Automorphic Forms Attached to Differential Equations and Relationships with the Picard Number of an Elliptic Surface PDF Download
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Author: Peter Stiller
Publisher:
ISBN:
Category : Algebraic functions
Languages : en
Pages : 0
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Book Description
Author: Peter Stiller
Publisher:
ISBN:
Category : Algebraic functions
Languages : en
Pages : 0
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Book Description
Author: Peter F. Stiller
Publisher: Springer Science & Business Media
ISBN: 3322907082
Category : Technology & Engineering
Languages : en
Pages : 201
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Book Description
In studying an algebraic surface E, which we assume is non-singular and projective over the field of complex numbers t, it is natural to study the curves on this surface. In order to do this one introduces various equivalence relations on the group of divisors (cycles of codimension one). One such relation is algebraic equivalence and we denote by NS(E) the group of divisors modulo algebraic equivalence which is called the N~ron-Severi group of the surface E. This is known to be a finitely generated abelian group which can be regarded naturally as a subgroup of 2 H (E,Z). The rank of NS(E) will be denoted p and is known as the Picard number of E. 2 Every divisor determines a cohomology class in H(E,E) which is of I type (1,1), that is to say a class in H(E,9!) which can be viewed as a 2 subspace of H(E,E) via the Hodge decomposition. The Hodge Conjecture asserts in general that every rational cohomology class of type (p,p) is algebraic. In our case this is the Lefschetz Theorem on (I,l)-classes: Every cohomology class 2 2 is the class associated to some divisor. Here we are writing H (E,Z) for 2 its image under the natural mapping into H (E,t). Thus NS(E) modulo 2 torsion is Hl(E,n!) n H(E,Z) and th 1 b i f h -~ p measures e a ge ra c part 0 t e cohomology.
Author: Peter Stiller
Publisher: American Mathematical Soc.
ISBN: 0821823000
Category : Mathematics
Languages : en
Pages : 123
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Book Description
In this paper we explore a relationship that exists between the classical cusp form for subgroups of finite index in [italic]SL2([double-struck capital]Z) and certain differential equations, and we develop a connection between the equation's monodromy representation and the special values in the critical strip of the Dirichlet series associated to the cusp form.
Author:
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 836
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Author:
Publisher: American Mathematical Society(RI)
ISBN:
Category : Number theory
Languages : en
Pages : 1084
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Book Description
These six volumes include approximately 20,000 reviews of items in number theory that appeared in Mathematical Reviews (MR) between 1984 and 1996. This is the third such set of volumes in number theory: the first was edited by W.J. LeVeque and included reviews from 1940-1972; the second was edited by R.K. Guy and appeared in 1984.
Author:
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 1884
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Book Description
Author: Friedrich Hirzebruch
Publisher: Springer Science & Business Media
ISBN: 3663107264
Category : Technology & Engineering
Languages : en
Pages : 216
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Book Description
This book provides a comprehensive introduction to the theory of elliptic genera due to Ochanine, Landweber, Stong, and others. The theory describes a new cobordism invariant for manifolds in terms of modular forms. The book evolved from notes of a course given at the University of Bonn. After providing some background material elliptic genera are constructed, including the classical genera signature and the index of the Dirac operator as special cases. Various properties of elliptic genera are discussed, especially their behaviour in fibre bundles and rigidity for group actions. For stably almost complex manifolds the theory is extended to elliptic genera of higher level. The text is in most parts self-contained. The results are illustrated by explicit examples and by comparison with well-known theorems. The relevant aspects of the theory of modular forms are derived in a seperate appendix, providing also a useful reference for mathematicians working in this field.
Author: Jan Hendrik Bruinier
Publisher: Springer Science & Business Media
ISBN: 3540741194
Category : Mathematics
Languages : en
Pages : 273
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Book Description
This book grew out of three series of lectures given at the summer school on "Modular Forms and their Applications" at the Sophus Lie Conference Center in Nordfjordeid in June 2004. The first series treats the classical one-variable theory of elliptic modular forms. The second series presents the theory of Hilbert modular forms in two variables and Hilbert modular surfaces. The third series gives an introduction to Siegel modular forms and discusses a conjecture by Harder. It also contains Harder's original manuscript with the conjecture. Each part treats a number of beautiful applications.
Author:
Publisher:
ISBN:
Category : Functional analysis
Languages : en
Pages : 808
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Book Description
Author: Henri Darmon
Publisher: American Mathematical Soc.
ISBN: 9780821889459
Category : Mathematics
Languages : en
Pages : 148
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Book Description
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.
