Friday, June 08, 2007

Seventh Exam: Elliptic Curves

I forgot this exam paper inside the exam room, so you'll only get summaries of problems. But still plenty of comments. =)

So I slept through all three of my alarms today. They go off at 8, 8:15, and 8:30 and I figured that at least one of them would get through to me. Apparently the stress and no sleep is getting to me, though, cause I woke up at 8:50 today. (And I'm really really out of shape... I was tired just sprinting to CMS.) My thought process went something like "Oooh, early... it's only 8:50... 8:50!!!! Crap crap crapcrapcrapcrap...." I wasn't late to the exam, though. Got there a minute ahead of time and wearing contacts and with my hair brushed. And wearing clean clothes. I was damn proud of myself... after I stopped panicking.

The exam was long. And by long I mean we should have had 4.5 hours instead of 3. The kind of long where you just want to turn in all of the class notes, and that wouldn't be too little writing. The kind where you're panicking the entire time because you won't be able to do all of the exam. The kind where everyone asks for more paper because the people setting up the room didn't prepare for that kind of exam.

1. Consider the elliptic curve X^3 + Y^3 + dZ^3 = 0 with identity (1:-1:0). Describe the group law on it geometrically. Find the points of inflexion. (I messed up on this part cause I was nervous. Not too badly, though.) Compute its Weierstrass equation. Prove that there are infinitely many rational x,y such that x^3 + y^3 = 7. And can I say "EEEWWWWWW"? EEEEEEEEWWWWWWW. Lots of computations, lots of icky worrying, and you have to find a point on the curve y^2 = x^3 - 432*7^2. 432??!! What kind of sadistic person gives a question with that kind of coefficient? On a test?! I'm surprised I made so few mistakes.

2. Let f be a power series in T with coefficients in R such that f(0) = 0 and f'(0) is a unit. Show that there exists a power series g in T such that f(g(T)) = T and g(f(T)) = 0. Outline where this fact is used in the proof of weak Mordell-Weil. OK, so first off, the computations in this proof are icky. Just cause there are a lot of coefficients to keep track of, and it was icky. On the other hand, it was pretty easy. But the second part was just brutal, cause this fact is SO far away from Mordell-Weil that it's unclear how much you're even supposed to write. How much can you assume? How precisely do they want us to point out where the fact is used? I just ended up writing a quick one-paragraph proof of weak Mordell-Weil and then pointing out where the fact is necessary... but maybe the question wanted us to prove that part in its entirety? What does "outline" mean? Does it mean "touch on the main points" or does it mean "write out in full excruciating detail"? Cause given the way things are structured in this place I wouldn't put it past them for it to mean the second. So I either did really well on this problem or really badly. I did well enough for me, anyway, so *pffffffft* to them if they don't think so.

3. EITHER write an essay on isogenies, including a proof that deg is a quadratic form, OR write an essay on heights and their place in the proof of Mordell-Weil. Almost everyone that I know did the second part because proving that deg is a quadratic form is just UGLY. And long. And did I mention ugly? And, once again, how much did they want in the essay? Heights are a pretty specialized construction, and there isn't really that much that they do... so is defining them, giving their properties and stating their use enough? Or do they actually want all of the gory details? I wrote two pages on this, and I'm not sure that it was enough.

4. Outline a procedure that often works to compute the rank of an elliptic curve over Q. For p a prime show that the rank of the group of rational points on y^2 = x^3-p^2x is at most 2. Given an example where the rank is exactly one. All of the same questions as above apply to the meaning of the word "outline", of course... and the computations were ugly. And p=3 didn't work, so I had to do the work over again to try p=5. (Which did work, thank God.) And I kept worrying that I didn't write enough for the stupid "outline."

So all of these worries that I didn't write enough, and I still ran out of time. I actually mostly finished everything, but I wanted to add a couple more sentences to the end of 3. This test was evil. And stressful. And long.

But I'm done!!!!! Done done done done done done done done done done done odone done done done odne done dondod done ond eond oend onedone do edn eod done ond eond odn eondoe odned. Right. =) I can go back to my usually scheduled life now. You know what the first step is? Throwing all of this out:

(That's actually not all of it. I also have two notebooks filled with notes that I made to studly, and my desk is also covered with notes. I'll be so glad to get it all out.)


Anonymous Anonymous said...

Enjoy your time, sleep, and food.

BTW, your room floor was not that bad at all :-)


9:39 AM  

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