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18.704 2004秋季課程:代數與數理論研討會:橢圓曲線上的有理點(Seminar in Algebra and Number Theory: Rational Points on Elliptic Curves, Fall 2004)

翻譯:左輕侯
編輯:陳玉侖

 

Adding rational points on an elliptic curve.
橢圓曲線上點加的弦切律法則。 (圖片由Daniel Rogalski博士提供。)
Adding rational points on an elliptic curve. (Image courtesy of Dr. Daniel Rogalski.)

課程重點

本課程包含相關閱讀材料部分,還有作業專題部分。

This course features a readings section. In addition to assignments, a project section is also available.

課程描述

本研討會的授課對象是數學系學生, 由學生主持講座,不要求學生有主持講座的經驗.
This is a seminar for mathematics majors, where the students present the lectures. No prior experience giving lectures is necessary.

師資

講師:
Daniel Rogalski 博士

上課時數

教師授課:
每週3節
每節1小時

程度

大學部

 

回應

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原文聲明

 

必備先修課:

修過任一線性代數課程,以及要有證明的經驗。

教科書:

Silverman, Joseph H. 和 John Tate. 《橢圓曲線上的有理點》New York: Springer-Verlag, 1992年8月1日. ISBN: 0-3879-7825-9.(譯注:國內沒出版)

最後兩講取材於:

Koblitz, Neal.《橢圓曲線和模型式之導論》,第二版 , New York: Springer-Verlag, 1993. ISBN: 0-3879-7966-2.(譯注:國內有出版ISBN:7-5062-0126-7)

課程描述:

本課程的主要的和最重要的目的是讓學生發展用數學進行交流的技巧,同時還可以學習一些有趣的數論內容!因為每個演講者都需要一定數量的聽眾,所以選修本課的學生必須來上課。每週有作業,最後要交期末論文。

問題集

堂上提交十個問題集,一份期末論文,沒有期末考試。

評分

項目 百分比
出勤,參與,努力 60%
作業 25%
期末論文 15%
Prerequisite

One subject in linear algebra and some experience with proofs.

Textbook

Most of the material for the class will follow the text:

Silverman, Joseph H., and John Tate. Rational Points on Elliptic Curves. New York: Springer-Verlag, 1 August 1992. ISBN: 0387978259.

The last two lectures are based on the textbook below:

Koblitz, Neal. Introduction to Elliptic Curves and Modular Forms. 2nd ed. New York: Springer-Verlag, 1993. ISBN: 0387979662.

Description

The goal of this course is first and foremost to allow students to develop skills in communicating mathematics to others. They also learn some interesting number theory along the way! Because it is important that every speaker have a supportive audience, attendance is mandatory. There are also weekly homeworks and a final paper.

Problem Sets

There are ten regular problem sets, due at lecture, and a final paper. There is no final exam.

Grading

ACTIVITIES PERCENTAGES
Attendance, Participation, Effort 60%
Homework 25%
Final Paper 15%

 

 

Calender 課 課程單元 重要日期
1 射影平面
The Projective Plane
 
2 射影平面上的曲線
Curves in the Projective Plane
 
3 圓錐曲線上的有理點
Rational Points on Conics
 
4 三次曲線的幾何學
Geometry of Cubic Curves
作業1到期
Homework 1 due
5 Weierstrass標準形
Weierstrass Normal Form
 
6 群定律的精確公式(座標公式)
Explicit Formulas for the Group Law
 
7 二階點和三階點
Points of Order Two and Three
作業2到期
Homework 2 due
8 判別式
The Discriminant

座標為整數的有限階點(1)
Points of Finite Order have Integer Coordinates - Part 1
 
9 座標為整數的有限階點(2)
Points of Finite Order have Integer Coordinates - Part 2
 
10 座標為整數的有限階點(3)
Points of Finite Order have Integer Coordinates - Part 3

Nagell-Lutz定理
The Nagell-Lutz Theorem
作業3到期
Homework 3 due
11 三次曲線的實點和複點
Real and Complex Points on Cubics
 
12 高函數和遞降法
Heights and Descent
 
13 點P+P0的高
Height of P + P_0
作業4到期
Homework 4 due
14 2P的高度
Height of 2P
 
15 一個有用的同態(1)
A Useful Homomorphism - Part 1
作業5到期
Homework 5 due
16 一個有用的同態(2)
A Useful Homomorphism - Part 2
 
17 Mordell定理(1)
Mordell's Theorem - Part 1
 
18 Mordell定理(2)
Mordell's Theorem - Part 2

例(1)
Examples - Part 1
作業6到期
Homework 6 due
19 例(2)
Examples - Part 2
 
20 例(3)
Examples - Part 3
 
21 奇異三次曲線
Singular Cubics
 
22 有限域上的有理點
Rational Points over Finite Fields
 
23 Gauss定理(1)
Gauss's Theorem - Part 1
 
24 Gauss定理(2)
Gauss's Theorem - Part 2
作業7到期
Homework 7 due
25 復習有限階點
Points of Finite Order Revisited
 
26 用橢圓曲線分解整數(1)
Factorization using Elliptic Curves - Part 1
 
27 用橢圓曲線分解整數(2)
Factorization using Elliptic Curves - Part 2
作業 8到期
Homework 8 due
28 三次曲線上的整點
Integer Points on Cubics

計程車(1)
Taxicabs - Part 1
 
29 計程車(2)
Taxicabs - Part 2

Thue定理(1)
Thue's Theorem - Part 1
 
30 Thue定理(2)
Thue's Theorem - Part 2
作業 9到期
Homework 9 due
31 輔助多項式的構造
Construction of an Auxiliary Polynomial
 
32 輔助多項式有界性
The Auxiliary Polynomial is Small
 
33 輔助多項式的非零性
The Auxiliary Polynomial Does Not Vanish
 
34 Diophantine逼近定理的證明
Proof of the DAT

進一步的發展
Further Developments
作業 10到期
Homework 10 due
35 同餘數和橢圓曲線I:Koblitz (1)
Congruent Numbers and Elliptic Curves I: Koblitz - Part 1
 
36 同餘數和橢圓曲線II:Koblitz (2)
Congruent Numbers and Elliptic Curves II: Koblitz - Part 2
 

 


本課程取材於以下教材: Silverman, Joseph H.和John Tate.《橢圓曲線上的有理點》. New York: Springer-Verlag, 1992. ISBN: 0-3879-7825-9.
Most of the material for the class will follow the text: Silverman, Joseph H., and John Tate. Rational Points on Elliptic Curves. New York: Springer-Verlag, 1 August 1992. ISBN: 0387978259.

