On Refined Conjectures of Birch and Swinnerton-Dyer Type for Hasse–Weil–Artin $L$-Series
Author: David Burns
Publisher: American Mathematical Society
Published: 2024-06-07
Total Pages: 168
ISBN-13: 1470469669
DOWNLOAD EBOOK →View the abstract.
Author: David Burns
Publisher: American Mathematical Society
Published: 2024-06-07
Total Pages: 168
ISBN-13: 1470469669
DOWNLOAD EBOOK →View the abstract.
Author: Cristian Popescu
Publisher: American Mathematical Soc.
Published:
Total Pages: 517
ISBN-13: 0821886983
DOWNLOAD EBOOK →Author: Xinyi Yuan
Publisher: Princeton University Press
Published: 2013
Total Pages: 266
ISBN-13: 0691155925
DOWNLOAD EBOOK →This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the Gross-Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross-Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross-Zagier formula is reduced to local formulas. The Gross-Zagier Formula on Shimura Curves will be of great use to students wishing to enter this area and to those already working in it.
Author: Carlos J. Moreno
Publisher: American Mathematical Soc.
Published: 2005
Total Pages: 313
ISBN-13: 0821842668
DOWNLOAD EBOOK →Since the pioneering work of Euler, Dirichlet, and Riemann, the analytic properties of L-functions have been used to study the distribution of prime numbers. With the advent of the Langlands Program, L-functions have assumed a greater role in the study of the interplay between Diophantine questions about primes and representation theoretic properties of Galois representations. This book provides a complete introduction to the most significant class of L-functions: the Artin-Hecke L-functions associated to finite-dimensional representations of Weil groups and to automorphic L-functions of principal type on the general linear group. In addition to establishing functional equations, growth estimates, and non-vanishing theorems, a thorough presentation of the explicit formulas of Riemann type in the context of Artin-Hecke and automorphic L-functions is also given. The survey is aimed at mathematicians and graduate students who want to learn about the modern analytic theory of L-functions and their applications in number theory and in the theory of automorphic representations. The requirements for a profitable study of this monograph are a knowledge of basic number theory and the rudiments of abstract harmonic analysis on locally compact abelian groups.
Author: Bjorn Poonen
Publisher: American Mathematical Society
Published: 2023-08-10
Total Pages: 357
ISBN-13: 1470474581
DOWNLOAD EBOOK →This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere. The origins of arithmetic (or Diophantine) geometry can be traced back to antiquity, and it remains a lively and wide research domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central subject: obstructions to existence of rational points. —Yuri Manin, Max-Planck-Institute, Bonn It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student. —Hendrik Lenstra, University Leiden Understanding rational points on arbitrary algebraic varieties is the ultimate challenge. We have conjectures but few results. Poonen's book, with its mixture of basic constructions and openings into current research, will attract new generations to the Queen of Mathematics. —Jean-Louis Colliot-Thélène, Université Paris-Sud A beautiful subject, handled by a master. —Joseph Silverman, Brown University
Author: H. P. F. Swinnerton-Dyer
Publisher: Cambridge University Press
Published: 2001-02-22
Total Pages: 164
ISBN-13: 9780521004237
DOWNLOAD EBOOK →Broad graduate-level account of Algebraic Number Theory, first published in 2001, including exercises, by a world-renowned author.