Let L be a full-rank lattice in the Euclidean spaceRn. Suppose L is aZ-lattice in the sense that ∥v∥2 ∈ Z, for all vector v ∈ L. Let φ : Zn −→ L be an isomorphism of abelian groups. Then Q(x) :=∥φ(x)∥2 is a positive-definite integral quadratic form in n variables. The form Q(x) depends on L as well as the choice of φ, while its integral
3 ESCALATION
equivalence class depends only on L. Conversely, if Q(x1, ...xn) is a positive-definite integral quadratic form with Gram matrix A, then the assignment (u, v) 7→ uTAv defines an inner product on Rn, and hence by the Gram Schmidt process, there is an orthogonal automorphism φ ofRnsuch that Q(x) =∥φ(x)∥2, for all x∈ Zn. Then we see that Q(x) is exactly the quadratic form induced by φ and the lattice L := φ(Zn).
Thus, we have a bijection between equivalence classes andZ-lattices with integral square norms. In this thesis, for convenience we will use the identification between a quadratic form Q(x) and the lattice L = L(Q) via certain implicitly chosen φ.
Therefore, when we say that a lattice has certain property, it means the correspond-ing quadratic form has such property. For example, the lattice Z4 is universal, by Largrange’s theorem.
Under the above correspondence, if e1, ..., en is the standard basis of Rn so that fi := φ(fi) is the corresponding Z-basis of L, then the entries of the Gram matrix A = (aij) of Q(x) can be expressed as
aij = (fi, fj). (2.4.1)
Suppose L⊂ Rn is a full Z-lattice and let f1, ..., fn be aZ-basis of L. By (2.4.1), the lattice 2L is contained in the dual lattice L∗ of L. Here
L∗ :={x ∈ Rn | (x, y) ∈ Z, for all y ∈ L} (2.4.2) which is spanned by the dual basis f1∗, ..., fn∗ satisfying
(fi∗, fj) = δi,j, (2.4.3) where δi,j denotes the Kronecker symbol.
3 Escalation
The proofs of the main theorems are built on the concept of escalation. We also refer to the rank of a lattice as its dimension.
3 ESCALATION
Definition 3.1. If a positive-definite integral quadratic form Q(x) is not universal, we call the smallest positive integer that Q(x) cannot represent the truant of Q(x).
We define an escalation of a non-universal L to be any integral lattice generated by L and a vector whose square-norm is the truant of L. An escalator lattice is a lattice which is obtained by consecutive escalations of the zero-dimensional lattice.
Note that if a non-universal L is a full lattice in Rn, then its escalation is a full lattice in either Rn or Rn+1.
Lemma 3.2. There are only finitely many escalator lattice of a given dimension n.
Proof. We prove by induction on n. The unique escalation of the one-dimensional lattice is the lattice Z ⊂ R corresponding to the quadratic form x2. Hence the case n = 1 is proved. Suppose L =Zf + L1 is an n-dimensional escalation of an escalator lattice L1 such that∥f∥2 equals the truant a of L1.
We first consider the case where L1 is of dimension n− 1 and let f1, ..., fn−1 be a Z-basis. Write f = αen+ f′, where en∈ Rn is the standard unit vector perpendicular to Rn−1 and f′ ∈ Rn−1. By (2.4.1), the value
(f′, fi) = (f, fi)
must be either an integer or a half integer. Hence by (2.4.3), the vector f′ ∈ 12L∗1. The discrete subgroup 12L∗1 ⊂ Rn−1 contains only finitely many vectors f′ with ∥f′∥2 < a.
For each such f′, the number α must equal±√
a− ∥f′∥2. Hence a given L1 can only have finitely many escalations. This together with the induction hypothesis implies that there are finitely many n-dimensional escalation of escalator lattice of dimension n− 1. Finally, we note that if L1 is n-dimensional with f1, ..., fn as a Z-basis and L is an n-dimensional integral lattice containing L1, then by (2.4.1), for every f ∈ L, the values (f, fi), i = 1, ..., n are in 12Z, and hence L ∈ 12L∗1, by (2.4.3). Therefore, L/L1 ⊂ 12L∗1/L1 which is finite, and hence there are only finitely many such L. In particular, there can only be finitely many n-dimensional escalator lattices containing L1.
