Representations of Integral Hermitian Forms by Sums of Norms

In 1770, Lagrange proved the famous four square theorem, which says that each positive integer a can be written as a sum of four squares. This theorem has been generalized in many directions since then. One interesting generalization is to consider the representation of positive definite integral quadratic forms in more variables by sums of squares. Let gZ(n) be the smallest number of squares whose sum represents all positive definite integral quadratic forms in n variables over Z that are represented by some sums of squares. In 1996, Icaza first proved the existence of gZ(n) and she also gave an explicit upper bound for it. An improved upper bound was obtained later by Kim and Oh in 2005. In this thesis, we consider the Hermitian analog of the above representation problem. Let E be an imaginary quadratic field and O be its ring of integers. For any positive integer m, let Im be the free Hermitian lattice over O with an orthonormal basis. Via the standard correspondence between free Hermitian lattices and Hermitian forms, Im corresponds to the integral Hermitian form x1x1 + · · · + xmxm over O. For any positive integer n, let SE(n) be the set consisting of all positive definite integral Hermitian lattices of rank n over O that are represented by some Im. We define gE(n) to be the smallest positive integer g such that every Hermitian lattice in SE(n) is represented by Ig. Our main result is an explicit upper bound for gE(n) for any imaginary quadratic field E and positive integer n.