Author: Ernst-Ulrich GekelerPublisher: Springer
ISBN: 3540473866
Category : Mathematics
Languages : en
Pages : 122
Book Description
Drinfeld Modular Curves
Author: Ernst-Ulrich GekelerPublisher: Springer
ISBN: 3540473866
Category : Mathematics
Languages : en
Pages : 122
Book Description
Drinfeld Modules, Modular Schemes And Applications
Author: M Van Der PutPublisher: World Scientific
ISBN: 9814546402
Category : Mathematics
Languages : en
Pages : 378
Book Description
In his 1974 seminal paper 'Elliptic modules', V G Drinfeld introduced objects into the arithmetic geometry of global function fields which are nowadays known as 'Drinfeld Modules'. They have many beautiful analogies with elliptic curves and abelian varieties. They study of their moduli spaces leads amongst others to explicit class field theory, Jacquet-Langlands theory, and a proof of the Shimura-Taniyama-Weil conjecture for global function fields.This book constitutes a carefully written instructional course of 12 lectures on these subjects, including many recent novel insights and examples. The instructional part is complemented by research papers centering around class field theory, modular forms and Heegner points in the theory of global function fields.The book will be indispensable for everyone who wants a clear view of Drinfeld's original work, and wants to be informed about the present state of research in the theory of arithmetic geometry over function fields.
CM Points on Products of Drinfeld Modular Curves
Author: Florian BreuerDrinfeld Modular Forms Modulo a Prime Ideal and Weierstrass Points on Drinfeld Modular Curves
Author: Publisher:
ISBN:
Category :
Languages : en
Pages : 190
Book Description
We first consider the action of the Hasse derivatives on Drinfeld modular forms, which were shown by Uchino and Satoh to act as differential operators on the algebra of Drinfeld quasi-modular forms. While these operators do not preserve modularity, we show that they do preserve modularity modulo a prime ideal. We also study the behavior of the filtration under the action of the first Hasse derivative, and obtain results analogous to those obtained by Serre and Swinnerton-Dyer about Ramanujan's Theta-operator in the classical setting. We then consider a family of modular curves constructed by Drinfeld, and we study their Weierstrass points, a finite set of points of geometric interest. These curves are moduli spaces for Drinfeld modules with level structure, which are the objects which in our setting play a role analogous to that of elliptic curves. Previous work of Baker shows that for each Weierstrass point of these curves, the reduction modulo a certain prime ideal of the underlying Drinfeld module is supersingular. We study a modular form W for this congruence subgroup whose divisor is closely related to the set of Weierstrass points, an idea first presented by Rohrlich in the classical setting. To this end, we first establish a one-to-one correspondence between certain Drinfeld modular forms on the congruence subgroup and forms on the full modular group. In certain cases we can then use knowledge about the action of the Hasse derivatives to compute explicitly a form that is congruent to W modulo our prime ideal. This allows us to obtain an analogue of Rohrlich's result, which is the first important step towards obtaining a more precise relationship between the supersingular locus and Weierstrass points on our modular curves, as illustrated by Ahlgren and Ono in the classical setting.
Drinfeld Modular Curves, Heegner Points and Interpolation of Special Values
Author: Ambrus PálCohomology of Drinfeld Modular Varieties, Part 1, Geometry, Counting of Points and Local Harmonic Analysis
Author: Gérard LaumonPublisher: Cambridge University Press
ISBN: 0521470609
Category : Mathematics
Languages : en
Pages : 362
Book Description
Originally published in 1995, Cohomology of Drinfeld Modular Varieties aimed to provide an introduction, in two volumes, both to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. The present volume is devoted to the geometry of these varieties, and to the local harmonic analysis needed to compute their cohomology. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated.
Invariants of Some Algebraic Curves Related to Drinfeld Modular Curves
Author: Ernst-Ulrich GekelerDrinfeld Modules
Author: Mihran PapikianPublisher: Springer Nature
ISBN: 3031197070
Category : Mathematics
Languages : en
Pages : 541
Book Description
This textbook offers an introduction to the theory of Drinfeld modules, mathematical objects that are fundamental to modern number theory. After the first two chapters conveniently recalling prerequisites from abstract algebra and non-Archimedean analysis, Chapter 3 introduces Drinfeld modules and the key notions of isogenies and torsion points. Over the next four chapters, Drinfeld modules are studied in settings of various fields of arithmetic importance, culminating in the case of global fields. Throughout, numerous number-theoretic applications are discussed, and the analogies between classical and function field arithmetic are emphasized. Drinfeld Modules guides readers from the basics to research topics in function field arithmetic, assuming only familiarity with graduate-level abstract algebra as prerequisite. With exercises of varying difficulty included in each section, the book is designed to be used as the primary textbook for a graduate course on the topic, and may also provide a supplementary reference for courses in algebraic number theory, elliptic curves, and related fields. Furthermore, researchers in algebra and number theory will appreciate it as a self-contained reference on the topic.
Optimal Elliptic Curves, Discriminants, and the Degree Conjecture Over Function Fields
Author: Mihran PapikianRational Points on Modular Elliptic Curves
Author: Henri DarmonPublisher: American Mathematical Soc.
ISBN: 0821828681
Category : Mathematics
Languages : en
Pages : 146
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.