This book provides a comprehensive account of the theory of moduli spaces of elliptic curves (over integer rings) and its application to modular forms.

Author: Haruzo Hida

Publisher: World Scientific

ISBN: 9789814368643

Category: Mathematics

Page: 454

View: 646

This book provides a comprehensive account of the theory of moduli spaces of elliptic curves (over integer rings) and its application to modular forms. The construction of Galois representations, which play a fundamental role in Wiles' proof of the Shimura?Taniyama conjecture, is given. In addition, the book presents an outline of the proof of diverse modularity results of two-dimensional Galois representations (including that of Wiles), as well as some of the author's new results in that direction.In this new second edition, a detailed description of Barsotti?Tate groups (including formal Lie groups) is added to Chapter 1. As an application, a down-to-earth description of formal deformation theory of elliptic curves is incorporated at the end of Chapter 2 (in order to make the proof of regularity of the moduli of elliptic curve more conceptual), and in Chapter 4, though limited to ordinary cases, newly incorporated are Ribet's theorem of full image of modular p-adic Galois representation and its generalization to ?big? ?-adic Galois representations under mild assumptions (a new result of the author). Though some of the striking developments described above is out of the scope of this introductory book, the author gives a taste of present day research in the area of Number Theory at the very end of the book (giving a good account of modularity theory of abelian ?-varieties and ?-curves).

Celebrating one of the leading figures in contemporary number theory – John H. Coates – on the occasion of his 70th birthday, this collection of contributions covers a range of topics in number theory, concentrating on the arithmetic of ...

Author: David Loeffler

Publisher: Springer

ISBN: 9783319450322

Category: Mathematics

Page: 492

View: 481

Celebrating one of the leading figures in contemporary number theory – John H. Coates – on the occasion of his 70th birthday, this collection of contributions covers a range of topics in number theory, concentrating on the arithmetic of elliptic curves, modular forms, and Galois representations. Several of the contributions in this volume were presented at the conference Elliptic Curves, Modular Forms and Iwasawa Theory, held in honour of the 70th birthday of John Coates in Cambridge, March 25-27, 2015. The main unifying theme is Iwasawa theory, a field that John Coates himself has done much to create. This collection is indispensable reading for researchers in Iwasawa theory, and is interesting and valuable for those in many related fields.

... K. TENT (ed) Non-equilibrium statistical mechanics and turbulence, J. CARDY,
G. FALKOVICH & K. GAWEDZKI Elliptic curves and big Galois representations, D.
DELBOURGO Algebraic theory of differential equations, M.A.H. MACCALLUM ...

Author: Fred Diamond

Publisher: Cambridge University Press

ISBN: 9781107693630

Category: Mathematics

Page: 390

View: 919

Part two of a two-volume collection exploring recent developments in number theory related to automorphic forms and Galois representations.

We may assume that k is big enough to contain all eigenvalues of o and o ' , say k
= Fg = Fp ( a ) , where a is of order q- 1. Let & q - 1 be a primitive ( q - 1 ) th root of
unity . The Brauer character x of o , defined on the p - regular elements of G , is ...

This is the fifth conference in a bi-annual series, following conferences in Besancon, Limoges, Irsee and Toronto. The meeting aims to bring together different strands of research in and closely related to the area of Iwasawa theory.

Author: Thanasis Bouganis

Publisher: Springer

ISBN: 9783642552458

Category: Mathematics

Page: 483

View: 364

This is the fifth conference in a bi-annual series, following conferences in Besancon, Limoges, Irsee and Toronto. The meeting aims to bring together different strands of research in and closely related to the area of Iwasawa theory. During the week before the conference in a kind of summer school a series of preparatory lectures for young mathematicians was provided as an introduction to Iwasawa theory. Iwasawa theory is a modern and powerful branch of number theory and can be traced back to the Japanese mathematician Kenkichi Iwasawa, who introduced the systematic study of Z_p-extensions and p-adic L-functions, concentrating on the case of ideal class groups. Later this would be generalized to elliptic curves. Over the last few decades considerable progress has been made in automorphic Iwasawa theory, e.g. the proof of the Main Conjecture for GL(2) by Kato and Skinner & Urban. Techniques such as Hida’s theory of p-adic modular forms and big Galois representations play a crucial part. Also a noncommutative Iwasawa theory of arbitrary p-adic Lie extensions has been developed. This volume aims to present a snapshot of the state of art of Iwasawa theory as of 2012. In particular it offers an introduction to Iwasawa theory (based on a preparatory course by Chris Wuthrich) and a survey of the proof of Skinner & Urban (based on a lecture course by Xin Wan).

