I'm beginning to doubt it.

I've spent the past five years slightly obsessed with elliptic curves having torsion subgroup Z_2 x Z_8. I proved last summer that such elliptic curves are birationally equivalent to



(This is probably in Ogg's paper where he listed families of elliptic curves for each of the fifteen known torsion subgroups.) Of course, there is a natural conjecture that as the rational parameter t varies, you get all kinds of ranks r. However, in the years I've been playing around with this curve, the highest rank found is r = 3. I find that too surprising, so I spent this past summer searching for larger ranks. My SUMSRI students worked hard writing code in Maple which used mwrank to compute ranks for some 500,000 elliptic curves. Still, we didn't find any curves of rank r > 3. Does this mean that no such curves exist?

Let me be a bit more precise about what's going on here. It's well-known that for each rational number t, the collection of rational points on the curve above forms a finitely generately abelian group:



This is the Mordell-Weil group, and r = r(t) is the rank of E_t. Using Galois cohomology, we have a short exact sequence



where the three objects are finite groups of orders





for some integers r = r(t) and s = s(t). The integer r is the (Mordell-Weil) rank, and the integer s is the 2-Selmer rank. Fortunately, software such as mwrank and MAGMA can compute s incredibly quickly, but unfortunately have trouble computing r. This is because the third group above, the 2-part of the Shafarevich-Tate group, is too mysterious to compute. Note that r <= s, so computing the 2-Selmer rank gives a nice upper bound for the Mordell-Weil rank.

I'm going to perform a "brute force" search for a rational number t such that r(t) > 3... I'm going to generate a list of candidate elliptic curves by considering t = a/b for |a|, |b| <= 5,000. I ran a Maple code on the machines at Purdue this afternoon and this computed 3.1 million(!) elliptic curves. Over the next couple of days, I'll use mwrank to compute the 2-Selmer ranks s = s(t). I'll present a histogram of the data (i.e., s(t) vs. t) to get an idea of how difficult it will be to find t such that r(t) > 3.

Stay tuned...