最後兩講的材料來自於: Koblitz, Neal.《橢圓曲線和模形式之導論》第二版. New York: Springer-Verlag, 1993. ISBN: 0-3879-7966-2.
The last two lectures are based on the textbook: Koblitz, Neal. Introduction to Elliptic Curves and Modular Forms. 2nd ed. New York: Springer-Verlag, 1993, ISBN: 0387979662.

Readings 課 課程單元 閱讀資料 註
1 射影平面
The Projective Plane
A.1部分:射影投影平面
Section A.1: The Projective Plane
前一半:A.1部分到p. 222第11行"coordinates for the point."
First Half: Section A.1, up through p. 222 line 11 "... coordinates for the point."

後一半:從p.222第12行"In order to motivate...""直到A.1部分結束。
Second Half: Continuing from p. 222 line 12 "In order to motivate..." through the end of section A.1.
2 射影平面上的曲線
Curves in the Projective Plane
A.2部分:射影平面上的曲線
Section A.2: Curves in the Projective Plane
前一半: A.2部分,講到p.228第4行 (方程式的部分)
First Half: Section A.2, up through p. 228 line 4 (the displayed equations.)

後一半:從p.228第5行"The preceding discussion..."直到p.230倒數第 4行(方框中的句子) A.2部分的其他部分我們不講。
Second Half: Continuing from p. 228 line 5 "The preceding discussion..." up through p. 230 line 4 from the bottom (the sentence in the box.) We will not cover the rest of section A.2.
3 圓錐曲線上的有理點
Rational Points on Conics
I.1:圓錐曲線上的有理點
Section I.1: Rational Points on Conics
前一半:I.1.講到第11頁的倒數第4行"...infinity for t!]"
First Part: Section I.1. up through line 4 from the bottom of p. 11 "...infinity for t!]"

後一半:11頁倒數第三行"These formulas can be used..."直到本節完了。為了在25分鐘內講完,需要刪除一些內容;我建議刪除第13頁的微積分應用和本節最後一段的Hasse定理。
Second Part: Line 3 from bottom of p. 11 "These formulas can be used..." through the end of the section. To fit in 25 minutes, you will have to omit some of this material; I suggest omitting one or both of the applications to Calculus which are on p. 13. Also the last paragraph of the section mentioning Hasse's theorem can be omitted.
4 三次曲線的幾何學
Geometry of Cubic Curves
I.2:三次曲線的幾何學
Section I.2: Geometry of Cubic Curves
前一半:I.2,直到18頁倒數第二段"...is shown in Figure 7." 如果這些內容對於25分鐘來講太多了,建議從16頁最後的3條三次曲線的定理開始,但不用證明此定理;換句話說,我們承認定理不去證明,並且省略17頁前三段的論述。
First Part: Section I.2, up through p. 18 second full paragraph, ending "...is shown in Figure 7." If this is too much for 25 minutes, consider stating but omitting the proof of the theorem concerning 3 cubic curves which is stated at the bottom of p. 16. In other words, you could say we accept this theorem without proof and omit the argument in the first 3 paragraphs on p. 17.

後一半:由於接下來主要是驗證第一部分定義的加法運算,所以我請大家回憶交換群的公理。接著來講I.2,由18頁倒數第一段(原文為倒數第二段,根據教材 修改-譯者注)"It is a little harder..."開始直到本節完。試著自己畫一下圖1.9,畫出大概的圖形就可以了。如果最後沒有時間講述Mordell定理的內容,可以之後再講。
Second Part: Please recall the axioms for a commutative group, since most of your part is to verify them for the addition operation defined in the first part. Then continue in Section I.2, from p. 18 second paragraph from the bottom "It is a little harder..." through the end of the section. Try to reproduce Figure 1.9 or draw something roughly equivalent. If there is no time to restate Mordell's theorem at the end, that can be postponed.
5 Weierstrass 標準形
Weierstrass Normal Form
I.3: Weierstrass 標準形
Section I.3: Weierstrass Normal Form
前一半:從I.3節開始,講到 p. 24倒數第三行 "...points on cubic curves in Weierstrass normal form."為了在25分鐘之內講完這些內容,可以省略某些內容。
First Part: Section I.3, up through p. 24 line 3 from the end "...points on cubic curves in Weierstrass normal form." You will probably want to omit some details here and there to make it fit to 25 minutes.

