebraic geometry. --- Number Theory. --- Algebraic Geometry. --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Geometry, algebraic. --- Algebraic geometry --- Geometry