4 THE ZP-THEORY
To determine escalator lattices of low dimensions, we follow the proof of Lemma 3.2. The only 1-dimensional escalator lattice is L1 =Z ⊂ R with with truant 2 and L∗1 = L1. Denote f1 = 1∈ L1. Since no vector in 12L1 represents 2, an escalation L of
The corresponding lattices are Z√
2e2 +Zf1, Z(√27e2 + 12f1) + Zf1, and Ze2 +Zf1
whose truants are respectively 5, 3, and 3. It turns out that all these three lattices have no 2-dimensional escalations. Escalating them, we obtain 34 three-dimensional nonisometric escalator lattices which have no three-dimensional escalations, and there are actually 6560 four-dimensional nonisometric escalations of these 34 lattices, [6].
We call these the basic escalators.
Lemma 3.3. Each universal positive-definite integral quadratic form must contain a universal escalator lattice. Conversely, the truant of any non-universal form is the same as the truant of some non-universal escalator lattice within it.
Proof. Let Q be a positive-definite integral quadratic form and denote L = L(Q).
There exists a maximal sequence of escalator lattices {0} ⊂ L1 ⊂ L2 ⊂ . . . ⊂ Lk
within L. Since Lk is maximal, it is either universal or having the same truant as that of L. Because Lk ⊂ L, if Lk is universal, so is L.
4 The Z
p-theory
To see if an integral quadratic form represents an integer m, we can first check if it represents m locally. In this section, we study the local theory of quadratic forms.
Let p be a finite prime number. A quadratic form Q(x) is called Zp-integral if and
4.1 The normalized form 4 THE ZP-THEORY
only if Q(x)∈ Zp[x]. Two Zp-integral quadratic forms Q and Q′ are Zp-equivalent if and only if their Gram matrices A and A′ satisfy
A′ = BTAB, for some B ∈ GL(n, Zp).
4.1 The normalized form
Definition 4.1. A quadratic form Q(x) is Zp-elementary if and only if either Q(x) is one-dimensional equal to ux2, u ∈ Z∗p, or p = 2 and Q(x) is 2-dimensional with Gram matrix
a11 a12 a12 a22
, such that a11, a22 ∈ Z2 and 2a12, 2a21 ∈ Z∗2.
Definition 4.2. An n-dimensional Zp-integral quadratic form Q(x) is a normalized form if and only if there is a partition P of {1, ..., n} such that
Q(x) = ∑
J∈P
pνJQJ(xJ), (4.1.1)
where each QJ is Zp-elementary and each νJ ≥ 0.
Lemma 4.3. Every quadratic form Q(x) is Zp-equivalent to a normalized form.
Proof. Let A = (aij) denote the Gram matrix of Q(x). We prove by induction on n, the dimension of Q(x). If dimQ = 1, the lemma obviously holds. We first consider the case in which p ̸= 2. Suppose min{ordp(aij)| 1 ≤ i, j ≤ n) occurs at some i = j.
We may assume it occurs at i = j = 1. By completing the squares, we can cancel a12, . . . , a1n as well as a21, . . . , an1. Then
Q(x)≃ pordpa11· x21⊕ Q′ (4.1.2) with dim Q′ = n− 1. Then the proof is completed by the induction hypothesis.
Suppose min{ordp(aij | 1 ≤ i, j ≤ n) does not occur at diagonal entries. We may assume it equals to ordpa12. Then we add the second column and the second row into the first ones. By this process, a11 is replaced by a11 + 2a12+ a22. Since ordp(a11+ 2a12+ a22) = ordp(a12) = min{ordp(aij)| 1 ≤ i, j ≤ n), the proof is reduced the previous case.
4.1 The normalized form 4 THE ZP-THEORY
Now we consider the p = 2 case. Similarly, if min{ordp(aij)| 1 ≤ i, j ≤ n) occurs at some diagonal entry, the lemma is proved by completing the squares. Otherwise, we may assume
min{ordp(aij)| 1 ≤ i, j ≤ n) = ordp(a12) = k.
Write the leading 2× 2 submatrix as
pk−1
Since its determinant is a unit in Z2, the matrix E is invertible. Write
A =
Lemma 4.4. Let Q(x) be an integral quadratic form. Then the following statements are equivalent:
(a) p| NQ, (b) p| D2Q,