The Fifteen Theorem for Universal Hermitian Lattices over Imaginary Quadratic Fields

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields over $\mathbb{Q}(\sqrt{-m})$ for all m. For each imaginary quadratic field $\mathbb{Q}(\sqrt{-m})$, we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13,14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneeberger's fifteen theorem and ours is the number 13.


introduction
The research for positive definite rational quadratic forms for which the represented integer set is as large as possible has its origins at the beginning of modern number theory. In 1770, Lagrange [20] found the famous four square theorem: the positive definite quadratic form x 2 1 + x 2 2 + x 2 3 + x 2 4 represents all positive integers. Since then, his theorem has been generalized in many directions. One of the generalizations is to find all positive definite quadratic forms that represent all positive integers, which we call universal quadratic forms. The first breakthrough in this direction was made by Ramanujan [24]. In 1917, he discovered all 55 quaternary diagonal universal forms, up to isometry. In 1927, Dickson [6] confirmed Ramanujan's list except one form which was not universal and extended Ramanujan's results to non-diagonal forms. It was Dickson who called those forms universal. In 1948, Willerding [26] found 124 quaternary classical non-diagonal universal forms, up to isometry, and claimed that the list was complete. But her list was incomplete with some mistakes.
More generally, a positive definite quadratic form over a totally real number field is said to be universal if every totally positive integer of the field is represented by it. To establish a general context for the following discussion, let F be a totally real algebraic number field with ring of integers O and let O + be the subring of totally positive elements of O. By the term O-lattice (or, lattice over F ), we will always mean an O-module on the totally positive definite quadratic space (V, Q) over F . As the similar way of defining an quadratic forms (over Z), we can define an quadratic lattices over O. So, an O-lattice L is said to be universal if every element of O + can be represented by L. In 1928, Götzky [8] proved that x 2 1 + x 2 2 + x 2 3 + x 2 4 is universal over Q( √ 5). In 1941, Maass [21] proved the three square theorem, which states: the quadratic form x 2 1 +x 2 2 +x 2 3 is universal over Q( √ 5). All positive definite ternary universal forms over real quadratic fields were determined in [3]. Further developments on universal forms over totally real number fields are established by B. M. Kim (see [13], [14] and [15]).
If a Hermitian lattice represents all positive integers, we call it universal. In 1997, Earnest and Khosravani [7] found 13 universal binary Hermitian forms over imaginary quadratic fields of class number 1. If the quadratic field over Q has the class number bigger than 1, Iwabuchi [11] determined all universal binary Hermitian lattices over this field. After that, J.-H. Kim and P.-S. Park [18] added three binary Hermitian forms to the Earnest-Khosravani-Iwabuchi's list and completed the list. Further generalizations of Hermitian lattices were made by P.-S. Park [23] and A. Rokicki [25].
In [4], Bhargava's generalization of the fifteen theorem was announced: for any infinite set S of positive integers, there is a finite subset S 0 of S such that any positive definite quadratic form that represents every element of S 0 represents all elements of S. And he found S 0 for some interesting sets S. In [17], B. M. Kim, M.-H. Kim and B.-K. Oh proved the finiteness theorem as a generalization of Bhargava's result: for any infinite set S of positive definite quadratic forms of bounded rank, there is a finite subset S 0 of S such that any positive definite quadratic form that represents every element of S 0 represents all of S. Nice survey papers related to these subjects are [19] and [9].
In this paper, first, we will suggest a matrix representation for non-free Hermitian lattices. Due to this matrix representation, we can do escalate to find candidates of universal Hermitian lattices including non-free lattices over imaginary quadratic fields Q( √ −m). Second, we obtain a Conway-Schneeberger-Bhargava type criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then L is universal. Hence we call this theorem the fifteen theorem for universal Hermitian lattices. As the language of finiteness theorem for representability, S is the set of all positive integers and S 0 = {1, 2, 3, 5, 6, 7, 10, 13, 14, 15}. We will call the set S 0 a set of critical numbers. Even though the fifteen theorem and 290-theorem gives us the rough upper bound for the critical numbers that lie in between 15 to 290, it is hard to figure out the set of critical numbers for each field. For each imaginary quadratic field Q( √ −m), we will give an optimal set of critical numbers by arithmetic calculation. For example, a Hermitian lattice L over Q( √ −39) is universal if and only if L represents 1, 2, 3, 5, 6, 7, 13.

