Drinfeld Modular Forms Modulo a Prime Ideal and Weierstrass Points on Drinfeld Modular Curves PDF Download
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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.
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.
Author: Mihran Papikian Publisher: 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.
Author: Henri Darmon Publisher: 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.
Author: Gary Cornell Publisher: Springer Science & Business Media ISBN: 1461219744 Category : Mathematics Languages : en Pages : 592
Book Description
This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.
Author: William A. Stein Publisher: American Mathematical Soc. ISBN: 0821839608 Category : Mathematics Languages : en Pages : 290
Book Description
This marvellous and highly original book fills a significant gap in the extensive literature on classical modular forms. This is not just yet another introductory text to this theory, though it could certainly be used as such in conjunction with more traditional treatments. Its novelty lies in its computational emphasis throughout: Stein not only defines what modular forms are, but shows in illuminating detail how one can compute everything about them in practice. This is illustrated throughout the book with examples from his own (entirely free) software package SAGE, which really bring the subject to life while not detracting in any way from its theoretical beauty. The author is the leading expert in computations with modular forms, and what he says on this subject is all tried and tested and based on his extensive experience. As well as being an invaluable companion to those learning the theory in a more traditional way, this book will be a great help to those who wish to use modular forms in applications, such as in the explicit solution of Diophantine equations. There is also a useful Appendix by Gunnells on extensions to more general modular forms, which has enough in it to inspire many PhD theses for years to come. While the book's main readership will be graduate students in number theory, it will also be accessible to advanced undergraduates and useful to both specialists and non-specialists in number theory. --John E. Cremona, University of Nottingham William Stein is an associate professor of mathematics at the University of Washington at Seattle. He earned a PhD in mathematics from UC Berkeley and has held positions at Harvard University and UC San Diego. His current research interests lie in modular forms, elliptic curves, and computational mathematics.
Author: Peter Sarnak Publisher: Cambridge University Press ISBN: 1316582442 Category : Mathematics Languages : en Pages : 124
Book Description
The theory of modular forms and especially the so-called 'Ramanujan Conjectures' have been applied to resolve problems in combinatorics, computer science, analysis and number theory. This tract, based on the Wittemore Lectures given at Yale University, is concerned with describing some of these applications. In order to keep the presentation reasonably self-contained, Professor Sarnak begins by developing the necessary background material in modular forms. He then considers the solution of three problems: the Ruziewicz problem concerning finitely additive rotationally invariant measures on the sphere; the explicit construction of highly connected but sparse graphs: 'expander graphs' and 'Ramanujan graphs'; and the Linnik problem concerning the distribution of integers that represent a given large integer as a sum of three squares. These applications are carried out in detail. The book therefore should be accessible to a wide audience of graduate students and researchers in mathematics and computer science.
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Author: Mihran Papikian
Author: Henri Darmon
Author: Gary Cornell
Author: William A. Stein
Author: Ahmad El-Guindy
Author: Ernst-Ulrich Gekeler
Author: John C. Baez
Author: Peter Sarnak
Author: Jean-Pierre Serre