Series:

Lecture Notes in Mathematics

5 primary works

Book 504

Cover of Frobenius Distributions in GL2-Extensions

Book 1433

Cover of Topics in Nevanlinna Theory
These are notes of lectures on Nevanlinna theory, in the classical case of meromorphic functions, and the generalization by Carlson-Griffith to equidimensional holomorphic maps using as domain space finite coverings of C resp. Cn. Conjecturally best possible error terms are obtained following a method of Ahlfors and Wong. This is especially significant when obtaining uniformity for the error term w.r.t. coverings, since the analytic yields case a strong version of Vojta's conjectures in the number-theoretic case involving the theory of heights. The counting function for the ramified locus in the analytic case is the analogue of the normalized logarithmetic discriminant in the number-theoretic case, and is seen to occur with the expected coefficient 1. The error terms are given involving an approximating function (type function) similar to the probabilistic type function of Khitchine in number theory. The leisurely exposition allows readers with no background in Nevanlinna Theory to approach some of the basic remaining problems around the error term. It may be used as a continuation of a graduate course in complex analysis, also leading into complex differential geometry.

Book 1564

Cover of Basic Analysis of Regularized Series and Products
Analytic number theory and part of the spectral theory of
operators (differential, pseudo-differential, elliptic,
etc.) are being merged under amore general analytic theory
of regularized products of certain sequences satisfying a
few basic axioms. The most basic examples consist of the
sequence of natural numbers, the sequence of zeros with
positive imaginary part of the Riemann zeta function, and
the sequence of eigenvalues, say of a positive Laplacian on
a compact or certain cases of non-compact manifolds. The
resulting theory is applicable to ergodic theory and
dynamical systems; to the zeta and L-functions of number
theory or representation theory and modular forms; to
Selberg-like zeta functions; andto the theory of
regularized determinants familiar in physics and other parts
of mathematics. Aside from presenting a systematic account
of widely scattered results, the theory also provides new
results. One part of the theory deals with complex analytic
properties, and another part deals with Fourier analysis.
Typical examples are given. This LNM provides basic results
which are and will be used in further papers, starting with
a general formulation of Cram r's theorem and explicit
formulas. The exposition is self-contained (except for
far-reaching examples), requiring only standard knowledge of
analysis.

Book 1625

Cover of Topics in Cohomology of Groups

Topics in Cohomology of Groups

by Serge Lang

Published 19 August 1996
The book is a mostly translated reprint of a report on cohomology of groups from the 1950s and 1960s, originally written as background for the Artin-Tate notes on class field theory, following the cohomological approach. This report was first published (in French) by Benjamin. For this new English edition, the author added Tate's local duality, written up from letters which John Tate sent to Lang in 1958 - 1959. Except for this last item, which requires more substantial background in algebraic geometry and especially abelian varieties, the rest of the book is basically elementary, depending only on standard homological algebra at the level of first year graduate students.

Book 1868

Cover of Posn(r) and Eisenstein Series

Posn(R) and Eisenstein Series provides an introduction, requiring minimal prerequisites, to the analysis on symmetric spaces of positive definite real matrices as well as quotients of this space by the unimodular group of integral matrices. The approach is presented in very classical terms and includes material on special functions, notably gamma and Bessel functions, and focuses on certain mathematical aspects of Eisenstein series.