The column number for 3-modular matrices

An integer-valued matrix $\mathbf{A}$ is $\Delta$-modular if each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix has determinant at most $\Delta$ in absolute value. The column number problem is to determine the maximum number of pairwise non-parallel columns of a rank-$r$, $\Delta$-modular matrix. Exact values for the column number are only known for $r \le 2$ or $\Delta \le 2$. We prove that if $r$ is sufficiently large, then the maximum number of pairwise non-parallel columns of a rank-$r$, $3$-modular matrix is $\binom{r+1}{2} + 2(r-1)$. This settles a conjecture by Lee, Paat, Stallknecht, and Xu on the column number in the case $\Delta = 3$. We complement this main result by showing that there are at least three $3$-modular matrices with pairwise non-isomorphic vector matroids that attain this upper bound. More generally, we show that if $r > \Delta$, then the number of $\Delta$-modular matrices with $\binom{r+1}{2} + (\Delta-1)(r-1)$ pairwise non-parallel columns and pairwise non-isomorphic vector matroids is at least exponential in $\sqrt{\Delta}$; previously only one matrix was known due to Lee et al.