橢圓曲線算術(shù)中的高等論題

前言

In the introduction to the first volume of The Arithmetic o/Elliptic Curves（Springer-Verlag, 1986）, I observed that "the theory of elliptic curves isrich, varied, and amazingly vast," and as a consequence, "many importanttopics had to be omitted." I included a brief introduction to ten additionaltopics as an appendix to the first volume, with the tacit understanding thateventually there might be a second volume containing the details. You arenow holding that second volume. Unfortunately, it turned out that even those ten topics would not fitinto a single book, so I was forced to make some choices. The followingmaterial is covered in this book:I. Elliptic and modular functions for the full modular group.II. Elliptic curves with complex multiplication.   III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers,Tate's algorithm, and Ogg's conductor-discriminant formula.V. Tate's theory of qcurves over p-adic fields.VI. Neron's theory of canonical local height functions.So what's still missing？ First and foremost is the theory of modularcurves of higher level and the associated modular parametrizations of ellip-tic curves. There is little question that this is currently the hottest topicin the theory of elliptic curves, but any adequate treatment would seem torequire （at least） an entire book of its own. （For a nice introduction, seeKnapp [1].） Other topics that I have left out in order to keep this bookat a manageable size include the description of the image of the g-adicrepresentation attached to an elliptic curve and local and global dualitytheory. Thus, at best, this book covers approximately half of the materialdescribed in the appendix to the first volume. I apologize to those who mayfeel disappointed, either at the incompleteness or at the choice of particulartopics.

內(nèi)容概要

　　美國哈佛大學(xué)從1977年開始，曾多次舉辦”橢圓曲線” 班，《橢圓曲線算術(shù)中的高等論題(英文版)》作者是該討論班成員之一。橢圓曲線是一個(gè)古老的數(shù)學(xué)課題，最近由于代數(shù)數(shù)論和代數(shù)幾何等現(xiàn)代數(shù)學(xué)的進(jìn)展，使它得到了新的活力?！稒E圓曲線算術(shù)中的高等論題(英文版)》是以1986年版的《橢圓曲線的算術(shù)理論》為藍(lán)本，但在知識(shí)體系上做了較大的改動(dòng)形成了這不教程，講述上也更加專業(yè)，但在思想上是作者前《橢圓曲線算術(shù)中的高等論題(英文版)》的延續(xù)。包括橢圓和模型函數(shù)；復(fù)乘方法；橢圓曲線；Néron模型；復(fù)域上的橢圓曲線等內(nèi)容。每章末都配有大量習(xí)題。目次：橢圓和模型函數(shù)；復(fù)乘方法；橢圓曲線；Néron模型；復(fù)域上的橢圓曲線?！　∽x者對(duì)象：適合數(shù)學(xué)專業(yè)的研究生和相關(guān)的科研人員。

書籍目錄

PrerepuisitesChapter 1　The standard Brownian motionChapter 2  Brownian local timesChapter 3  The general 1-dimensional diffusionChapter 4  GeneratorsChapter 5  Time changes and killing Chapter 6  Local and inverse local timesChapter 7  Brownian motion in several dimensionsChapter 8  A general view of diffusion in several dimensionsBibliography List of notationsIndex

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