y^2=x^3+Dz^12,

I have my student working on finding primitive integer solutions to some Diophantine equations of the form $C:y^2=x^3+Dz^{12}$. This is a genus two curve (in the appropriate weighted projective space), and it is a double cover of an elliptic curve, so computation of the Mordell-Weil rank is usually pretty straightforward. Say we find a $D$ so that the elliptic curve $y^2=x^3+D$ has rank $2$, but we suspect that there are no non-primitive solutions there. Can we prove this? I suspect that a simple a Mordell-Weil sieving should give us the desired result, and it seems pretty successful so far. However, it seems the trivial solutions
$(1,1,0)$ is causing some trouble for us. Right now, we're trying to figure out if there is a way of getting pass this problem.

Coin removing problem,

Here is a classic problem from combinatorial game theory/high school math contest:

There are 100 coins on the table, and there are two players that take turn, each taking 1, 2, or 3 coins from the pile. The person who can not move, loses. Who wins and what is the winning strategy?

If you haven't seen this before, I'm sure you can solve it after a little bit of playing around. Of course the 100 coins is just a red herring, and the winning strategy is all that matters.Furthermore, if the possible moves were 1,2,3,...,n coins, then the problem is about the same level of difficulty. However, I think this problem is a lot more tricky if the possible moves are chosen randomly. Say, each player can remove 3,5 or 8 coins. So, here is a problem that I like to know how to solve:

Given a set S, the set of possible moves, describe the set of positive integers so that player 1 has a winning strategy.

I'm not sure if there is a nice answer to this, however, I'm still curious if there is a way of computing this.

installing sage on sharcnet,

I've been trying to install sage on sharcnet for the past little while. There are few problems that come up when I try doing that, and I'm storing the experience here, in case I succeed, I can try it again.

• sharcnet uses pathscale as the default compiler, and it is recommended to me to compile using pathscale. pathscale has version 2.x.x, which is less than gcc 4.1.x that sage recommends. As result, I have to edit prereq-0.7/configure to get rid of it. (To install prereq, one should run {}/base/prereq-0.7-install)
• I need to have CFLAGS, CXXFLAGS, FCFLAG set, otherwise SAGE decides to set them to bunch of flags that causes problem on pathscale.
• after prereq, the next problem shows up in termcap. The -D flag for cc and gcc seem to be different. (This might be sharcnet preprocessing things.) Getting rid of the (unnecessary) offending -D flag makes it work. While making a patch, I notice that CFLAG is just over ridden. I'm changing that.

testing latex javascrip code.

I came across watchmath, and supposedly there are instructions to make latex code work on blogger. Specifically, something like $f(x)=x^2$ should look nice right now. I'm not sure if it is working though. Hmm... what about $$f(x)=x^2$$? What about ${f(x)=x^2}$?

I see, my template was causing some problem. Appologies for the ugly template, but now I can use latex code.

WSJ article

Wall Street Journal had an article with the title: Russia's Conquering Zero. There is also a big picture of $\zeta(s)=\sum_{n=1}^\infty {1 \over n^s}$. Now, what would you suppose the article is about? Here is the link for the curious.
http://online.wsj.com/articl/SB10001424052748703740004574513870490836470.html

Frey-Mazur's conjecture.

In his rational isogenies paper, Mazur asks if there are any pairs of elliptic curves E and F and prime l>5 such that E[l] \isom F[l] as Galois module. This question lead to the conjecture that no such pair of elliptic curves exist (I'm not sure where this was first published as a conjecture). Of course, with more computation power, we can actually find such pairs for l=7, 11, and even 13 just by looking at small Cremona's table. Dave Brown pointed out to me that the moduli space of such (ordered) pairs of elliptic curves for l=7 is in birational to P^2, and Sander Dahman and I spent a few weeks trying to come up with an explicit birational map for this moduli space. Similarly the moduli space of unordered pairs for l=11 and 13 have infinite number of rational points for similarly good geometric reasons. The natural question is what happens for 17? Doing a slow exhaustive search on Cremona's table doesn't seem to find any example for level less than 25000. This search is slowing down considerably, so I might need to come up with faster way of searching it right now. Right now, I go through all elliptic curves in the table, check their discriminant, see if we can do level lowering modulo 17 or so, if so, I look at all possible lower levels, and check if the two modular forms are congruent modulo 17. I guess the last step can be sped up by just checking if they are congruent, vs calculating the congruence number. I will start doing that soon.

J(11)[2],

In Poonen, Schaefer, and Schtoel, they find all rational points of $X_E(7)$ for various $E$'s by first finding ranks of $J_E(7)$ and applying Chabauty to find the rational points on $X_E(7)$ (and more stuff as well). They find the ranks of $J_E(7)$ by doing two descent, specially since they have an explicit generators for
$J_E(7)[2]$ in terms of the cusps of $X_E(7)$.
Now, it is natural to ask, does $J(p)[2]$ also have such an explicit set of generators? It seems unlikely, since, for example, $J(11)$ is 26 dimensional, and hence you would need 52 generators for it, by there are only 60 generators. This doesn't rule it out explicitly, but doesn't seem to be very likely, specially when one digs a bit deeper on how the cusps of $X_E(7)$ generated the two torsion points.