LOCAL DENSITIES AND EXPLICIT BOUNDS FOR

REPRESENTABILITY BY A QUADRATIC FORM

Jonathan Hanke

§1 Introduction

One of the oldest questions in number theory is the question of when a number m is represented by an integral quadratic form Q in n variables, meaning that Q(x⃗) = m for some x⃗ ∈ ℤ^{n}. In fact, it suffices to answer the question for any form Q̃ which differs from Q by some invertible integral change of variables, since they represent the same numbers. We call the set of such forms the class of Q. To begin to answer this question, people first considered the weaker condition of m being locally represented by Q, meaning that m is represented by Q mod p^{α} for all α ≥ 0 and also that it is represented over the real numbers ℝ. The condition that m is locally represented by Q leads to finitely many congruence conditions on m. Since these are easy to check, the question then becomes when is some locally represented m actually represented by Q?

However, in general there are finitely many classes of quadratic forms which are locally equivalent to Q (called the genus of Q), so local conditions are not enough to determine the numbers represented by Q. To answer this question, one would need to somehow distinguish the class of Q from among these finitely many classes.

The first major quantitative result along these lines was that of Siegel, who gave a description of a certain weighted average of representations by forms in the genus of Q as an infinite product of local factors. He further showed that these averages are the Fourier coefficients of the Eisenstein series appearing in the theta function

(1.1)  Θ_{Q}(z) = ∑_{m∈ℤ} r_{Q}(m)e^{2πimz},

where r_{Q}(m) denotes the number of representations of m by Q (suitably normalized in the case where the r_{Q}(m) are infinite). It is this interpretation of the Eisenstein series that allows one to make precise effective statements, provided enough is known about the local factors.

There is a more refined local equivalence one can use which divides the genus into spinor genera. It is an amazing fact, again due to Siegel, that his weighted averages above only depend on the spinor genus of Q when n ≥ 4.

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The author would like to thank Ken Ono for suggesting to him that one could obtain an effective bound within a fixed square class when n = 3, which is how this project began. The author would also like to thank Manjul Bhargava for many useful conversations and his enduring interest in the case n = 4, and Rainer Schulze-Pillot for several helpful comments.

In the case where n ≥ 3 and the form Q is indefinite, meaning that it locally represents both positive and negative numbers (over ℝ), one can show that each form in the genus of Q is its own spinor genus. So by Siegel’s weighted average result, we have local conditions for the representability of a given number by Q.

If Q is not indefinite, then we say Q is positive definite or negative definite depending on whether Q represents positive or negative numbers. By replacing Q by −Q, we may assume Q is positive definite. In this case, the spinor genus remains a useful notion, but it does not provide as complete of an answer as when Q is indefinite.

In this paper, we seek to give explicit lower bounds for an integer m which ensure that m is represented by a given integral positive definite quadratic form Q in n ≥ 3 variables. Our main interest is in the cases n = 3 and 4, since a reasonable bound in these cases was not previously known and they allow us to compute which (spinor) locally represented numbers are not represented by Q. These bounds also help us to describe the general representation behavior of Q when n = 3. In this case, Duke and Schulze-Pillot [Du-SP] showed an ineffective lower bound over ℚ, avoiding discussion of the primes p | N and the spinor exceptional-type square classes. We give an effective version of their result, including an effective asymptotic statement for these square classes. For completeness, we actually give bounds for all n ≥ 3. In fact, we work quite generally over a totally real number field F and only restrict to F = ℚ to avoid mentioning Hilbert modular forms. However, using Hilbert modular forms, our results could easily be extended to any totally real F.

