Effective structure theorems for symplectic spaces via height

TL;DR

This paper establishes the existence of bounded-height symplectic bases and decompositions for symplectic spaces over global fields, extending classical results with explicit bounds and applicability to various fields.

Contribution

It provides a symplectic analogue of Siegel's lemma with explicit height bounds, applicable over any field with a product formula, including algebraically closed fields.

Findings

Existence of symplectic bases of bounded height.

Small-height decompositions into hyperbolic planes.

Generation of flags of totally isotropic subspaces.

Abstract

Given a $2k$-dimensional symplectic space $(Z,F)$ in $N$ variables, $1 < 2k \leq N$, over a global field $K$, we prove the existence of a symplectic basis for $(Z,F)$ of bounded height. This can be viewed as a version of Siegel's lemma for a symplectic space. As corollaries of our main result, we prove the existence of a small-height decomposition of $(Z,F)$ into hyperbolic planes, as well as the existence of two generating flags of totally isotropic subspaces. These present analogues of known results for quadratic spaces. A distinctive feature of our argument is that it works simultaneously for essentially any field with a product formula, algebraically closed or not. In fact, we prove an even more general version of these statements, where canonical height is replaced with twisted height. All bounds on height are explicit.