## Wanlin Li, “Vanishing of hyperelliptic L-functions at the central point”

My Ph.D. student Wanlin Li has posted her first paper!  And it’s very cool.  Here’s the idea.  If chi is a real quadratic Dirichlet character, there’s no reason the special value L(1/2,chi) should vanish; the functional equation doesn’t enforce it, there’s no group whose rank is supposed to be the order of vanishing, etc.  And there’s an old conjecture of Chowla which says the special value never vanishes.  On the very useful principle that what needn’t happen doesn’t happen.

Alexandra Florea (last seen on the blog here)  gave a great seminar here last year about quadratic L-functions over function fields, which gave Wanlin the idea of thinking about Chowla’s conjecture in that setting.  And something interesting developed — it turns out that Chowla’s conjecture is totally false!  OK, well, maybe not totally false.  Let’s put it this way.  If you count quadratic extensions of F_q(t) up to conductor N, Wanlin shows that at least c N^a of the corresponding L-functions vanish at the center of the critical strip.  The exponent a is either 1/2,1/3, or 1/5, depending on q.  But it is never 1.  Which is to say that Wanlin’s theorem leaves open the possibility that o(N) of the first N hyperelliptic L-functions vanishes at the critical point.  In other words, a density form of Chowla’s conjecture over function fields might still be true — in fact, I’d guess it probably is.

The main idea is to use some algebraic geometry.  To say an L-function vanishes at 1/2 is to say some Frobenius eigenvalue which has to have absolute value q^{1/2} is actually equal to q^{1/2}.  In turn, this is telling you that the hyperelliptic curve over F_q whose L-function you’re studying has a map to some fixed elliptic curve.  Well, that’s something you can make happen by physically writing down equations!  Of course you also need a lower bound for the number of distinct quadratic extensions of F_q(t) that arise this way; this is the most delicate part.

I think it’s very interesting to wonder what the truth of the matter is.  I hope I’ll be back in a few months to tell you what new things Wanlin has discovered about it!

## LMFDB!

Very happy to see that the L-functions and Modular Forms Database is now live!

When I was a kid we looked up our elliptic curves in Cremona’s tables, on paper.  Then William Stein created the Modular Forms Database (you can still go there but it doesn’t really work) and suddenly you could look at the q-expansions of cusp forms in whatever weight and level you wanted, up to the limits of what William had computed.

The LMFDB is a sort of massively souped up version of Cremona and Stein, put together by a team of dozens and dozens of number theorists, including too many friends of mine to name individually.  And it’s a lot more than what the title suggests:  the incredibly useful Jones-Roberts database of local fields is built in; there’s a database of genus 2 curves over Q with small conductor; it even has Maass forms!  I’ve been playing with it all night.  It’s like an adventure playground for number theorists.

One of my first trips through Stein’s database came when I was a postdoc and was thinking about Ljunggren’s equation y^2 + 1 = 2x^4.  This equation has a large solution, (13,239), which has to do with the classical identity

$\pi/4 = 4\arctan(1/5) - \arctan(1/239)$.

It turns out, as I explain in an old survey paper, that the existence of such a large solution is “explained” by the presence of a certain weight-2 cuspform in level 1024 whose mod-5 Galois representation is reducible.

With the LMFDB, you can easily wander around looking for more such examples!  For instance, you can very easily ask the database for non-CM elliptic curves whose mod-7 Galois representation is nonsurjective.  Among those, you can find this handsome curve of conductor 1296, which has very large height relative to its conductor.  Applying the usual Frey curve trick you can turn this curve into the Diophantine oddity

3*48383^2 – (1915)^3 = 2^13.

Huh — I wonder whether people ever thought about this Diophantine problem, when can the difference between a cube and three times a square be a power of 2?  Of course they did!  I just Googled

48383 Diophantine

and found this paper of Stanley Rabinowitz from 1978, which finds all solutions to that problem, of which this one is the largest.

Now whether you can massage this into an arctan identity, that I don’t know!

## Alexandra Florea on the average central value of hyperelliptic L-functions

Alexandra Florea, a student of Soundararajan, has a nice new paper up, which I heard about in a talk by Michael Rubinstein.  She computes the average of

$L(1/2, \chi_f)$

as f ranges over squarefree polynomials of large degree.  If this were the value at 1 instead of the value at 1/2, this would be asking for the average number of points on the Jacobian of a hyperelliptic curve, and I could at least have some idea of where to start (probably with this paper of Erman and Wood.)  And I guess you could probably get a good grasp on moments by imitating Granville-Soundararajan?

But I came here to talk about Florea’s result.  What’s cool about it is that it has the a main term that matches existing conjectures in the number field case, but there is a second main term, whose size is about the cube root of the main term, before you get to fluctuations!

The only similar case I know is Roberts’ conjecture, now a theorem of Bhargava-Shankar-Tsimerman and Thorne-Taniguchi, which finds a similar secondary main term in the asymptotic for counting cubic fields.  And when I say similar I really mean similar — e.g. in both cases the coefficient of the secondary term is some messy thing involving zeta functions evaluated at third-integers.

My student Yongqiang Zhao found a lovely geometric interpretation for the secondary term the Roberts conjecture.  Is there some way to see what Florea’s secondary term “means” geometrically?  Of course I’m stymied here by the fact that I don’t really know how to think about her counting problem geometrically in the first place.