Author: Australian Mathematical SocietyPublish On: 2008

Elliptic Curves and Big Galois Representations . Cambridge University Press ,
London Mathematical Society Lecture Note Series 356 , The London
Mathematical Society . Murdoch University • Clarke , B . ( 2008 ) . Linear Models :
The Theory ...

Author: Lawrence C. WashingtonPublish On: 2003-05-28

Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem.

Author: Lawrence C. Washington

Publisher: CRC Press

ISBN: 0203484029

Category: Computers

Page: 440

View: 819

Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to

Graduate Texts in Mathematics . 83 . I References on the 3rd generation of
Iwasawa theory VIA LL ( Del08 ) D. Delbourgo , Elliptic curves and big Galois representations , London Mathematical Society Lecture Note Series 356. 2008 . (
DO ) M.

In this case , Theorem B reduces to THEOREM C . Let E be a modular elliptic curve over Q with ordinary reduction at a ... There are several possible
approaches to Theorem A , all of which use the existence of a “ big Galois representation " T ...

Author: Cornell University. Dept. of MathematicsPublish On: 2000

Most of the results are consequences of stability for the The big challenge for the
future for these problems is corresponding variational problem . understanding
three dimensions . ... I am also very interested in elliptic surfaces of high rank and
constructing elliptic curves of high rank . ... I have been studying boundedness
properties of certain On elliptic units and p - adic Galois representations attached
...

Soon to answer one of the subject's afterwards , D. R. Heathbig riddles -
problems that Brown proved that Fermat's have ... in which jectures were true ,
then so was Fermat's . elliptic curves , and Galois representations ” . either x , y or
z is zero .

GALOIS REPRESENTATIONS AND ALGEBRAIC EQUATIONS ARISING FROM
MWL We now work in the following situation . Let ko be a perfect field ... E be an elliptic curve defined over Ko , and consider the MWL of E / K . ( For what follows ,
see [ S6 , III , S14 ] . ) Clearly G acts on E ( K ) ... the contrapuntal theme “ ( a ) big Galois representation versus ( b ) small Galois representation ” . Let us recall the
...

... DIStle— “ Modular forms , elliptic curves , garded mathematician , and he spent
and Galois representations " -- Wiles's ... DRAW PEOPLE TO maticians primed by
rumors that some well - established mathematics . thing big was in the works .

Mordell - Weil Lattices and Galois Representation . ... Galois representation
arising from the Mordell - Weil lattices . ... Let E be an elliptic curve defined over Q
( t ) , t being a variable over Q , and let f : S be its Kodaira - Néron model , which is
an ... The second one is also interesting , because if the image of p is trivial , then
we have El ( t ) ) = E ( Q ( t ) ) so that the rank of E over Q ( t ) can be relatively big
.

CH ] J . Coates and S . Howson , Euler characteristics and elliptic curves II ,
Journal Math . Soc . Japan 53 ( 2001 ) ... ( DS ) D . Delbourgo and P . Smith ,
Kummer theory for big Galois representations , to appear in Math . Proc . of the
Camb .

Author: I. Fesenko

Publisher: Amer Mathematical Society

ISBN: UVA:X030104019

Category: Mathematics

Page: 828

View: 839

This volume is dedicated to Professor John H. Coates, an outstanding contributor to number theory, both through his pioneering and fundamental mathematical works and through the magnificent mathematical school he has established. It contains 24 articles written by 38 authors on a wide range of topics in the cutting edge of research in number theory, algebraic geometry and analysis: zeta functions and $L$-functions, automorphic and modularity issues, Galois representations,arithmetic of elliptic curves, Iwasawa theory, noncommutative Iwasawa theory, and $p$-adic analysis. This volume will be of interest to researchers and students in these and neighboring fields. Information for our distributors: A publication of the Documenta Mathematica. The AMS distributes this series,beginning with volume 3, in the United States, Canada, and Mexico.

Galois Representations Arising from MWL From now on , we consider the
following situation . ... Now let E denote an elliptic curve defined over K , and
consider E / K . The associated elliptic surface f : S → C is now defined ...
Determine the image of Q . In particular , we ask : ( i ) How big or ( ii ) how small
can Im ( Q ) be ?

It will be seen how diophantine properties of a family of curves over the rational
numbers depends on the diophantine properties of a ... We want to estimate how big solutions of diophantine equations can be . ... are independent of the final
three sections which give applications of the abc conjecture to elliptic curves , via
the Szpiro conjecture . ... conjecture , for instance another conjecture which he
had made , concerning the modularity of Galois representations over the
rationals .