Isospectral definite ternary Fq[t]-lattices

Abstract

We prove that the representations numbers of a ternary definite integral quadratic form defined over Fq[t], where Fq is a finite field of odd characteristic, determine its integral equivalence class when q is large enough with respect to its successive minima. Equivalently, such a quadratic form is determined up to integral isometry by its theta series.

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arXiv:0905.3779v1 [math.NT] 22 May 2009
ISOSPECTRAL DEFINITE TERNARY Fq[t]-LATTICES
JEAN BUREAU AND JORGE MORALES
Abstract. We prove that the representations numbers of a ternary
definite integral quadratic form defined over Fq[t], where Fq is a finite
field of odd characteristic, determine its integral equivalence class when
q is large enough with respect to its successive minima. Equivalently,
such a quadratic form is determined up to integral isometry by its theta
series.
1. Introduction
It has been a long-standing question to determine whether integral definite
Z-lattices are determined up to isometry by their theta series. In 1979,
Watson [17] proved that definite binary Z-lattices are determined by their
primitive representations. The case of ternary lattices had to wait until
1997 to be solved by Schiemann [16] by means of extensive computations.
He proved that definite ternary Z-lattices are indeed determined by their
representation numbers. This is not the case for forms of rank ≥ 4, where
counterexamples have been found (cf. [15], [9] and [4]).
In this paper we prove the analogue of Schiemann’s theorem for definite
ternary Fq[t]-lattices, where Fq is a finite field of odd characteristic. We
show first that the representation numbers determine invariants such as the
successive minima and the genus (Sections 3 and 4). Our proof that the
representation numbers determine the equivalence class requires different
arguments according to different configurations of the successive minima
(Section 6). When the successive minima have alternating parity, we use a
theorem of Carlitz based on Fourier inversion and we are able to conclude
equivalence under the hypothesis that the ground field Fqis large enough
(see Theorem 6.17 for a precise statement). This condition is not required
in the two other cases (Theorems 6.5 and 6.9).
We thank the referee for his/her careful reading and useful comments.
2. Notation and terminology
The following notation will be in force throughout the paper:
Fq: The finite field of order q. We always assume q odd.
A : The polynomial ring Fq[t]
K : The field of rational functions Fq(t)
δ : A fixed non-square of F×
q.
2000 Mathematics Subject Classification. 11T06, 11E12, 11E16, 11E20.
1

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2JEAN BUREAU AND JORGE MORALES
Let L be an A-lattice of finite rank n and let Q be a quadratic form on
L. The form Q is definite if it is anisotropic over the field K∞= Fq((1/t)).
This implies in particular that n ≤ 4.
Let B(x,y) = Q(x + y) − Q(x) − Q(y) be the associated symmetric
bilinear form.Djokovi´ c [6] showed that if (L,Q) is definite, then there
exists an A-basis v1,...,vnof L such that the Gram matrix M = (mij),
where mij=1
2B(vi,vj), satisfies
(1)degmii≤ degmjjfor i ≤ j and degmij< degmiifor i < j.
Such a basis is called reduced. Gerstein [7, Theorem 2] showed that if
1,...,v′nis another reduced basis for (L,Q), then
n
?
where uij∈ Fq.
In particular, the increasing sequence of degrees of the diagonal terms of
a reduced Gram matrix
v′
v′
j=
i=1
uijvi,
(degm11,degm22,...,degmnn)
is an invariant of the equivalence class of the quadratic form. This sequence
is called the sequence of successive minima of Q and will be denoted by
(µ1(L,Q),µ2(L,Q),...,µn(L,Q)).
The representation numbers of (L,Q) are defined by
(2)R(L,Q,a) = |{v ∈ L : Q(v) = a}| (a ∈ K).
It is easy to see that if (L,Q) is definite, the above numbers are finite.
Clearly they depend only on the isometry class of (L,Q).
Definition 1. Two definite quadratic forms (L,Q) and (L′,Q′) are called
isospectral1if R(L,Q,a) = R(L′,Q′,a) for all a ∈ K.
Following Conway’s [5] terminology, we shall call an invariant of (L,Q)
audible if it is determined by the representation numbers. The main goal of
this paper is to show that the equivalence class of a ternary definite quadratic
form over A is audible. We shall do this in several steps.
3. The successive minima
Let (L,Q) be a definite quadratic form over A. For m ∈ Z, define
(3)Lm= {v ∈ L : degQ(v) ≤ m}.
1The terminology comes from the fact that for quadratic forms over Z the representa-
tion numbers are naturally the dimensions of the eigenspaces of a Laplace operator, see
[10].

