# Numbers of points in lattice

Let $$\omega_1,\cdots,\omega_n$$ be $$n$$ elements of $$\overline{\mathbb F_q(T)}$$ that are $$\mathbb F_q(T)$$ linearly independant. Denote by $$\Lambda$$ the lattice $$\Lambda=\mathbb F_q[T]\omega_1+\cdots+\mathbb F_q[T]\omega_n$$. Can one estimate (with at least two significant terms) the number of elements of $$\Lambda$$ whose degree ($$\overline{\mathbb F_q(T)}$$ is considered embeded in the completion of $$\overline{\mathbb F_q\left(\left(\frac1T\right)\right)}$$) is less than $$r$$ ($$r$$ is a positive integer)?

Surely Riemann-Roch theorem (or Riemann hypothesis for curves) could help, but I do not see how.

There are two possibilities. If $$\omega_1,\dots, \omega_n$$ are not $$\mathbb F_q((\frac{1}{T}))$$-linearly independent, then the count is infinite for all $$r$$ sufficiently large. Indeed, if $$\sum_{i=1}^n a_i \omega_i$$ is a relation, then for $$f$$ in $$\mathbb F_q[T]$$ any polynomial, if we take the power series $$fa_1,\dots, f a_n$$ and remove all terms with negative powers of $$T$$ to obtain polynomials $$g_1,\dots, g_n$$ then $$\sum_{i=1}^n g_i \omega_i$$ will equal $$\sum_{i=1}^n (g_i -f a_i) \omega_i$$ and thus will have degree less than the maximum degree of the $$\omega_i$$. This gives infinitely many elements of $$\Lambda$$ with bounded degree.
Otherwise, $$\omega_1,\dots, \omega_n$$ generate a $$\mathbb F_q((\frac{1}{t}))$$-submodule of $$\overline{\mathbb F_q((\frac{1}{t}))}$$ of rank $$n$$. Consider the intersection of this submodule with the ring of integers, which is a $$\mathbb F_q[[\frac{1}{T}]]$$-module, necessarily free of rank $$n$$, and choose a generating set $$\alpha_1,\dots, \alpha_n$$. We can write $$\alpha_i =\sum_{j=1}^n c_{ij} \omega_j$$ for a unique $$c_{ij} \in \mathbb F_q((\frac{1}{T}))$$. Let $$C$$ be the $$n\times n$$ matrix with entries $$c_{ij}$$.
It is easy to see from Riemann-Roch that the number of elements of $$\Lambda$$ with degree $$\leq r$$ is equal to $$q^{ (r+1) n + \deg \det C}$$ for all $$r$$ sufficiently large.
Indeed, this is equal to the number of vectors $$a_1,\dots, a_n \in \mathbb F_q[T]^n$$ such that $$C^{-1} \begin{pmatrix} a_1 \\ \dots \\ a_n \end{pmatrix}$$ is a tuple of vectors of degree $$\leq r$$. We construct a vector bundle $$V$$ on $$\mathbb P^1$$ by gluing $$\mathcal O_{\operatorname{Spec} \mathbb F_q[T]}^n$$ to $$\mathcal O_{\operatorname{Spec} \mathbb F_q[[ \frac{1}{T} ]] }^n$$ using the gluing data $$C$$ on their intersection $$\operatorname{Spec} \mathbb F_q(( \frac{1}{T} ))$$. The desired count is equal to the number of sections of $$H^0(\mathbb P^1, V(r) )$$, which by Riemann-Roch for vector bundles is $$q^{n + \deg V(r)} = q^{n + nr + \deg V }= q^{n + nr + \deg \det V} =q^{ n + nr + \deg \det C}$$ as soon as $$H^1(\mathbb P^1, V(r))=0$$, which happens for all $$r$$ sufficiently large.