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A sequel to Lectures on Riemann Surfaces (Mathematical Notes, 1966), this volume continues the discussion of the dimensions of spaces of holomorphic cross-sections of complex line bundles over compact Riemann surfaces. Whereas the earlier treatment was limited to results obtainable chiefly by one-dimensional methods, the more detailed analysis presented here requires the use of various properties of Jacobi varieties and of symmetric products of Riemann surfaces, and so serves as a further introduction to these topics as well. The first chapter consists of a rather explicit description of a canonical basis for the Abelian differentials on a marked Riemann surface, and of the description of the canonical meromorphic differentials and the prime function of a marked Riemann surface. Chapter 2 treats Jacobi varieties of compact Riemann surfaces and various subvarieties that arise in determining the dimensions of spaces of holomorphic cross-sections of complex line bundles. In Chapter 3, the author discusses the relations between Jacobi varieties and symmetric products of Riemann surfaces relevant to the determination of dimensions of spaces of holomorphic cross-sections of complex line bundles. The final chapter derives Torelli's theorem following A. Weil, but in an analytical context. Originally published in 1973. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

An unabridged, unaltered printing of the Second Edition (1920), with original format, all footnotes and index: The Series of Natural Numbers - Definition of Number - Finitude and Mathematical Induction - The Definition of Order - Kinds of Relations - Similarity of Relations - Rational, Real, and Complex Numbers - Infinite Cardinal Numbers - Infinite Series and Ordinals - Limits and Continuity - Limits and Continuity of Functions - Selections and the Multiplicative Axiom - The Axiom of Infinity and Logical Types - Incompatibility and the Theory of Deductions - Propositional Functions - Descriptions - Classes - Mathematics and Logic - Index

2014 Reprint of 1959 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. Serge Lang was a French-born American mathematician. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He was a member of the Bourbaki group.

2014 Reprint of 1955 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. Reprint of the 3rd Edition. Weyl was a German American mathematician who, through his widely varied contributions in mathematics, served as a link between pure mathematics and theoretical physics, in particular adding enormously to quantum mechanics and the theory of relativity. Hermann Weyl (1885-1955) was perhaps the most important and, above all, the most multifaceted of David Hilbert's students. His life's work encompassed such varied disciplines as number theory, complex analysis, mathematical physics, and geometry. His youthful work "The Concept of a Riemann Surface," which was published in 1913 by Teubner, in Leipzig, quickly achieved acclaim as an epochal work, a work that exerted lasting influence on several branches of mathematics.