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](http://dx.doi.org/10.4064/aa-56-4-329-340) 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?



Lower bounds for class number of function fields with fixed $q$, growing $g$
$\def\FF{\mathbb{F}}$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](https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/acta-arithmetica/all/169/4/83945/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} $.