In §2 we introduce some basic facts and notation. In §3 we describe an explicit reduction procedure which helps us to compute the number of solutions Q(x⃗) ≡ m (mod 𝔭^{k}), the main idea being that either Hensel’s Lemma applies or we can divide the representation by a power of the local uniformizer π_{𝔭}. In §4 we review some essential facts about modular forms that we will need, with special attention to the Shimura lift when n = 3 where it plays a crucial role because of its close relationship with spinor genera. In §5 we establish explicit lower bounds for the main term of our asymptotic for r_{Q}(m). When n ≥ 4 this comes from the Fourier coefficients of an Eisenstein series, while when n = 3 there is an additional spinor term which comes into play. In §6 we state our main results, which provide effective lower bounds on m which, when satisfied, guarantee that m is represented by Q. Here our main emphasis is on n = 3 and n = 4, where such bounds provide theoretically interesting and computationally useful information about the representation behavior of Q respectively. In §7.1 we use the results of §5 to describe the asymptotics of the main term, and in §7.2 we describe how to modify our results find the numbers represented by a quadratic polynomial. In §8 we apply these results to answer affirmatively the long-standing conjecture of Kneser (popularized by Kaplansky) that for Q = x^{2} +3y^{2} +5z^{2} +7w^{2} the only integers ≥ 0 which fail to be represented are 2 and 22. In §9, for completeness we include some straightforward Gauss sum computations, originally due to Siegel, which are used to understand the generic local factors β_{𝔭}(m).

§2 Setup and Notation

We will use the standard notation ℂ, ℝ, ℚ, ℤ, ℕ, p, ℚ_{v}, ℤ_{v} to denote the complex numbers, real numbers, rational numbers, rational integers, natural numbers, a positive prime number, the completion of ℚ at the valuation v, and ring of integers in ℚ_{v}.

Similarly, we let F, 𝔬_{F} , 𝔭, F_{v}, 𝔬_{v} denote a totally real number field, its ring of integers, a prime ideal in 𝔬_{F}, the completion of F at the valuation v, and the ring of integers in F_{v}. We note that if v is archimedian then 𝔬_{v} = F_{v}. We also let q = N_{F/ℚ}(𝔭) = #(𝔬_{F} /𝔭) and π_{𝔭} denote a choice of uniformizer at 𝔭. When discussing divisibility we may abuse notation slightly, writing 𝔭 | m when strictly speaking we should write π_{𝔭} | m. We denote by 𝔸_{F} and 𝔸_{F}^{×} respectively the adeles and ideles of F, and let 𝔸_{𝔬_{F}} = ∏_{v} 𝔬_{v}.

We abbreviate e^{2πiz} as e(z), let ℂ^{1} denote the unit circle in ℂ, and let μ_{N} denote the N^{th} roots of unity. We also write a | b, gcd {x_{i}}, lcm {x_{i}}, ⌊x⌋, τ(m), σ(m), φ(m), μ(m) for a divides b, the greatest common divisor and least common multiple of finitely many numbers x_{i}, the greatest integer ≤ x, the number of positive divisors of m, the sum of the positive divisors of m, the Euler phi and Möebuis function, respectively. We write k >> 1 to mean k is sufficiently large (i.e., there is some positive constant M such that the following statements are true when k > M).

If x⃗ ∈ ℤ^{n} and P is a partition of {1, ⋯, n}, then for each j ∈ P we let x⃗_{j} denote the vector whose components are x_{i} for all i ∈ j. Similarly, for any 𝕊 ⊆ P we take x⃗_{𝕊} to be the vector whose components are x_{i} for all i ∈ ∪_{j∈𝕊} j. Implicit in this notation is a fixed ordering of 𝕊, which we always take to be the natural ordering on {1, ⋯, n}.

Throughout this paper we will be considering an integral totally definite quadratic form Q of dimension n over F, by which we mean a function

(2.1)  Q(x⃗) = ∑_{i,j=1}^{n} c_{ij}x_{i}x_{j},  with c_{ij} = c_{ji} ∈ F

such that Q(x⃗) ∈ 𝔬_{F} for all x⃗ in some fixed 𝔬_{F} -lattice L of rank n, and for which Q(F_{v}) ⊆ ℝ is either ≥ 0 or ≤ 0 (but not both) for all archimedian places v of F. Since L is locally free, at each place v of F we may choose a local basis {y_{i}} so that

(2.2)  L = ∑_{i=1}^{n} 𝔬_{v} y_{i}.