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ISOSPECTRAL DEFINITE TERNARY Fq[t]-LATTICES3
It is easy to see that the Lmare finite-dimensional Fq-subspaces of L and
that they form an increasing sequence whose union is L. We encode their
successive dimensions into the formal power series
(4)
SL(u) =
?
m∈Z
dim(Lm/Lm−1)um
and
TL(u) =
?
m∈Z
dim(Lm)um.
Notice that both SL(u) and TL(u) are Laurent series in u since Lm= {0}
for m ≪ 0 (we do not assume that Q takes integral values on L). It is clear
from their definition that both series are audible.
Proposition 3.1. With the notation above we have
SL(u) =uµ1+ uµ2+ uµ3
1 − u2
and
TL(u) =uµ1+ uµ2+ uµ3
(1 − u2)(1 − u),
where (µ1,µ2,µ3) are the successive minima of (L,Q). In particular the
sequence (µ1,µ2,µ3) is audible.
Proof. Let v2,v2,v3be a reduced basis of L. Notice that since Q is definite,
µ1,µ2,µ3cannot all have the same parity.
Suppose first that µ1 ≡ µ2 (mod 2).
trivial.When m ≡ µ1 (mod 2) and µ1 ≤ m < µ2, the quotient space
Lm/Lm−1 is 1-dimensional (with basis {t(m−µ1)/2v1}).
(mod 2) and m ≥ µ2, the quotient Lm/Lm−1is 2-dimensional (with basis
{t(m−µ1)/2v1,t(m−µ2)/2v2}). When m ≡ µ3 (mod 2), the quotient Lm/Lm−1
is trivial if m < µ3and 1-dimensional if m ≥ µ3(with basis {t(m−µ3)/2v3}).
Putting this information into the series, we get
If m < µ1, then clearly Lm is
When m ≡ µ1
SL(u) =
(µ2−µ1)/2−1
?
=uµ1+ uµ2+ uµ3
1 − u2
k=0
uµ1+2k+ 2
∞
?
k=0
uµ2+2k+
∞
?
k=0
uµ3+2k
.
The case when µ1 ?≡ µ2 (mod 2) is computed similarly. We spare the
reader the details. The second identity follows from the obvious relation
SL(u) = (1 − u)TL(u).
?
4. The genus
Let p be a prime ideal of A and let ξ be a root (in an algebraic closure
of Fq) of a generator of p. The canonical character χp: Kp→ C×is the
homomorphism defined by
χp(f) = exp(2πiTr(Resξ(f))/p),
where Resξ(f) ∈ Fq(ξ) is the residue of f at ξ (i.e. the coefficient of (T−ξ)−1
in the Laurent series expansion of f at ξ) and Tr : Fq(ξ) → Fpis the trace
to the prime field Fp. Clearly the definition is independent of the choice of
the root ξ, since residues at different roots are conjugate over Fq. Notice

