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!

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Very nice! If we consider the large q limit instead (hyperelliptic curves of fixed genus), then the proportion of vanishing does tend to 0, and there’s a quantitative estimate (saving a small power of q depending on g). I don’t know if lower bounds have been studied in that case.