BSD and congruent numbers

Question

Let $n$ be a positive integer, and let $E_n$ denote the elliptic curve $y^2=x^3-n^2x$. By work of Tunnell, it's known that if $E_n$ satisfies the BSD conjecture, then there is an algorithm to decide whether $n$ is a congruent number or not. By work of Bhargava and Shankar, we also know that a positive proportion of elliptic curves satisfy BSD. Is it known whether a positive proportion of the curves $E_n$ satisfy BSD?

Answer 1

Bhargava and Shankar's work is only part of the story.

In order to say anything about BSD, we certainly have to be able to say something about the analytic rank. Sometimes we can get lucky and access the analytic rank directly: in quadratic twist families, Waldspurger's formula relates critical $L$-values to coefficients of half-integral weight modular forms. Tunnell's theorem is proved using Waldspurger's result, and for some elliptic curves, this can go further and be used to say that a positive proportion of twists have non-vanishing $L$-value. For example, if $E$ is the elliptic curve $X_0(14)$, then Kevin James (http://www.math.clemson.edu/~kevja/PAPERS/JAMS.pdf) showed that at least 7/64 of the (negative) quadratic twists have analytic rank 0. By Kolyvagin's theorem, analytic rank 0 implies arithmetic rank 0, so BSD is satisfied for these twists. James's work requires the associated half-integral weight modular forms to satisfy certain properties modulo 3, which I believe are not satisfied for the congruent number elliptic curve. (I didn't check this carefully.)

One other way in which we know how to control the analytic rank is via results on $p$-adic $L$-functions, using work of Skinner, Urban, and W. Zhang. Roughly speaking, these results say that if $E$ is moderately well-behaved at an odd prime $p$, and its $p$-Selmer group has rank either 0 or 1, then analytic rank = $p$-Selmer rank = arithmetic rank. Bhargava and Shankar study the average size of the $p$-Selmer group in families defined by congruence conditions, $p\leq 5$, and they're able to find a large, well-behaved family of curves for which the average size of the $p$-Selmer group is $p+1 < p^2$. Thus, a positive proportion of these curves have $p$-Selmer rank either 0 or 1, and we're able to apply the results of Skinner, Urban, and Zhang. (More care is needed, but this is the gist. A souped up version of this argument is in Bhargava's paper with Skinner and Zhang.)

You should now ask whether we can understand the distribution of $p$-Selmer groups in families of quadratic twists. This set is too small for Bhargava and Shankar's approach to reach it, but if $p=2$, there's another way! The key here is that, for any quadratic twist, $E_n[2] \simeq E[2]$, as Galois modules. Now, the $2$-Selmer group of $E_n$ sits inside $H^1(\mathbf{Q},E_n[2])$, but this $H^1$ is isomorphic to $H^1(\mathbf{Q},E[2])$, so we can view the varying $2$-Selmer groups as landing in the same target space. This gives us some traction, and indeed, if $E$ has full $2$-torsion, we can use this to find the full probability distribution on $\dim_{\mathbf{F}_2}\mathrm{Sel}_2(E_n/\mathbf{Q})$. This was carried out for the congruent number elliptic curve by Heath-Brown, and for general curves with full two-torsion by Dan Kane. In particular, for $5/16$ of the twists, the rank of $E_n(\mathbf{Q})$ is provably 0. This says nothing about the analytic rank, though, other than that it's even (this follows from the proof). To understand the analytic rank via Skinner, Urban, and Zhang, we'd need to be able to access $p$-Selmer for odd primes $p$, and here, we lose the fact that the $p$-Selmer groups of the twists all sit inside the same space. Thus, we've lost our traction, and we have no idea how to start.

Thus, it seems very likely that this is still not known. That said, I'm not familiar with the results of Shou-Wu Zhang that Jeremy mentioned, so I'd happily be corrected!

Answer 2

This really should be a comment, but it seems too long to be contained in the comment box.

I am only familiar with the details of Bhargava and Shankar's paper on binary quartic forms (which is not enough to imply BSD holds for a positive proportion of elliptic curves), but it seems that their work cannot say anything about the type of curve in your question. In particular, your curve has a rational $2$-torsion point. In the binary quartic forms paper, Bhargava and Shankar essentially proved that one can ignore such curves. Indeed, if we order elliptic curves defined over $\mathbb{Q}$ by naive height up to $X$, then there are $CX^{5/6}$ for some positive constant $C$ many elliptic curves but only $O(X^{3/4 + \epsilon})$ many curves with a rational $2$-torsion point. Therefore, this set is ignored in their counting theorems as part of the error term. I suspect that their later papers, which use ternary cubic forms to calculate the average of the $3$-Selmer group and another representation to calculate the average of the $5$-Selmer group, still have this quality; i.e., they simply ignore the curves with additional structure such as rational torsion points.

Therefore, using their technology, even if one succeeds in proving that 100% of elliptic curves satisfy BSD, it will still say almost nothing about the special curves with additional structure.