Numbers of points in lattice

Question

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.

Answer 1

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.

I am not sure what you mean by at least two significant terms.