In this basis we may represent Q by a matrix Q_{v} as in (2.1) (denoted as Q_{𝔭} if v corresponds to 𝔭). At every non-archimedian place corresponding to 𝔭, we may use Lemma 1.1 below to write Q in the local normalized form

(2.3)  Q(x⃗) ≅_{𝔬𝔭} ∑_{j} π_{𝔭}^{ν_{j}}Q_{j}(x⃗_{j})

with dim(Q_{j}) ≤ 2. When 𝔭 ∤ 2 we have dim(Q_{j}) = 1, and when v is archimedian,

we may take Q ≅_{ℝ} ∑_{i=1}^{n} x_{i}^{2}.

Lemma 2.1. Over 𝔬_{𝔭} we may locally write the integral quadratic form Q as a direct sum of forms π_{𝔭}^{ν_{j}}Q_{j} where either Q_{j}(x) = ux^{2} for some u ∈ 𝔬_{𝔭}^{×} or Q_{j}(x, y) = ax^{2} + bxy + cy^{2} with b ∈ 𝔬_{𝔭}^{×}, a, c ∈ 𝔬_{𝔭}, and ord_{𝔭}(a) = ord_{𝔭}(c). If 𝔭 ∤ 2 then the Q_{j} appearing are all 1-dimensional.

Proof. Follwing the method in [C-S, pp369–70] we consider the symmetric matrix of the bilinear form associated to Q with half-integral entries and integral diagonal.

If the least 𝔭 divisible entry is on the diagonal, then by elementary symmetric row-column operations we may clear its row and column isolating it as a direct summand. If the (strictly) least 𝔭 divisible entry is not on the diagonal, we may reorder our variables so that it is adjacent to the diagonal. Since Q is integral, this

gives an upper left 2 × 2 submatrix of the form [ \begin{matrix}π_{𝔭}^{ν}α & π_{𝔭}^{ν}β \\ π_{𝔭}^{ν}β & π_{𝔭}^{ν}γ \end{matrix} ] with β ∈ \frac{1}{2}𝔬_{𝔭}^{×} and

α, γ ∈ 𝔭𝔬_{𝔭}. Further, since the determinant αγ − β^{2} is in \frac{1}{4}𝔬_{𝔭}^{×}, these form a basis for the 𝔬_{𝔭}-module \frac{1}{2}𝔬_{𝔭} and we may clear the two associated rows and columns as above to isolate it as a direct summand. If 𝔭 ∤ 2 or ord_{𝔭}(α) > ord_{𝔭}(γ), we may add the second row-column to the first, replacing α by α + 2β + γ. When 𝔭 ∤ 2 this reduces us to the diagonal case, and when 𝔭 | 2 this ensures ord_{𝔭}(α) = ord_{𝔭}(γ). Passing to the associated quadratic form gives the desired normalized form for Q with a = α, b = 2β, and c = γ in the 2 × 2 blocks. □

We now define the determinant D = D_{Q} and level N = N_{Q} of Q. We take D_{Q} to be the fractional ideal of F generated locally at each prime 𝔭 by the elements D_{Q,𝔭} = D_{𝔭} = det(Q_{𝔭}), and N_{Q} as the least ideal N_{𝔭} ⊆ 𝔬_{𝔭} so that N_{𝔭}(2Q_{𝔭})^{−1} is a matrix of integral ideals whose diagonal entries lie in 2𝔬_{F}. When the class number of F is 1 (e.g. F = ℚ), we may choose a global basis for L in which case we can take D_{Q} = det(Q) as an element of F, which is unique up to multiplication by 𝔬_{F}^{×}. When F = ℚ we will understand D_{Q} to be such an element. We also let Gen(Q) and Spn(Q) denote the genus and spinor genus of Q respectively.

