Let $V$ be an algebraic affine set, defined over $\mathbb{F}_q$. Lang-Weil tells me that if $V$ has $C$ top-dimensional irreducible components of dimension $D$ then $$(*) V(\mathbb{F}_{q^n}) = C \cdot q^{n \cdot D} + O_{V}(q^{n \cdot D-\frac{n}{2}})$$
for any $n$. I can also go in the other direction: If $(*)$ holds for all $n$ for some $C$ and $D$, then $V$ has $C$ top-dimensional components of dimension $D$. In particular, if $V(\mathbb{F}_{q^n}) = q^{n \cdot D} + O_{V}(q^{n \cdot D-\frac{n}{2}})$ then $V$ is irreducible of dimension $D$ (as a variety over $\overline{F}_q$).
My question is, what can I learn about the geometry of $V$ from more refined information on $V(\mathbb{F}_{q^n})$? I will be more concrete in the following two questions:
- Suppose that $V(\mathbb{F}_{q^n}) = q^{n \cdot D} + O(q^{n \cdot m})$ for some $m<D$. What can I deduce about the geometry of $V$, except for dimension and irreducibility?
- Suppose that $V(\mathbb{F}_{q^n}) = q^{n \cdot D} + A \cdot q^{n \cdot m} + O(q^{nm-\frac{n}{2}})$ for some $m<D$ and integer $A$. What can I deduce about the geometry of $V$, except for dimension and irreducibility?