後一半: 從I.3,p. 24最後一段"The transformations we used..."開始講到本節最後。如果在25分鐘之內難以講完,不需著墨(或者跳過)25頁關於"橢圓曲線"這一名稱的來歷。
Second Part: Section I.3, continuing from p. 24 last paragraph "The transformations we used..." through the end of the section. If it has trouble fitting in 25 minutes, don't linger on (or skip) the motivation for the name "elliptic curve" on p. 25.
6 群定律的精確公式(座標公式)
Explicit Formulas for the Group Law
I.4: 群定律的精確公式(座標公式)
Section I.4: Explicit Formulas for the Group Law
在大約30分鐘盡量講述I.4的內容。可以著重於不同點的加法,如果有時間,可以討論兩倍點的公式。在本節必須要講述一個實例。
Please cover as much of Section I.4 as seems reasonable in 30 minutes or so. You can concentrate on the formula for adding distinct points, and not worry about the doubling formula, unless you have time. A specific example is also called for.
7 二階點和三階點
Points of Order Two and Three
II.1: 二階點和三階點
Section II.1: Points of Order Two and Three
可以在50分鐘內輕鬆地講完第II.1節。不用著急。
You should be able to cover the entire contents of Section II.1 comfortably in 50 minutes. Take your time.
8 判別式
The Discriminant

座標為整數的有限階點(1)
Points of Finite Order have Integer Coordinates - Part 1
II.3: 判別式
Section II.3: The Discriminant

II.4: 座標為整數的有限階點(1)
Section II.4: Points of Finite Order have Integer Coordinates - Part 1
第 II.3 節全部,然後從第II.4節開始,講到第50頁的第二段在"...in every C(p^{nu})."處結束。
Cover Section II.3 in its entirety, then Section II.4 up through p. 50 second full paragraph, ending "...in every C(p^{nu})."
9 座標為整數的有限階點(2)
Points of Finite Order have Integer Coordinates - Part 2
II.4: 座標為整數的有限階點(2)
Section II.4: Points of Finite Order have Integer Coordinates - Part 2
這一講包括第II.4節中大部分的技術內容 (抱歉。) 從回顧第50頁第二段C(p^{nu})的定義開始(如果之前的人沒講述過該定義,可以在這裏提出),直到第54頁第四段完"....t(P)= x(P)/y(P)."
This lecture contains a lot of technical material in the middle of Section II.4 (sorry.) Start by reviewing the definition of C(p^{nu}) in the second full paragraph on p. 50 (or make this definition for the first time if the previous speaker didn't get to it.) Then continue through to p. 54 up through paragraph 4 ending "....t(P)= x(P)/y(P)."
10 座標為整數的有限階點(3)
Points of Finite Order have Integer Coordinates - Part 3

Nagell-Lutz 定理
The Nagell-Lutz Theorem
II.4: 座標為整數的有限階點(3)
Section II.4: Points of Finite Order have Integer Coordinates - Part 3

II.5: Nagell-Lutz定理
Section II.5: The Nagell-Lutz Theorem
從第54頁第II.4節第5段"This last formula..."開始。注意:我認為大家不一定一眼就可以看出p^{nu} R/p^{3nu}R是一個階為p^{2nu}的迴圈群,所以需要證明。結束第II.4節,講完第II.5節(如果時間少,可以刪掉第57 頁的部分內容。)
Begin in Section II.4 on p. 54 with paragraph 5, beginning "This last formula..." Note: I don't think the audience will think the fact that the group p^{nu} R/p^{3nu}R is cyclic of order p^{2nu} is so obvious. Maybe you can say something about that. Then finish Section II.4 and do all of Section II.5 (there are things on p. 57 that can be omitted if necessary for time.)
11 三次曲線的實和複點
Real and Complex Points on Cubics
II.2:三次曲線的實和複點
Section II.2: Real and Complex Points on Cubics
本節較難,而且以後用到的機會也不多,但是概念非常重要,因此我希望每個人都要稍微接觸,雖然不一定可以完全理解這裏的內容。
This section is kind of difficult, and we will not really use it much later. But its ideas are very important so I want everyone to be exposed to them, even if not everyone understands everything here.

由於本節內容包括許多圖,所以在講課的時候也要畫示意圖。記住這個課堂上的一些同學沒有學過複分析,所以可以多提供一些複分析的解釋。
There is a certain amount of hand-waving in this section, so you will have to do that in your lecture too. Remember that not everybody in the class has had Complex analysis, so any extra explanation of the facts from complex analysis you are using would be welcome.

最後,如果無法涵蓋所有內容也沒關係,重點是要提到複數域上的橢圓曲線是一個環面,曲線簇中的有限階點的簡單幾何描述。
Finally, if you don't cover everything here, that's fine. The most important idea is that over the complex numbers, an elliptic curve looks like a torus, and the points of finite order in the group of the curve have a very simple geometric description. Make sure you get to that part.
12 高函數和遞降法
Heights and Descent
III.1:高函數和遞降法
Section III.1: Heights and Descent
第 III.1 節, 對於內容有點長。要在黑板上詳細列述遞降法。如有必要,第 67頁和第68頁最後的一段可以刪去。如果內容還是太多,請告知教授。
This section, Section III.1, is possibly a bit long for one lecture. Please state the descent theorem fully on the board. You can definitely cut out the last paragraph on p. 67 and p. 68 if necessary. Let me know if it still runs long.
13 點P +P0的高
Height of P + P_0
III.2: 點P + P0的高
Section III.2: Height of P + P_0
第III.2節全部。本節內容看起來難,但實際上不難,主要原因是因為這裏有太多的符號。
Please cover Section III.2 entirely. The argument looks harder than it is, because it has so much notation.

如果時間不容許,可以略去某些細節,或者減略第一部分中講到x和y有m/e^2和n/e^3的形式。 本節的內容和第II.4節的內容相似,事實上是緊接上節的論點。如果有時間可以詳細討論。
It may be possible to omit the details or abbreviate the first part of the argument, that x and y have the form m/e^2 and n/e^3, if necessary for time. This argument is very similar to one done in section II.4, and in fact quickly follow from that previous argument. But if you have time, you might as well do it.

要反覆提醒聽眾基本論點,以免大家迷失在細節中。
Try to remind the audience occasionally of the overall plan of the argument so they don't get lost in the details.
14 2P的高度
Height of 2P
III.3:2P的高度
Section III.3: Height of 2P
講完III.3全部。內容比較多,但是我覺得其中的技術內容不是很有趣,因此不值得用兩節課。這是一些建議。
Please cover section III.3. This is a lot of material for one lecture, but I don't think the techniques are interesting enough to deserve being spread over 2 lectures. So here's some advice.