Preliminaries
The notation and terminology of O'Meara's book [22] will be adopted here. For the terminology specific to the Hermitian case, the paper [12] can be referred to. We begin by setting some additional notations that will remain in effect throughout this paper. Let F denote the imaginary quadratic field Q( √ −m) for a positive squarefree integer m with nontrivial Q-involution and let O be the ring of integers of F . It is well-known that O is generated by if m ≡ 3 (mod 4). By the term O-lattice L (or integral lattice L over F ), we will mean a finitely generated O-module on the Hermitian space (V, H) over F , where V is an n-dimensional vector space over F with the nondegenerate Hermitian form H. All lattices considered here will be assumed to be integral and positive definite in the sense that H(x, y) ∈ O for all x, y ∈ L and H(x) := H(x, x) > 0 for all x = 0. It follows from these assumptions that H(x), called a (Hermitian) norm, is in Z for all x ∈ L.
As the ring of integers of an imaginary quadratic field is not generally a principal ideal domain, lattices do not need to be free. Let {v 1 , . . . , v n } be an O-basis for L. In case that O is a principal ideal domain, every Hermitian lattice is free. Therefore L = Ov 1 +· · ·+Ov n and there is a function f : Such a function will be referred to as a Hermitian form associated to L. And we can obtain an associated Hermitian matrix for L by taking the n × n-matrix whose entry is H(v i , v j ). If a basis {v 1 . . . , v n } for L is orthogonal then the associated matrix of L is denoted by H(v 1 ), . . . , H(v n ) . Similarly, in case that O is not a principal ideal domain, there is a fractional ideal A such that L = Ov 1 + · · · + Ov n−1 + Av n by [22, 81:5]. Since any ideal in O is generated by at most two elements, we can write L = Ov 1 +· · ·+Ov n−1 +(α, β)Ov n for some α, β ∈ O. Therefore, we have a Hermitian form f : . Also we have a Hermitian (n + 1) × (n + 1)-matrix associated to L as follows:  Note that this matrix is positive semi-definite, but this represents an n-ary positive definite Hermitian lattice. Considering a 2n-dimensional vector space V over Q corresponding to V as defined in [12], we can regard (V, H) over F as a 2n-dimensional quadratic space ( V , B H ) such that B H (x, y) = 1 2 [H(x, y) + H(y, x)] = 1 2 Tr F/Q (H(x, y)). Analogously, by viewing L as a Z-lattice on ( V , B H ) we can obtain a quadratic Z-lattice L on V associated to a Hermitian O-lattice L on V and also f (x 1 , y 1 , . . . , x n , y n ) = f (x 1 + ωy 1 , . . . , x n + ωy n ) is an associated quadratic form in 2n-variables corresponding to this lattice L. For the convenience, we say that f is an associated quadratic form of L. For example, the associated quadratic form of the Hermitian lattice 1 over Q( √ −m) is To distinguish the matrix of a quadratic Z-lattice L from the matrix of a Hermitian lattice L, we will add subscript Z to the matrix of the quadratic Z-lattice L. And a Hermitian O-lattice L represents a Z-lattice M if and only if the associated quadratic form of L represents M . If L is a universal lattice of rank n, then there are infinitely many universal lattices of rank k (k > n) which contain L. Thus in order to obtain any meaningful finiteness result for such lattices, we should consider a new universal lattice. A universal lattice is called new when it does not contain any universal lattice of smaller rank.
For m ≡ 1, 2 (mod 4), the associated quadratic lattice of a Hermitian lattice over Q( √ −m) is a classical Z-lattice. Thus we can determine the universality of a Hermitian lattice over Q( √ −m) via applying the fifteen theorem. On the contrary, for m ≡ 3 (mod 4), the associated quadratic lattice of a Hermitian lattice over Q( √ −m) is a nonclassical Z-lattice. Therefore, the recent big result of Bhargava and Hanke, the 290-conjecture, can be applied to prove the universality.
We adopt some notations from Conway-Sloane [4]. The notation p d (resp. p e ) denotes an odd (resp. even) power of p; if p = 2, u k denotes a unit of the form 8n + k (k = 1, 3, 5, 7) and if p is odd, u + (resp. u − ) denotes a unit which is a quadratic residue (resp. non-residue) modulo p.

