Representations of integers by positive definite quadratic forms
Winfried Kohnen
1. Introduction
An old and classical problem in number theory is to find closed and exact formulas for the number of representations of a positive integer n by a given positive definite integral quadratic form Q. As is well-known, of course, a powerful tool in the investigation of this problem is the theory of modular forms. Indeed, if say the rank of Q is even and at least 4, then the associated generating series, usually called a theta series, is a modular form of integral weight at least 2 and some level, hence can be expressed as the sum of a modular form in the space generated by Eisenstein series and a corresponding cusp form. The Fourier coefficients of Eisenstein series of integral weight ≥ 2 on congruence subgroups are given explicitly in terms of modified elementary divisor sums. Thus if the space of cups forms happens to be zero (which is often the case for very small levels), this leads to precise and finite formulas for the number of representations of n by Q as finite linear combinations of sums of the above type. In general, however, one gets only asymptotic results when n is large, since the coefficients of cusp forms are mysterious and in general no simple arithmetic expressions are known for them.
A very prominent special case of the above problem is to find closed formulas for the number of representations rs(n) of n as a sum of an even number s of integral squares.
For smalls, there are classical explicit finite formulas due to a variety of people, including Gauss, Lagrange, Jacobi, Hardy, Mordell, Rankin and many others, as is well-known.
In the first part of this small paper, we would like to report shortly on recent joint work with ¨O. Imamoglu [1] in which we expressrs(n) for eachsdivisible by 8 as a finite rational linear combination of products of two modified elementary divisor functions. Although the coefficients of these linear combinations are not uniquely determined, for each given sthey all can be obtained by solving a certain system of linear equations. No pre-knowledge of any special values of rs(n), e.g. for small n, is needed for this. The proof uses both the Rankin-Selberg method and the Eichler-Shimura theory of periods of cusp forms.
It is much easier to prove a similar result for the theta series of an even integral unimodular lattice (so the theta series then is on the full modular group) that is extremal.
However, as is well-known there are only finitely many classes of extremal lattices of the above type and so the above method seems to be confined to work only in a finite number of cases on the full modular group. However, in the second part of this paper we will shortly report on recent joint work with R. Salvati-Manni in which we generalize the result of [1] to the theta series of certain infinite families of even integral unimodular lattices that are not necessarily extremal.
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2. Sums of integral squares
We will denote by Mk (resp. Sk) the space of modular forms (resp. cusp forms) of even integral weight k on the full modular group. We write Mk(2) (resp. Sk(2)) for the space of modular forms (resp. cusp forms) on the usual Hecke congruence subgroup Γ0(2).
All the results given below (and more details) and also complete proofs can be found in [1].
We denote by
Ek(z) = 1− 2k Bk
X
n≥1
σk−1(n)qn (z ∈ H, q =e2πiz)
the normalized Eisenstein series inMk. We let
Eki∞(z) := 1
2k−1(2kEk(2z)−Ek(z)) (z ∈ H) and
Ek0(z) :=Eki∞|kW2(z) = 2k/2
2k−1(Ek(z)−Ek(2z)) (z ∈ H)
be the normalized Eisenstein series of weightk on Γ0(2) for the cusp infinity and the cusp zero, respectively.
As is well-known, dimCMk(2) = [k4] + 1 and dimCSk(2) = [k4]−1.
The functions Ek−2`0 E2`i∞(`= 2,3, . . . ,k2 −2) clearly are in Sk(2).
Theorem 1. The products Ek−2`0 E2`i∞(`= 2,3, . . . ,k2 −2) generate Sk(2).
Now suppose thatk ≥8 and denote byk(n;`) (n∈N; `= 2,3, . . . ,k2−2) the Fourier coefficients of Ek−2`0 E2`i∞. Define a rational ([k4]−1, k2 −3)-matrix by
Ak := k(n;`)
n=1,...,[k4]−1;`=2,...,k2−2.
Proposition 1. The matrix Ak has maximal rank.
We denote bye0k(n) (n∈N) the Fourier coefficients of Ek0. Let us write
θ(z) := X
n∈Z
qn2 (z ∈ H) 2
for the basic theta function of weight 12. Thus θs(z) =X
n≥0
rs(n)qn.
Theorem 2. Suppose that s ∈N with 8|s, s≥16. Put
e0 := (e0s/2(1), . . . , e0s/2(s/8−1))∈Qs/8−1 and let
λ = (λ2, . . . , λs/4−2)∈Qs/4−3 be any rational vector such that
As/2λ0 =−e00. Then in the above notation
θs(z) =Es/2i∞(z+ 1 2) +
s/4−2
X
`=2
λ`Es/2−2`i∞ (z+ 1
2)E2`0 (z+ 1 2).
2. A generalization
Here we shall give a result saying that the theta series ϑD+m(z) = X
x∈Dm+
eπi(x·x)z (z ∈ H)
attached to the D+m–lattice for anym∈Nwith 8|mstill can be expressed in a similar way as above.
Recall that for m even one defines
D+m:=Dm ∪ (Dm+ (1
2,· · ·,1 2)) where
Dm:={x∈Zm|x1+· · ·+xm≡0 (mod 2)}.
If 4|m, one has the description
Dm+ ={x∈Rm|xµ ≡0 (mod 1
2) ∀µ, xµ ≡xν (mod 1)∀ µ, ν, x1+· · ·+xm ≡0 (mod 2)}
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andD+mfor 8|mis an integral even unimodular lattice (note thatD8+is theE8–root lattice).
In particular, ϑDm+ is a modular form of weight m2 on the full modular group if 8|m.
We will keep the same notation as in sect. 1. Furthermore, iff(z) =P
n≥0a(n)e2πinz (z ∈ H) is a Fourier series, we set
(f|U2)(z) :=X
n≥0
a(2n)e2πinz (z ∈ H).
We then have
Theorem [2]. Suppose that m∈N with 8|m. Then in the above notation
(1) ϑD+m = 2m/4−1
Em/20 +
m/4−2
X
`=2
λ`Em/20 −2`E2`i∞
+
Em/2i∞ +
m/4−2
X
`=2
λ`Em/2−2`i∞ E2`0
|U2.
For details of the proof, we refer to [2].
References
[1] ¨O. Imamoglu and W. Kohnen: Representations of integers as sums of an even number of squares. To appear in Math. Ann.
[2] W. Kohnen and R. Salvati-Manni: On the theta series attached to Dm+-lattices.
To appear in IJNT
Winfried Kohnen, Universit¨at Heidelberg, Mathematisches Institut, INF 288, D-69120 Heidelberg, Germany
e-mail: [email protected]
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