最重要的一點是如果我們可以證明引理3'(比較容易證明),則可以直接推出引理3。引理3'是一個純粹代數的結果,所以稍微有點難,在課上不一定非要完整證明。
The most important part is to show that Lemma 3 will follow if we can prove Lemma 3' (and that part is quick.) Lemma 3' is a bit of a detour since it is essentially a result in pure algebra. So if we don't see a full and complete proof of Lemma 3' in class that's OK.

我想可以先寫下引理3'的完整證明,看需要多少時間,如果多於一個小時(我覺得會),就只給(a)或(b)的完整證明,至於其他部分就稍微提一下,但至少要約略說明證明使用的方法。
I think you can be optimistic at first and write up a lecture that includes a full proof of Lemma 3', and then practice it once and see how long it takes. If it is too long for an hour (I suspect so), then give a careful proof of only one of the parts (a) or (b), and for the other part just give some highlights of the proof or at least a general idea of what sort of techniques it uses.
15 一個有用的同態(1)
A Useful Homomorphism - Part 1
III.4:一個有用的同態(1)
Section III.4: A Useful Homomorphism - Part 1
這裡會慢下來,每節用兩堂課的時間來講述,因為這一節較長而且所談的技巧方法比較有意思。
Starting with this section we will slow down a bit and take two lectures per section, both because the sections are longer and the techniques they introduce are more interesting.

本講講完III.4節,直到第79頁的命題 (如果有時間, 可以講述命題的內容。)
In this lecture, please cover section III.4 up through the Proposition on p. 79 (if you have time to state this proposition, please do.)

本節從純計算的角度介紹橢圓曲線的一些重要性質。本節介紹的映射φ和ψ是"同源"的實例。上課的時候,會用意想不到的方法來描述正在進行的內容(有些在書中第79頁有提到)
What this section does is introduce in a purely computational way some important facts about elliptic curves. The maps phi and psi which are introduced here are examples of "isogenies". When we meet I can let you in on some of the fancier ways of describing what is happening here (the book already says a bit about this on p. 79.)
16 一個有用的同態(2)
A Useful Homomorphism - Part 2
III.4:一個有用的同態(2)
Section III.4: A Useful Homomorphism - Part 2
請敘述第79頁的命題且證明,講完III.4節。作者留下一點證明給讀者,如果你有興趣,可以補上。
Please state the proposition on p. 79 and complete the proof of it, taking you up through the end of section III.4. The authors leave some bits of the proof to the reader. You can present some of those if you want.

我可以再告訴你一些關於第79頁內容中"看來很深奧"的方法,這可以用於課堂講述上。準備講課內容前可以找我討論。
I can tell you a bit more about the "highbrow" approach to all of this which is mentioned on page 79, and you can use some of this in your lecture. You might want to come see me to discuss this before you prepare your lecture.
17 Mordell定理(1)
Mordell's Theorem - Part 1
III.5:Mordell定理(1)
Section III.5: Mordell's Theorem - Part 1
講完第III.5節的Mordell定理,講到第85節的命題以及證明的一部分。不要太急,可以先講命題的(a)和 (b) 部分,其餘部分留給下一個人。由於時間較多,可以把(a)和(b)部分講得比教材上詳細。
Please cover section III.5 on Mordell's Theorem, through the statement of the Proposition on p. 85 and part of the proof. I don't want to rush this, so just prove parts (a) and (b) of the proposition and leave the rest for the next person. This may give you time to be more detailed in your proof of parts (a) and (b) than the book.
18 Mordell定理(2)
Mordell's Theorem - Part 2

例(1)Examples - Part 1
III.5:Mordell定理(2)
Section III.5: Mordell's Theorem - Part 2

III.6:例(1)
Section III.6: Examples - Part 1
再次?述第85頁的命題,其中的第(a),(b)部分已經在上堂課證明過了,該命題也論述過。證明(c)和(d),講完第III.5節。(提一下最後幾周的關鍵內容。)
Restate the Proposition on p. 85, which will also be stated in the previous lecture and parts (a), (b) already proved. Then prove parts (c) and (d) and finish section III.5 (You get to give the punchline of the last several weeks!)

從III.6,講到90頁的中間"...of rank r."
Then begin section III.6, through the middle of p. 90, ending "...abelian group of rank r."
19 例(2)
Examples - Part 2
III.6:例(2)
Section III.6: Examples - Part 2
接著講III.6,從90頁的中間"In our case what are the possibilities..."開始,先簡單提過上次講課內容, 包括符號#?[2] 的意思,和計算公式,這些在上堂課末有提過。
Continue Section III.6, beginning in the middle of p. 90 with the paragraph "In our case what are the possibilities..." Remind everyone a little of what happened last time, including what the notation #Gamma[2] means, and the formula for it which was the end of the last lecture.

