18.704 | Fall 2004 | Undergraduate

Seminar in Algebra and Number Theory: Rational Points on Elliptic Curves

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

lec # Topics Readings NOTES
1 The Projective Plane Section A.1: The Projective Plane First Half: Section A.1, up through p. 222 line 11 “… coordinates for the point.”

Second Half: Continuing from p. 222 line 12 “In order to motivate…” through the end of section A.1.

2 Curves in the Projective Plane Section A.2: Curves in the Projective Plane First Half: Section A.2, up through p. 228 line 4 (the displayed equations.)

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 Rational Points on Conics Section I.1: Rational Points on Conics First Part: Section I.1. up through line 4 from the bottom of p. 11 “…infinity for t!]”

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 Geometry of Cubic Curves Section I.2: Geometry of Cubic Curves 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.

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 Normal Form Section I.3: Weierstrass Normal Form 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.

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 Explicit Formulas for the Group Law Section I.4: Explicit Formulas for the Group Law 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 Points of Order Two and Three Section II.1: Points of Order Two and Three You should be able to cover the entire contents of Section II.1 comfortably in 50 minutes. Take your time.
8 The Discriminant

Points of Finite Order have Integer Coordinates - Part 1

Section II.3: The Discriminant

Section II.4: Points of Finite Order have Integer Coordinates - Part 1

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 Points of Finite Order have Integer Coordinates - Part 2 Section II.4: Points of Finite Order have Integer Coordinates - Part 2 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 Points of Finite Order have Integer Coordinates - Part 3

The Nagell-Lutz Theorem

Section II.4: Points of Finite Order have Integer Coordinates - Part 3

Section II.5: The Nagell-Lutz Theorem

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 Real and Complex Points on Cubics Section II.2: Real and Complex Points on Cubics 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.

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.

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 Heights and Descent Section III.1: Heights and Descent 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 Height of P + P_0 Section III.2: Height of P + P_0 Please cover Section III.2 entirely. The argument looks harder than it is, because it has so much notation.

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.

Try to remind the audience occasionally of the overall plan of the argument so they don’t get lost in the details.

14 Height of 2P Section III.3: Height of 2P 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.

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.

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 A Useful Homomorphism - Part 1 Section III.4: A Useful Homomorphism - Part 1 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.

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.)

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 A Useful Homomorphism - Part 2 Section III.4: A Useful Homomorphism - Part 2 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.

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’s Theorem - Part 1 Section III.5: Mordell’s Theorem - Part 1 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’s Theorem - Part 2

Examples - Part 1

Section III.5: Mordell’s Theorem - Part 2

Section III.6: Examples - Part 1

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!)

Then begin section III.6, through the middle of p. 90, ending “…abelian group of rank r.”

19 Examples - Part 2 Section III.6: Examples - Part 2 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.

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 Examples - Part 3 Section III.6: Examples - Part 3 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 Singular Cubics Section III.7: Singular Cubics 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 Rational Points over Finite Fields Section IV.1: Rational Points over Finite Fields 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’s Theorem - Part 1 Section IV.2: Gauss’s Theorem - Part 1 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’s Theorem - Part 2 Section IV.2: Gauss’s Theorem - Part 2 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.

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.

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.)

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 Points of Finite Order Revisited Section IV.3: Points of Finite Order Revisited 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.

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 Factorization using Elliptic Curves - Part 1 Section IV.4: Factorization using Elliptic Curves - Part 1 Cover Section IV.4, up through the displayed line “p-1 = product of primes to small powers” on p. 132.

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.

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 Factorization using Elliptic Curves - Part 2 Section IV.4: Factorization using Elliptic Curves - Part 2 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.

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 Integer Points on Cubics

Taxicabs - Part 1

Section V.1: Integer Points on Cubics

Section V.2: Taxicabs - Part 1

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 Taxicabs - Part 2

Thue’s Theorem - Part 1

Section V.2: Taxicabs - Part 2

Section V.3: Thue’s Theorem - Part 1

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.)

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.

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’s Theorem - Part 2 Section V.3: Thue’s Theorem - Part 2 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.

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 Construction of an Auxiliary Polynomial Section V.4: Construction of an Auxiliary Polynomial 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.

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.

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.

Take heart in the knowledge that this is the last time you have to present this semester.

32 The Auxiliary Polynomial is Small Section V.5: The Auxiliary Polynomial is Small 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.

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 The Auxiliary Polynomial Does Not Vanish Section V.6: The Auxiliary Polynomial Does Not Vanish 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.

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 Proof of the DAT

Further Developments

Section V.7: Proof of the DAT

Section V.8: Further Developments

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.

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 Congruent Numbers and Elliptic Curves I: Koblitz - Part 1 Congruent Numbers and Elliptic Curves I: Koblitz, Section I.Intro, I.1. and I.2 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.

In your lecture, I ask you to introduce the congruent number problem, and show exactly how it translates into a problem about elliptic curves.

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.

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 Congruent Numbers and Elliptic Curves II: Koblitz - Part 2 Congruent Numbers and Elliptic Curves II: Koblitz, Section I.9 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.

Much of the material in I.9 preceding Proposition 17 can be skipped; we either know it already or it’s not worth mentioning.

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.

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.

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.

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Fall 2004
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Lecture Notes
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