main results
Note that xx+yy +zz +uu represents all positive integers even when all variables take values in Z. Thus it is obviously a quaternary universal Hermitian lattice over all imaginary quadratic fields. Since classical universal quadratic forms over Z was already classified, we do not need to investigate this kind of lattices. A Hermitian lattice is called inherited if its coefficients are all rational integers. If a Hermitian lattice is not inherited, we call it uninherited. We will mainly consider uninherited universal Hermitian lattices.
If a lattice L is not universal, define the truant of L to be the smallest positive integer not represented by L. An escalation of a nontrivial lattice L is defined to be any lattice which is generated by L and a vector whose norm is equal to the truant of L. Conway, Schneeberger and Bhargava suggested this escalation method for free lattices (see [5], [1]). In this article, we suggested the method for a matrix representation for non-free Hermitian lattices (see section 2) and we use the escalation method to find candidates of universal Hermitian lattices including non-free Hermitian lattices.
If ℓ does not represent 15, then we can obtain a universal pro forma quinary Hermitian lattice by attaching a vector of norm 15 to ℓ. If ℓ is inherited and it is not universal, then ℓ is one of the following 4 lattices whose truants are all 15: 1 ⊥ In this case, we can obtain a universal lattice by attaching a vector of norm 15. Note that these quinary universal lattices are inherited if m ≥ 129. Case III-4 b(2). m ≡ 3 (mod 4): From the positive semi-definiteness, the escalation lattices ℓ's are all inherited for m ≥ 203. We may assume that m = 35, 43, 47, 51, . . . , 199. Note that 1, 2, 5 Z represents all positive integers except the form 5 d u − . If n = 5 d u − ≥ 3m, then at least one of n − m, n − 2m and n − 3m is not of the form 5 d u − . Thus n → 1, 2, 5 . If ℓ represents all positive integers n < 3m, then ℓ is universal. If ℓ is not universal and uninherited, then ℓ is one of the followings and their conjugates: We can check that ℓ represents all positive integers except only 15. Hence we can obtain a universal lattice by attaching a vector of norm 15 to ℓ. If ℓ is inherited and ℓ is not universal, then we also obtain a universal lattice by the same process.
Case III-5 a. 7 → 1 ⊥ 2 1 1 4 : Then we only have m = 6. Since 1 ⊥ 2 1 1 4 represents a universal quadratic lattice 1, 6 Z ⊥ 2 1 1 4 Z , 1 ⊥ 2 1 1 4 is universal. From now through the Case III-5, we may assume that m = 6. There are 30 quadratic lattices of this type up to isometry. These quadratic lattices are universal except the following lattice Note that the truant of ℓ ′ is 10 and ℓ ′ represents all numbers 1 to 15 except 10. Assume that ℓ represents ℓ ′ . If m = 10, then the Hermitian lattice ℓ represents 10, hence ℓ is a quaternary universal Hermitian lattice. From the positive semidefiniteness, if m ≥ 17, then ℓ is inherited. If m = 13, 14 and ℓ is uninherited, then Since ℓ represents all positive integers 1 through 15 except only the truant 10 of ℓ, the next escalation lattice of ℓ is a pro forma quinary universal Hermitian lattice which can be obtained by attaching a vector of norm 10. In this case, the universal Hermitian lattice is inherited if m ≥ 53. If ℓ is inherited and it is not universal, then ℓ is the following lattice whose truant is 10: Hence we can obtain a universal lattice via attaching a vector of norm 10. Case III-6. 1 ⊥ 2 1 1 5 → L: Note that 1 ⊥ 2 1 1 5 represents a quadratic lattice 1 Z ⊥ 2 1 1 5 Z whose truant is 7.
From now through the Case III-6, we may assume that m = 6.
Case III-6 b. 7 → 1 ⊥ 2 1 1 5 : The next escalation lattice ℓ of 1 ⊥ 2 1 1 5 is of the form for some α, β, γ. Case III-6 b(1). m ≡ 3 (mod 4): From the positive semi-definiteness, ℓ is inherited if m ≥ 37. And ℓ represents a quadratic lattice  There are 16 quadratic lattices of this type up to isometry. These quadratic lattices are universal except the following lattice Note that the truant of ℓ ′ is 15 and ℓ ′ represents all numbers 1 to 14. Assume ℓ represents ℓ ′ . If ℓ is uninherited, then m = 10, 13, 14, 17, 21 and Note that if m = 10, 13, 14, 21, then ℓ represents 15. Hence ℓ is a quaternary universal lattice. If m = 17, then it is isometric to the lattice in the Case III-4 b(1). If ℓ is inherited, and it is not universal, then ℓ is the following lattice whose truant is 15: Case III-7. Now we investigate the lattices It is known that all the binary Hermitian lattices in Table 1 are universal (see [11]) except for the last two over Q( √ −39) and they are listed in the end of this paper (see Table 3). If m ≡ 3 (mod 4), i.e., m = 6, 10, then the universalities of ternary lattices in Table 1 are checked by the fifteen theorem. Now assume that m ≡ 3 (mod 4), i.e., m = 15, 23, 31, 35, 39. Since it is enough to check the universality of ℓ ∼ = 1 ⊥ 2 ω ω b . Each associated quadratic lattice of ℓ has a sublattice ℓ ′ of class number 1 (see Table 2). The universality of each lattice ℓ ′ is proved by the method similar to the previous case.  The last exceptional lattice is The associated quadratic form of this lattice is N : x 2 1 + x 1 x 2 + 10x 2 2 + 2y 2 1 + y 1 z 1 + 5z 2 1 . Since 2N → N , we only need to show that N represents all odd positive integers. If we set x 2 = 2t for some t ∈ Z, then N represents For sufficiently large n, we will show that n − 39s 2 → N ′ = 1 Z ⊥ 2 1/2 1/2 5 Z for suitable s. It implies the desired result: n → ℓ over Q( √ −39). Note that N ′ represents two quadratic sublattices which are in the same genus by Brandt-Intrau table [2]: Hence if we show that n − 39s 2 is represented by the genus, then we can say n → ℓ.

Remark 2.
We computed all quaternary escalation lattices and corresponding truants (See Table 4). We can obtain pro forma quinary universal Hermitian lattices by attaching a vector whose norm is its truant, to each quaternary escalation lattice. But these are too numerous to list all.
From the above results we obtain the following theorems.  109,110,113,114,118,122,123,129,130,133,134,137,138,141,142,145,146,149,154,157,158,161,163,165,166,170,173,174,177,178,181,182,185,186,190,193,194,195,197,201,202,203,205,206,209