講到第94頁 "...we will now illustrate this procedure with several examples."也就是說講到例1之前,第92-93頁的步驟很重要,不要太拘泥於第. 90-91的細節。
Then go through p. 94 up through "...we will now illustrate this procedure with several examples." i.e. stop just before Example 1. The procedure on pages 92-93 is especially important, so try not to get bogged down in the technicalities on pp. 90-91.
20 例(3)
Examples - Part 3
III.6: 例(3)
Section III.6: Examples - Part 3
講94-98頁的例1-4。我會給一些作業,計算橢圓曲線的rank,所以請仔細講解這些例子,也可以只講其中一部分例子(比如去掉例4)。
Present the examples 1-4 on pages 94-98. I will likely assign some homework exercises which require you all to do calculations like these to find the rank of certain elliptic curves. So take your time to try to make it clear what is going on in these examples. If that means only doing some of the examples (for example omitting example 4), that's OK.
21 奇異三次曲線
Singular Cubics
III.7:奇異三次曲線
Section III.7: Singular Cubics
講III.7節關於奇異三次曲線的內容。本節內容有點偏離主題,因為我們主要是研究非奇異橢圓曲線,但是為了完整起見,還是要搞清楚奇異三次曲線的性質。一堂課就夠了。(刪掉主定理的(b)部分)。
Cover all of section III.7 on Singular Cubics. This is essentially a detour, since the nonsingular curves are the most interesting ones for us, but for completeness it is worthwhile to see what happens in the singular case. You should be able to cover this material in one class (omitting, just as the book, the proof of part (b) of the main theorem.)
22 有限域的有理點
Rational Points over Finite Fields
IV.1:有限域上橢圓曲線的有理點
Section IV.1: Rational Points over Finite Fields
講完書上的第IV.1節。本章我們將研究"具有p個元素的有限域Fp", 其中p是個素數。需要注意到此處模p的算術屬於初等數論。
Cover Section IV.1 in the book. In this chapter we will be working over the "finite field of p elements F_p" for some prime p. It is worth noting that all this means is that we are doing "arithmetic modulo p" exactly as in a first number theory course.
23 Gauss定理(1)
Gauss's Theorem - Part 1
IV.2:Gauss定理(1)
Section IV.2: Gauss's Theorem - Part 1
講第IV.2節,從開始到114頁頂部的公式Mp = 9[RTS]/m。內容有點多,注意在50分鐘內講完。如果講不完,可以直接接受論述而不用證明,比如可以刪掉Fp* 是一個迴圈群的證明。
Cover Section IV.2 from the beginning up through the formula M_p = 9[RTS]/m on the top of p. 114. This is kinda dense, so time yourself beforehand to make sure it fits in 50 minutes. If not, you can assume some facts without proof, for example you could omit the proof that F_p^* is a cyclic group.
24 Gauss定理(2)
Gauss's Theorem - Part 2
IV.2:Gauss定理(2)
Section IV.2: Gauss's Theorem - Part 2
接著第IV.2節,從第114頁的第一節"Now we just have to find..."直到本節完。
Continue in section IV.2 from the first full paragraph on p. 114 "Now we just have to find..." to the end of that section.

我沒有要求講完全部內容,你也不要試圖全講。本節主要內容是講Gauss定理的證明。因為這裏包括太多的公式和計算,所以可以刪掉部分內容,比如可以刪掉證明最後部分A的唯一性。
I do not expect you to include all of this material, nor would I advise you to try. So the main goal is to finish the proof of Gauss' theorem. This already involves a lot of formulas and calculations, so if it is itself too much you can omit some things like the very end of the proof where it is shown that A is unique.

第117頁有個印刷錯誤:第四個公式應該是β1β2β3 = (3k-2)p (漏了p)。
There is a typo on p. 117: the fourth displayed equation should read beta_1 beta_2 beta_3 = (3k-2)p (the p is missing.)

如果你有時間證明後面的例子和猜想,那非常好。如果將這些內容留給大家自學,那也不錯。
If you have time to say anything about the various examples and conjectures following the proof, fine, but it's also fine to leave that stuff for people to read themselves at home.
25 復習有限階點
Points of Finite Order Revisited
IV.3:復習有限階點
Section IV.3: Points of Finite Order Revisited
講IV.3節,本節內容非常重要,因為本節提出(另一個)計算有限階有理點的改良方法,對計算很有用。
Do Section IV.3. This is an important section, since it gives us (yet another) improved method for finding rational points of finite order, which is useful for calculations.

你可能沒有時間講完全部的例子,但是至少要講其中的一到兩個。
You will probably not have time for all of the examples at the end. It should suffice to cover one or two of them.
26 用橢圓曲線分解整數(1)
Factorization using Elliptic Curves - Part 1
IV.4:用橢圓曲線分解整數(1)
Section IV.4: Factorization using Elliptic Curves - Part 1
IV.4,講到132頁中間的這一行:"p-1 = product of primes to small powers"。
Cover Section IV.4, up through the displayed line "p-1 = product of primes to small powers" on p. 132.

總共有7頁,但沒有多少內容。所以要講述這7頁中重要內容,精簡上課內容,比如可以省略計算時間的估計,只講最後的結果。
This is 7 pages, but it is 7 pages in which very little is happening. So you should condense it into the most important points. You can probably skip the detailed calculations of the running times of raising to powers and of the euclidean algorithm; just say what the final estimate is.

最重要的部分是分解整數問題,要提到所有要點,其他的部分能省則省。
The important part is the problem of factoring integers. So concentrate that stuff (starting on mid p. 129) and streamline or omit as much of the rest as necessary.
27 用橢圓曲線分解整數(2)
Factorization using Elliptic Curves - Part 2
IV.4:用橢圓曲線分解整數(2)
Section IV.4: Factorization using Elliptic Curves - Part 2
講述第IV.4節關於整數分解的內容,特別是從第132頁中下面段落"Now we are ready to describe Lenstra's idea..."開始,到本節的結尾。
Describe the elliptic curve factoring algorithm in section IV.4. Specifically, begin on p. 132 with the paragraph "Now we are ready to describe Lenstra's idea..." and go to the end of the section.

為了全講完,必需簡化部份內容,儘量簡化本節末尾n = 1715761513 的例子(起碼省掉所有計算kP的所有計算細節,以及表中所有冗長的數字。)
Getting through this will require some condensing or cutting. Streamline the example n = 1715761513 at the end of the section as much as possible (at the very least leave out all of details of calculating kP with all of the hideous numbers in the table).
28 三次曲線上的整點
Integer Points on Cubics

計程車(1)
Taxicabs - Part 1
V.1:三次曲線上的整點
Section V.1: Integer Points on Cubics

V.2:計程車(1)
Section V.2: Taxicabs - Part 1
第V.1節(關於三次曲線的整點論題的一個引論)和第V.2,直到第149頁的命題。這些都是直接的內容。
Do Section V.1 (an introduction to the subject of integer points on cubics, and Section V.2 up through and including the statement of the Proposition on page 149. This is pretty straightforward material.
29 計程車 (2)
Taxicabs - Part 2

Thue定理(1)
Thue's Theorem - Part 1
V.2:計程車 (2)
Section V.2: Taxicabs - Part 2

V.3:Thue定理(1)
Section V.3: Thue's Theorem - Part 1
從第V.2節中間開始,也復習一下第149頁的命題 (這個命題以及證明都在以前講過了)。
Begin in the middle of Section V.2, by reminding us what the proposition on P. 149 said (it will be stated and proved in the previous lecture.)

