Abstract. An integer matrix [Formula: see text] is [Formula: see text]-modular if the determinant of each [Formula: see text] submatrix of [Formula: see text] has absolute value at most [Formula: see text]. The study of [Formula: see text]-modular matrices appears in the theory of integer programming, where an open conjecture is whether integer programs defined by [Formula: see text]-modular constraint matrices can be solved in polynomial time if [Formula: see text] is considered constant. The conjecture is known to hold true only when [Formula: see text]. In light of this conjecture, a natural question is to understand structural properties of [Formula: see text]-modular matrices. We consider the column number question, how many nonzero, pairwise nonparallel columns can a rank-[Formula: see text] [Formula: see text]-modular matrix have? We prove that for each positive integer [Formula: see text] and sufficiently large integer [Formula: see text], every rank-[Formula: see text] [Formula: see text]-modular matrix has at most [Formula: see text] nonzero, pairwise nonparallel columns, which is tight up to the term [Formula: see text]. This is the first upper bound of the form [Formula: see text] with [Formula: see text] a polynomial function. Underlying our results is a partial list of matrices that cannot exist in a [Formula: see text]-modular matrix. We believe this partial list may be of independent interest in future studies of [Formula: see text]-modular matrices.

We explore the lifting question in the context of cut-generating functions. Most of the prior literature on this question focuses on cut-generating functions that have the unique lifting property. We develop a general theory for understanding the lifting question for cut-generating functions that do not necessarily have the unique lifting property.