Computations with Modular Forms

Computations with Modular Forms
Author: Gebhard Böckle
Publisher: Springer Science & Business Media
Total Pages: 377
Release: 2014-01-23
Genre: Mathematics
ISBN: 3319038478

This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A key theme of the Conference and Summer School was the interplay between theory, algorithms and experiment. The 14 papers offer readers both, instructional courses on the latest algorithms for computing modular and automorphic forms, as well as original research articles reporting on the latest developments in the field. The three Summer School lectures provide an introduction to modern algorithms together with some theoretical background for computations of and with modular forms, including computing cohomology of arithmetic groups, algebraic automorphic forms, and overconvergent modular symbols. The 11 Conference papers cover a wide range of themes related to computations with modular forms, including lattice methods for algebraic modular forms on classical groups, a generalization of the Maeda conjecture, an efficient algorithm for special values of p-adic Rankin triple product L-functions, arithmetic aspects and experimental data of Bianchi groups, a theoretical study of the real Jacobian of modular curves, results on computing weight one modular forms, and more.

Abelian l-Adic Representations and Elliptic Curves

Abelian l-Adic Representations and Elliptic Curves
Author: Jean-Pierre Serre
Publisher: CRC Press
Total Pages: 203
Release: 1997-11-15
Genre: Mathematics
ISBN: 1439863865

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one

Serre's Problem on Projective Modules

Serre's Problem on Projective Modules
Author: T.Y. Lam
Publisher: Springer Science & Business Media
Total Pages: 412
Release: 2010-05-17
Genre: Mathematics
ISBN: 3540345752

An invaluable summary of research work done in the period from 1978 to the present

Number Theory

Number Theory
Author: Kağan Kurşungöz
Publisher: Walter de Gruyter GmbH & Co KG
Total Pages: 112
Release: 2021-11-08
Genre: Mathematics
ISBN: 3110761114

Three major branches of number theory are included in the volume: namely analytic number theory, algebraic number theory, and transcendental number theory. Original research is presented that discusses modern techniques and survey papers from selected academic scholars.

Rational Points on Modular Elliptic Curves

Rational Points on Modular Elliptic Curves
Author: Henri Darmon
Publisher: American Mathematical Soc.
Total Pages: 146
Release: 2004
Genre: Mathematics
ISBN: 0821828681

The book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and the Heegner point construction. The next three chapters introduce variants of modular parametrizations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. The main new contributions are found in Chapters 7-9, which survey the author's attempts to extend the theory of Heegner points and complex multiplication to situations where the base field is not a CM field. Chapter 10 explains the proof of Kolyvagin's theorem, which relates Heegner points to the arithmetic of elliptic curves and leads to the best evidence so far for the Birch and Swinnerton-Dyer conjecture.

Quadratic Forms, Linear Algebraic Groups, and Cohomology

Quadratic Forms, Linear Algebraic Groups, and Cohomology
Author: Skip Garibaldi
Publisher: Springer Science & Business Media
Total Pages: 344
Release: 2010-07-16
Genre: Mathematics
ISBN: 1441962115

Developments in Mathematics is a book series devoted to all areas of mathematics, pure and applied. The series emphasizes research monographs describing the latest advances. Edited volumes that focus on areas that have seen dramatic progress, or are of special interest, are encouraged as well.

Arithmetic Geometry

Arithmetic Geometry
Author: Clay Mathematics Institute. Summer School
Publisher: American Mathematical Soc.
Total Pages: 570
Release: 2009
Genre: Mathematics
ISBN: 0821844768

Based on survey lectures given at the 2006 Clay Summer School on Arithmetic Geometry at the Mathematics Institute of the University of Gottingen, this tile is intended for graduate students and recent PhD's. It introduces readers to modern techniques and conjectures at the interface of number theory and algebraic geometry.

Advanced Topics in the Arithmetic of Elliptic Curves

Advanced Topics in the Arithmetic of Elliptic Curves
Author: Joseph H. Silverman
Publisher: Springer Science & Business Media
Total Pages: 482
Release: 2013-12-01
Genre: Mathematics
ISBN: 1461208513

In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.