然後接著從"Next we might ask for..."開始直到本節結束。注意書上沒有明確說Lang猜想是什麼,實際上它是本節末尾的Silverman定理的延伸。別管它,就講Silverman定理。
Then continue starting with "Next we might ask for..." up through the end of that section. Note that the authors never say exactly what Lang's conjecture is. It is some sort of generalization of Silverman's Theorem which is stated at the end of the section. Don't worry about Lang's conjecture, just state Silverman's Theorem.

從V.3開始,講到第153頁的"...which is due to Thue"也就是說講到Diophantine逼近定理。如果你有時間就談Diophantine逼近定理的內容,這個定理在以後肯定還會提到。
Then begin section V.3, up through p. 153 "...which is due to Thue", i.e. end right before the Diophantine approximation theorem. You can state the DAT if you have time, although it will be restated next time in any case.
30 Thue定理(2)
Thue's Theorem - Part 2
V.3:Thue定理(2)
Section V.3: Thue's Theorem - Part 2
講153頁的Diophantine逼近定理(如果在上堂課已經講過就再講一次),然後講到本節的末尾,並大略證明。
State the Diophantine Approximation Theorem in Section V.3 on p. 153 (restate it if it is stated in the previous lecture), and continue through to the end of the section, which gives a sketch of the proof.

大略證明是重要的,這是因為一旦開始就容易迷失在證明的細節裏,所以我們要記住真正的目的。
The sketch is important, because it will be very easy to get lost in the details of the proof once we start it, so we need to keep reminding ourselves where we're going.
31 輔助多項式的構造
Construction of an Auxiliary Polynomial
V.4:輔助多項式的構造
Section V.4: Construction of an Auxiliary Polynomial
開始Diophantine逼近定理證明的第一步,也就是V.4節所講的輔助多項式的結構。這一節問題很多,首先是章節較長,還有就是符號多,最要命的是這節內容很複雜。但是我們不想分兩節課講這些內容,因為這樣的話第二堂就會在感恩節後了。
You will handle the first step of the proof of the DAT, which is the construction of an Auxiliary polynomial in Section V.4. This section is problematic. It is very long, and there is a lot of notation, but overall it's actually not doing anything all that complicated. Also, we don't want to spread it over two lectures, because then the second lecture will be after Thanksgiving break.

計畫是這樣的:Siegel引理比較容易,所以講述內容,用比較小的N和M舉個例子,不一定要證明。可以說證明就是運用初等矩陣理論、三角不等式和鴿籠原理。
So here's the plan. Siegel's Lemma is actually quite straightforward. So state it, and maybe calculate an example with small N and M to make it clear why it's reasonable, but probably don't try to do the proof. You can say that the proof uses nothing beyond elementary matrix theory, repeated use of the triangle inequality, and the pigeonhole principle.

接著仔細講輔助多項式的構造,從160頁中間開始,最後講輔助多項式定理。講述書上的例題就可以結束該單元,不需要講太多,有用的例題是不錯,但我不確定這例題的細節會很有用。如果還有時間就提一下章節的最後一段,這是下一章的重點。
Then do carefully the construction of the auxiliary polynomial, beginning on mid-page 160, finally stating the Auxiliary Polynomial Theorem. Refer the class to the worked example in the book that ends the section, but don't try to do much or any of it. Usually examples are good, but I'm not so sure how useful seeing the details of this one is. If you have a chance to mention the very last paragraph of the section, though, that sets up the next section nicely.

記住這是同學這學期最後一次講課了。
Take heart in the knowledge that this is the last time you have to present this semester.
32 輔助多項式有界性
The Auxiliary Polynomial is Small
V.5:輔助多項式有界性
Section V.5: The Auxiliary Polynomial is Small
V.5中關於輔助多項式有界性。本節不長,但是符號較多,要仔細講講,時間足夠。最好把輔助多項式有界性定理掛在黑板上,這樣就不會忘記我們的目的。
Do Section V.5 on the smallness of the Auxiliary polynomial. This is not really a long section, but it has a lot of notation. I think it's worth doing in painful detail. Explain each step carefully, since you have time to do so. It's probably worth keeping the Smallness Theorem up on the board, so we don't forget where we're going.

如果確實還有時間,先復習Diophantine逼近定理,因為剛放假回來,可以重溫有界性定理在Diophantine逼近定理的證明中有什麼作用。
If you think you will have time, see if you can start with a brief review of the DAT we are trying to prove, since we will just be coming back from break. Remind us how the Smallness Theorem fits into the plan of the proof.
33 輔助多項式的非零性
The Auxiliary Polynomial Does Not Vanish
V.6:輔助多項式的非零性
Section V.6: The Auxiliary Polynomial Does Not Vanish
V.6節中關於輔助多項式非零性的定理。對於50分鐘來說這不是太重的負擔,所以有時間講解這個定理在Diophantine逼近定理的證明中的作用。
Do Section V.6 on the non-vanishing of the auxiliary polynomial. This is not a huge amount of material for 50 minutes. So you should have time to begin with a reminder of how the Non-Vanishing Theorem generally fits in to our plan of proof of the DAT.

