On a Waring's problem for Hermitian lattices
Introduction
A positive integer a is said to be represented by a quadratic form f, written as , if there are some integers such that . A typical question is to determine those positive integers that can be represented by the quadratic form for a positive integer m. In light of the celebrated work of Fermat, Euler, Legendre and Lagrange on sums of squares, it is well-known that every positive integer can be written as a sum of at most four squares, and the number four is optimal.
This result has been extended in several different directions. For example, Waring's Problem seeks the smallest positive integer such that every positive integer is a sum of r powers of order k. There is also a higher dimensional generalization in terms of representations of positive definite integral quadratic forms, where a quadratic form is said to be represented by another quadratic form with if one finds an integral matrix A such that ; f is said to be equivalent to if A is a unimodular matrix.
In general, a quadratic form can be represented by the quadratic form if and only if it is a sum of m squares of integral linear forms. The renowned Lagrange's Four-Square Theorem implies that all unary positive definite integral quadratic forms can be represented by . In the 1930s, Mordell [16] and Ko [14] successively extended this result to quadratic forms in more variables, and they showed that all positive definite integral quadratic forms in n variables can be represented by if . All the preceding results can be derived directly from the observations that every positive definite integral quadratic form in n variables is represented by locally everywhere, and that there is exactly one equivalence class in the genus of when . Here, the genus of f, written as , is the set of all quadratic forms that are equivalent to f over the ring of p-adic integers at each prime number p and over the field of real numbers.
This approach does not work when , since in this case there are more than one equivalence classes in the genus of ; even worse, Mordell [17] found that the quadratic form corresponding to the root system cannot be represented by any sum of squares. This subtlety leads to the consideration of the set consisting of all quadratic forms in n variables that can be represented by some sums of squares.
Define the g-invariant to be the smallest positive integer g such that represents every quadratic form in . Icaza [8] first found the existence of and also derived an explicit upper bound for it by computing the constant c in the Hisa-Kitaoka-Kneser Theorem [7, Theorem 1] with respect to . Kim and Oh [11] later improved the upper bound for to an exponential of n through the theory of neighbors of quadratic forms. Recently, in [1], with my coauthors, we further improved the upper bound for to an exponential of and also observed that our method can be used to study a similar problem of Hermitian forms over an imaginary quadratic field with class number 1. Unfortunately, we cannot apply our method to general imaginary quadratic fields due to some emerged complications if the class numbers of those fields are bigger than 1; for instance, when the ring of integers is not a principle ideal domain, then a positive definite integral Hermitian form may not attain its minimum at a primitive vector.
In this paper, we shall consider the Hermitian analog of this representation problem in general situations. Let be an imaginary quadratic field with ℓ a square-free positive integer, and let be its ring of integers. Because is not necessarily a principle ideal domain in general, a Hermitian form corresponds to a Hermitian lattice but not vice visa; so, instead of considering only Hermitian forms, the more general notion of Hermitian lattices will be utilized. For each integer , denote by the free Hermitian lattice of rank m having an orthonormal basis. From the standard correspondence between Hermitian forms and free Hermitian lattices, corresponds to the Hermitian form . For each integer , let be the set consisting of all Hermitian lattices of rank n that can be represented by some , and write The finiteness of was observed in [1, Appendix A].
In this paper, we derive an explicit upper bound for that only depends on the imaginary quadratic field E and the rank n, and our main result reads as follows.
Theorem 1.1 Let be an imaginary quadratic field for a square-free positive integer ℓ, and let be an integer. Then, for defined in (1.1), we have Here, , is the smallest inert prime number in the extension , , χ is the Dirichlet character associated with E, is the third Bernoulli number twisted by χ, and is an absolute constant.
In Section 2, we introduce the geometric language of Hermitian spaces and Hermitian lattices. In Section 3, we consider the local density of a positive integer represented by a quadratic lattice (which is associated with a Hermitian lattice K in the genus of ) at each prime number p. In Section 4, we find a positive integer such that an integer t is represented by every Hermitian lattice when . In Section 5, we briefly review the theory of neighbors of Hermitian lattices over E following Schiemann [20]. In Section 6, we prove our main Theorem 1.1. Notice that the main scheme of Kim and Oh [11] is utilized in proving Theorem 1.1, but there are a number of difficulties that need to be overcome in the current setting based on [1], [15].
Section snippets
Preliminaries
Throughout this paper, we adopt the notations and terminologies for lattices from the classic monograph O'Meara [18]. For background information and terminologies specific to the Hermitian case, the reader may consult the work of Shimura [21], Gerstein [5] and Johnson [10].
Let be an imaginary quadratic field for a square-free positive integer ℓ, and let be the ring of integers of E. Then, one has with if and if . For each , denote
Local density
For a Hermitian lattice L on V, denote by the dual lattice of as a quadratic lattice over , and by the associated quadratic lattice of the dual lattice of L. In general, and are different. The next result describes the relation between these two quadratic lattices.
Lemma 3.1 Assume L is a Hermitian lattice, and write to be the inverse of the different of E over . Then, one has
Proof We only consider the case where
The smallest universal number
For a Hermitian lattice , let be the smallest positive integer l such that for all integers . Define to be the smallest universal number. Let be the smallest positive integer l such that for all (even, when is even) integers . Define accordingly. Then, we shall have if and if .
Given a positive definite integral quadratic lattice J of rank n with the
Neighborhood of Hermitian lattices
In this section, for completeness, we briefly review the theory of neighbors of Hermitian lattices over following Schiemann [20], which is considered as the Hermitian analogue of that of quadratic lattices developed by Kneser [13], and which is needed in Section 6. This theory enables us to identify a condition under which a Hermitian lattice L of rank n is represented by . Other helpful resources include Gerstein [5] and Johnson [10].
In the subsequent discussions, let L be an integral
Proof of Theorem 1.1
Let p be a prime number, and let be a prime ideal in lying above p. By abuse of notations, we regard in this section individual lattices in as representatives of their isometry classes, and simply say that all Hermitian lattices in the isometry class of N are -neighbors of M if N is a -neighbor of M. Given two Hermitian lattices , define the distance between M and N as the minimal number of edges between M and N.
If p remains inert in E, then is the unique
Declaration of Competing Interest
There exists no conflict of interest.
Acknowledgement
The author warmly thanks the anonymous referees and Professor David Harari for several valuable suggestions, sincerely appreciates her advisor Professor Wai Kiu Chan for the introduction of this topic, and acknowledges the partial financial support of the 2019–2020 Faculty Research Fellowships from College of Arts & Sciences at Texas A&M University-San Antonio.
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