Lower bounds for class number of function fields with fixed $q$, growing $g$

Question

Let $X$ be a smooth project curve of genus $g$ over the finite field with $q$ elements. Let $h$ be $\# \mathrm{Pic}^0(X)(\mathbb{F}_q)$. Weil showed that $h \geq (\sqrt{q}-1)^{2g}$. Lachaud and Martin-Descamps established the bound $h \geq \tfrac{q^g}{g+1} \tfrac{(q-1)^2}{q(q+1)} \approx \tfrac{q^g}{g}$, which is better for $q$ fixed and $g \to \infty$.

I was playing around with the estimates I was using in a recent question about finding function fields with class number $1$, and I seem to have come up with an argument that proves a bound of the form $h \geq c \tfrac{q^g}{\log g}$, for a small positive constant $c$.

I am trying to figure out if this is interesting. Of course, I'm doing a literature search, but I don't know this area, so maybe someone can tell me. What are the best current bounds in this area, and how does mine compare? Or, I suppose, does someone know a counterexample showing that $\tfrac{q^g}{\log g}$ is too good to be true?

Answer 1

$\def\FF{\mathbb{F}}$Easy results are usually not original, and this wasn't an exeption. What I was doing was almost exactly the same as On the number of rational points of Jacobians over finite fields, Lebacque and Zykin (2013). The one thing that they are missing is a simple asymptotic analysis, so I'll publish that here rather than trying to make it into a paper.

For any $N \geq 1$, Corollary 2.5 of Lebacque and Zykin implies that $$\#J(\FF_q) \geq q^g \exp\left( - \sum_{n=1}^N \frac{1}{n} - \sum_{n=1}^N \frac{q^{-n}}{n} - \frac{2g}{(\sqrt{q}-1)(N+1) q^{N/2}} \right).$$ $$= q^g \exp \left( - \log N + O(1) + O \left( \tfrac{g}{N q^{N/2}} \right) \right).$$ If we choose $N$ such that $q^{N/2} \approx 2g$, then we get a lower bound of $c \tfrac{q^g}{\log g} $.