整個講課期間要把非零性定理放在黑板上。如果有時間,請詳細描述證明的細節。
Please leave the Non-Vanishing Theorem on the board for the whole lecture if possible. In any case you will have time to cover the details of all of the steps carefully.
34 Diophantine逼近定理的證明
Proof of the DAT

進一步的發展
Further Developments
V.7:Diophantine逼近定理的證明
Section V.7: Proof of the DAT

V.8:進一步的發展
Section V.8: Further Developments
V.7節,也就是Diophantine逼近定理的證明。這個證明結合前面所講的引理,有充分時間仔細講解。
Do Section V.7, which finishes up the proof of the DAT. This basically collects together all of the Lemmas we prove earlier and shows how they fit together. You'll have time to do this carefully.

看看上面用了多長時間,剩下的時間可以講講V.8節Diophantine逼近定理證明的延伸,不必講得很詳細,只要講延伸和效率問題即可。
Depending on how long that part takes, you can use the remaining time to survey some of the generalizations of the DAT that are mentioned in Section V.8. You don't need to try to be comprehensive here, just mention some of the possible generalizations and the problem of effectivity.
35 同餘數和橢圓曲線I: Koblitz (1)
Congruent Numbers and Elliptic Curves I: Koblitz - Part 1
同餘數和橢圓曲線I:Koblitz教材的第I.1和 I.2節
Congruent Numbers and Elliptic Curves I: Koblitz, Section I.Intro, I.1. and I.2
最後這兩堂課討論先前的橢圓曲線怎麼和數論裏的一個有趣命題-同餘數理論-之間的關係。
In the last two lectures we will show how the material on elliptic curves we have already learned is related to an interesting problem in number theory, the congruent number problem.

在講課中,要介紹同餘數問題,以及它如何轉化成橢圓曲線問題。
In your lecture, I ask you to introduce the congruent number problem, and show exactly how it translates into a problem about elliptic curves.

特別注意,內容要從Koblitz的書中第一章引論、第I.1和I.2節開始。多數介紹性的內容可以略去,所以如果有時間可以講講Tunnell定理,很可惜我們以後不會再講這個定理了,所以恐怕以後都會繼續保持神秘感.
Specifically, derive your lecture from the Chapter I introduction, section I.1, and section I.2 from Koblitz's book. Most of the introduction can be skipped, though if you think you will have time please mention Tunnell's theorem. Unfortunately, we won't actually talk about that theorem again, so it is going to remain mysterious, I'm afraid.

有時間講完第I.1和 I.2節的大部分內容。完整地講命題1的證明。如果想就討論不同的(X,Y,Z)可以導出同一個x(我肯定你可以推導出來,但我不知道為什麼書上沒有講)。然後證明命題2。這兩個命題都是初級的,甚至連橢圓曲線的群定律都不用。
You should have time to cover the majority of Sections I.1, I.2. Prove Proposition 1 in full. If you want to mention also why no two distinct triples X,Y,Z can lead to the same x, please do (I'm sure you can figure this out, I don't know why the book omits it.) Then prove also Proposition 2. This is all pretty elementary, and doesn't even refer to the group law on the elliptic curve.
36 同餘數和橢圓曲線II: Koblitz (2)
Congruent Numbers and Elliptic Curves II: Koblitz - Part 2
同餘數和橢圓曲線II:Koblitz教材的第I.9節
Congruent Numbers and Elliptic Curves II: Koblitz, Section I.9
以Koblitz的書第I.9節為基礎,詳細講述橢圓曲線理論和同餘數問題之間的關係。
In your lecture, you will complete the picture of exactly how the congruent number problem is related to the theory of elliptic curves, basing your lecture primarily on Section I.9 of Koblitz's book.

I.9節命題17以前的內容可以跳過,因為不是講過了的內容,就是不值得講的內容。
Much of the material in I.9 preceding Proposition 17 can be skipped; we either know it already or it's not worth mentioning.

命題17實際上在作業8裏證明過了。但是並不是每個人都做了這個習題,只有少數人做對了,而且這個習題還靠另一個沒證明的習題做基礎,所以值得再提一次。
We actually proved Proposition 17 already in a homework exercise on problem set #8. But not everyone did this problem, and fewer did it completely correctly, and it relied on an unproved exercise. So it's worth repeating here.

要證明命題17,首先要證明I.8節的命題16(很簡單)。接著主要講命題17的證明第一段和最後一段,證明中間的引理,主要是證明橢圓曲線上有限階有理 點到有限域上橢圓曲線的點之間的嵌入映射的存在性,其中點的階不整除兩倍的判別式2D。我們已經知道了這個結果,至少知道有限階點的情形,而這正是我們需 要的。也可以看看作業8的答案。
To prove Proposition 17, first prove Proposition 16 from Section I.8 (it's easy), which I included in the handout. Then I think all you need to do is the first paragraph of the proof of Proposition 17 and the last paragraph. The stuff in between, including the internal Lemma, is there only to prove there exists an injective map from the group of rational points on an elliptic curve to the group of points over each finite field of prime order not dividing 2D where D is the discriminant. We already know this, at least in the case of points of finite order, which is all we need. See also the solutions to Homework #8.

先跳過命題18,直接證明命題19和命題20,這會佔去大部分的時間,尤其是命題20有點棘手(雖然是初級的).最後講命題18,這是命題19和命題17 的推論。這樣比直接證明命題18要好些。最後作結論,證明命題18把同餘數問題轉化為純粹橢圓曲線問題,並判別橢圓曲線的rank是不是為零。不幸地,在 橢圓曲線理論裏這也是個困難問題,我們已經在以前的講述中看到了。
Then skip Proposition 18 for the moment, and prove Propositions 19 and 20. This will take most of your time, since the proof of Proposition 20 is a little involved (though still elementary.) Finally, mention Proposition 18, and show it is a quick consequence of Propositions 19 and 17. This is better than the given proof of Proposition 18, which relies on an earlier exercise in Koblitz's book. Finally, conclude by saying that Proposition 18 turns the congruent number problem into a problem purely about elliptic curves; decide if the rank of such curves is zero or not. Unfortunately, that itself is also a very difficult problem, as we have seen before.