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4JEAN BUREAU AND JORGE MORALES
that χpis trivial on Ap; in fact Apis the largest fractional ideal of Kpon
which χpis trivial.
Let (W,Q) be a definite quadratic space over K and let L ⊂ W be an
A-lattice, not necessarily integral with respect to Q.
Define
µ(L,Q,χp) = lim
m→∞
1
|Lm|
?
x∈Lm
χp(Q(x)).
We shall see below that this is a stabilizing limit. We first notice that this
“average”, µ(L,Q,χp), is audible. Indeed, we have
µ(L,Q,χp) = lim
m→∞
1
|Lm|
?
?
x∈Lm
χp(Q(x))
= lim
m→∞
1
|Lm|
deg(a)≤m
R(L,a)χp(a).
We now express µ(L,Q,χp) in terms of local data. Let L♯be the dual of L
with respect to Q. Since L is the union of the Lm, for m large enough, the
restriction Lm→ L/(L♯∩ L) of the canonical projection is surjective. Thus
(5)
1
|Lm|
?
x∈Lm
χp(Q(x)) =|Lm∩ L♯|
|Lm|
?
?
x∈Lm/Lm∩L♯
χp(Q(x))
=
1
|L : L ∩ L♯|
x∈L/L∩L♯
χp(Q(x))
=
1
|Lp: Lp∩ L♯
p|
?
x∈Lp/Lp∩L♯
p
χp(Q(x)).
Theorem 4.1. Let π be a monic generator of p. The sequence µ(L,π−kQ,χp)
(k = 0,1,2,...) determines completely the local class (Lp,Q).
Proof. Let (Lp,Q) = (M1,Q1) ⊥ (M2,Q2) ⊥ ... ⊥ (Mr,Qr) be the Jordan
decomposition. Each Miis p-modular, i.e. M♯
ν1< ν2< ··· < νr. We define µ for local lattices using the last line of (5).
Then we have
i= π−νiMi, and we assume
µ(Mi,π−kQi,χp) =





[Mi: πk−νiMi]−1
?
x∈Mi/πk−νiMi
χp(π−kQ(x)) ifk ≥ νi;
1 ifk < νi.
We can express this further using the invariant γpdefined in [14, Ch. V, §8]
(see also [18, §24]). Then we have

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ISOSPECTRAL DEFINITE TERNARY Fq[t]-LATTICES5
(6)µ(Mi,π−kQi,χp) =
?
[Mi: πk−νiMi]−1/2γp(π−kQi)
1
if
if
k ≥ νi;
k < νi.
The invariant γp(π−kQi) is a 4th root of unity and depends only on the
class of π−kQiover the field Kp(actually, only on its Witt class) [14, Chapter
5]. In particular |γp(π−kQi)| = 1, thus
logq|µ(Mi,π−kQi,χp)| = −midegπ
2
sup{0,k − νi},
where miis the rank of Miand logqis the logarithm in base q. Using the
obvious fact that µ is compatible with orthogonal sums, we get
(7) logq|µ(L,π−kQ,χp)| = −
r
?
i=1
midegπ
2
sup{0,k − νi}.
As observed earlier, the left-hand side of (7) is audible as a function of k,
then so is the right-hand side. The functions fν: Z → R given by fν(k) =
sup{0,k−ν} are linearly independent, so the expression of logq|µ(L,π−kQ,χp)|
in (7) as linear combination of these functions is unique; it follows that the
numbers ν1,ν2,...,νrand the ranks m1,m2,...,mrof the Jordan factors of
Lpare audible.
It is left to show that det(Mi,Qi) is audible. Consider the case i = 1 and
k = ν1+ 1. Let F = π−ν1Q1(note that F is unimodular on M1). By (6),
we have
(8)
µ(L,π−ν1−1Q,χp) = µ(M1,π−1F,χp)
= q−m1deg π/2γp(π−1F).
The invariant γpsatisfies γp(< a >)γp(< b >) = γp(< ab >)(a,b)p, where
(a,b)pis the Hilbert symbol [18, §28 p. 176]. Applying this identity, we get
γp(π−1F) = γp(?π?)m1γp(F)(detF,π)p,
where ?π? is the rank-one form πX2. Since F is unimodular on M1, γp(F) =
1, so γp(π−1F) = γp(?π?)m1(detF,π)p. It follows from this and (8) that
the class of (M1,Q1) is audible. We continue similarly taking successively
k = ν2+ 1,...,νr+ 1.
?
Theorem 4.1 has two immediate consequences:
Corollary 4.2. The genus of (L,Q) is audible.
Corollary 4.3. The discriminant of (L,Q) is audible.