本書是Boston大學(xué)舉辦的數(shù)論和代數(shù)會(huì)議的講義擴(kuò)張而成。書中介紹和擴(kuò)充講述了Wiles的許多觀點(diǎn)和技巧，并闡述了他的結(jié)果是如何與Ribets定理、Frey，Serre思想的結(jié)合，來證明費(fèi)馬最后定理。從一個(gè)完整的證明開始，緊接著用一些章節(jié)介紹了雙曲線、模函數(shù)、曲線、伽羅瓦上同調(diào)和有限群的基本概念。表示理論是整個(gè)證明的核心，在一章有關(guān)自同構(gòu)表示論和Langlands-Tunnell定理給出，緊隨其后深度介紹Serres猜想、伽羅瓦變形、一般變形環(huán)、Hacke代數(shù)。本書以回顧和展望作為結(jié)束，既反映了這個(gè)問題的歷史，又將Wiles定理放在了一個(gè)更加一般的Diophantine背景，給出了預(yù)期應(yīng)用。數(shù)學(xué)專業(yè)的學(xué)生和老師將會(huì)發(fā)現(xiàn)這本書是一部很難得參考書。
Preface
Contributors
Schedule of Lectures
Introduction
CHAPTER Ⅰ
An Overview of the Proof of Fermat's Last Theorem GLENN STEVENS
A remarkable elliptic curve
Galois representations
A remarkable Galois representation
Modular Galois representations
The Modularity Conjecture and Wiles's Theorem
The proof of Fermat's Last Theorem
The proof of Wiles's Theorem
References
CHAPTER Ⅱ
A Survey of the Arithmetic Theory of Elliptic Curves JOSEPH H. SILVERMAN
Basic definitions
The group law
Singular cubics
Isogenies
The endomorphism ring
Torsion points
Galois representations attached to E
The Weil pairing
Elliptic curves over finite fields
Elliptic curves over C and elliptic functions
The formal group of an elliptic curve
Elliptic curves over local fields
The Selmer and Shafarevich-Tate groups
Discriminants, conductors, and L-series
Duality theory
Rational torsion and the image of Galois
Tate curves
Heights and descent
The conjecture of Birch and Swinnerton-Dyer
Complex multiplication
Integral points
References
CHAPTER Ⅲ
Modular Curvcs, Hecke Correspondences, and L-Functions DAVID E.ROHRLICH
Modular curves
The Hcckc corrospondences
L-functions
Rcfcrcnccs
CHAPTER Ⅳ
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