 


這是對學生在準備課程內容時的一些建議指導。(PDF)
These are directions suggested to students on how to prepare their lectures. (PDF)

Lecture Notes Table 課 課程單元 課堂講稿
1 射影平面
The Projective Plane
(PDF 1)

(PDF 2)
2 射影平面上的曲線
Curves in the Projective Plane
(PDF)
3 圓錐曲線上的有理點
Rational Points on Conics
(PDF)
4 三次曲線的幾何學
Geometry of Cubic Curves
(PDF)
5 Weierstrass標準形
Weierstrass Normal Form
(PDF)

(PDF)
6 群定律的精確公式(座標公式)
Explicit Formulas for the Group Law
(PDF)
7 二階點和三階點
Points of Order Two and Three
(PDF)
8 判別式
The Discriminant

座標為整數的有限階點(1)
Points of Finite Order have Integer Coordinates - Part 1
(PDF)
9 座標為整數的有限階點(2)
Points of Finite Order have Integer Coordinates - Part 2
(PDF)
10 座標為整數的有限階點(3)
Points of Finite Order have Integer Coordinates - Part 3

Nagell-Lutz定理
The Nagell-Lutz Theorem
(PDF)
11 三次曲線的實和複點
Real and Complex Points on Cubics
(PDF)
12 高函數和遞降法
Heights and Descent
(PDF)
13 點P+ P0的高
Height of P + P_0
(PDF)
14 2P的高度
Height of 2P
(PDF)
15 一個有用的同態(1)
A Useful Homomorphism - Part 1
(PDF)
16 一個有用的同態(2)
A Useful Homomorphism - Part 2
(PDF)
17 Mordell定理(1)
Mordell's Theorem - Part 1
(PDF)
18 Mordell定理(2)
Mordell's Theorem - Part 2

例(1)
Examples - Part 1
(PDF)
19 例(2)
Examples - Part 2
(PDF)
20 例(3)
Examples - Part 3
(PDF)
21 奇異三次曲線
Singular Cubics
(PDF)
22 有限域上的有理點
Rational Points over Finite Fields
(PDF)
23 Gauss定理(1)
Gauss's Theorem - Part 1
(PDF)
24 Gauss定理(2)
Gauss's Theorem - Part 2
(PDF)
25 復習有限階點
Points of Finite Order Revisited
(PDF)
26 用橢圓曲線分解整數(1)
Factorization using Elliptic Curves - Part 1
(PDF)
27 用橢圓曲線分解整數(2)
Factorization using Elliptic Curves - Part 2
(PDF)
28 三次曲線上的整點
Integer Points on Cubics

計程車(1)
Taxicabs - Part 1
(PDF)
29 計程車(2)
Taxicabs - Part 2

Thue定理(1)
Thue's Theorem - Part 1
(PDF)
30 Thue定理(2)
Thue's Theorem - Part 2
(PDF)
31 輔助多項式的構造
Construction of an Auxiliary Polynomial
(PDF)
32 輔助多項式有界性
The Auxiliary Polynomial is Small
(PDF)
33 輔助多項式的非零性
The Auxiliary Polynomial Does Not Vanish
(PDF)
34 Diophantine逼近定理的證明
Proof of the DAT

進一步的發展
Further Developments
(PDF)
35 同餘數和橢圓曲線I:Koblitz (1)
Congruent Numbers and Elliptic Curves I: Koblitz - Part 1
(PDF)
36 同餘數和橢圓曲線II:Koblitz (2)
Congruent Numbers and Elliptic Curves II: Koblitz - Part 2
(PDF)

 


作業在下表標明的課堂上提交。
Homework assignments are due in the lecture sessions noted in the table below.

Assignments 課 作業 解答
4 作業1(PDF)
Homework 1 (PDF)
(PDF)
7 作業2(PDF)
Homework 2 (PDF)
(PDF)
10 作業3(PDF)
Homework 3 (PDF)
(PDF)
13 作業4(PDF)
Homework 4 (PDF)
(PDF)
15 作業5(PDF)
Homework 5 (PDF)
(PDF)
18 作業6(PDF)
Homework 6 (PDF)
(PDF)
24 作業7(PDF)
Homework 7 (PDF)
(PDF)
27 作業8(PDF)
Homework 8 (PDF)
(PDF)
30 作業9(PDF)
Homework 9 (PDF)
(PDF)
34 作業10(PDF)
Homework 10 (PDF)
(PDF)

 

 

這裏是一些供學生參考的指導和可能的專題題目。(PDF)
These are directions and possible project topics suggested to students. (PDF)

學生專題
Student Projects

一般二次曲線的Hasse定理和有理點(PDF)(Dilip Das提供。獲得許可使用。)
Hasse's Theorem and Rational Points on the General Conic (PDF) (Courtesy of Dilip Das. Used with permission.)

橢圓曲線的初步結果(PDF)
A Few Elementary Facts about Elliptic Curves (PDF)

橢圓曲線密碼學(PDF)
Elliptic Curve Cryptography (PDF)

一維群代數簇(PDF)
On 1-dimensional Group Variety (PDF)

最優保密性:橢圓曲線和現代密碼學(PDF) (Thomas Coffee提供。獲得使用許可。)
Best Kept Secrets: Elliptic Curves and Modern Cryptosystems (PDF) (Courtesy of Thomas Coffee. Used with permission.)

Waring的問題,計程車牌問題以及其他方冪和問題(PDF)
Waring's Problem, Taxicab Numbers, and Other Sums of Powers (PDF)

橢圓曲線密碼學: Diffie-Hellman密鑰交換協定與橢圓曲線離散對數問題之間的關係(PDF)
Elliptic Curves in Public Key Cryptography: The Diffie Hellman Key Exchange Protocol and its relationship to the Elliptic Curve Discrete Logarithm Problem (PDF)

 

 

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