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.

testing

Just want to see how this works. Hmmm... well, I should put something mathematical in here anyways. So here is a question I'm wondering about right now. Let X_E(N) be the moduli space of elliptic curves $F$ along with an isomorphims $E[N]->F[N]$, ane let J_E(N)=Jac(X_E(N)). Note that elliptic curves isogeneous to E correspond to rational points on X_E(N), so we call these the trivial points on X_E(N). It is conjectured that for any E there is a constant c(E) such that for N>c(E) we have X_E(N) has only trivial rational points. We call this Frey-Mazur conjecture (although, Frey-Mazur conjecture is usually stated with c(E)=23 or so, and certainly independant of E). Unfortunately the standard methods used in proving X_1(N)(Q) has only trivial solutions don't apply here, since we can't find a quotient of J_E(N) with rank 0. In fact generically we expect J_E(N) to be irreducible over rationals, and there is no obvious reason why J_E(N) should have small rank. In fact, in the cases done by Poonen-Schaefer-Stoll for J_E(7) they already found an example with rank 3. I think Brown's work in extending their work to J_E(11) and J_E(13) seems to find other seemingly innnocent examples with fairly high ranks.

So here's my question: for a given E, how large can the rank of J_E(N)(Q) get as N gets larger? Is this value bounded? Are there examples where it grows as N gets larger? In theory, if the rank J_E(N)(Q) is less than genus of X_E(N) we can find all rational points of X_E(N)(Q), so if they don't grow too fast we should be fine.