Since in general D_{Q} is not an integral ideal, it is often more convenient to discuss its integrality properites in terms of D_{2Q}. However, N_{Q} is always an integral ideal.

Lemma 2.2. Let Q be an integral quadratic form over a number field F. Then

𝔭 | N_{Q} ⟺ 𝔭 | D_{2Q} ⟺ for some j either ν_{j} ≥ 1 or 𝔭|2 and dim(Q_{j}) = 1,

where N_{Q} and D_{Q} are respectively the level and determinant of Q, and for each prime ideal 𝔭 of F we define the ν_{j} as in (2.3) above.

Proof. From the local normal form (2.3) and Lemma 2.1, we see that

(2.4) ord_{𝔭}(D_{2Q,𝔭}) = ∑_{j} λ′_{j} where λ′_{j} = \{ \begin{matrix}ν_{j} + ord_{𝔭}(2) & if dim(Q_{j}) = 1, \\ ν_{j} & if dim(Q_{j}) = 2, \end{matrix}

and

(2.5) ord_{𝔭}(N_{𝔭}) = max_{j}\{λ′′_{j}\} where λ′′_{j} = \{ \begin{matrix}ν_{j} + 2 ord_{𝔭}(2) & if dim(Q_{j}) = 1, \\ ν_{j} & if dim(Q_{j}) = 2. \end{matrix}

Our Lemma follows by using these formulas to check when ord_{p}(·) ≥ 1. □

We say that an integer m ∈ 𝔬_{F} is represented by Q if Q(x⃗) = m has a solution with x⃗ ∈ L, and that m is locally represented by Q if it has a solution with x⃗ ∈ L_{v} for all places v of F. A prime 𝔭 is said to be anisotropic (resp. isotropic) with respect to Q when Q is anisotropic (resp. isotropic) over 𝔬_{𝔭}. We often omit the explicit mention of Q when our meaning is clear.

For m ∈ ℤ we let (m)_{𝕊} denote the maximal positive divisor of m divisible only by primes p ∈ 𝕊. For m ∈ 𝔬_{F}, we let N_{F/ℚ}(m)_{𝕊} denote the norm of the maximal integral ideal dividing m𝔬_{F} divisible only by primes 𝔭 ∈ 𝕊. We let Iso and Aniso respectively denote the set of isotropic and anisotropic primes of F. We also say that m ∈ 𝔬_{F} is supported on some set 𝕊 of primes when |m|_{𝔭} = 1 for all 𝔭 ∉ 𝕊, and take m → ∞_{𝕊} to mean m → ∞ in such a way that ord_{𝔭}(m) → ∞ for all 𝔭 ∈ 𝕊.

We define

R_{Q}(m) = {x⃗ ∈ L | Q(x⃗) = m}
R_{𝔭^{k},Q}(m) = {x⃗ ∈ L/𝔭^{k}L | Q(x⃗) ≡ m (mod 𝔭^{k})}

and let r_{Q}(m) = #R_{Q}(m), r_{𝔭^{k},Q}(m) = #R_{𝔭^{k},Q}(m). Solutions (mod 𝔭^{k}) satisfying additional congruence conditions are denoted as above with the relevant conditions as a superscript (e.g., R_{𝔭^{k},Q}^{Good}(m), r_{𝔭^{k},Q}^{Good}(m), etc.). We adopt the convention that x⃗_{𝕊} ≡ 0 mod 𝔭 ⟺ either 𝕊 = ∅ or 𝕊 ≠ ∅ and x⃗_{𝕊} ≡ 0 mod 𝔭. Therefore x⃗_{𝕊} ≢ 0 mod 𝔭 implies 𝕊 ≠ ∅ and x⃗_{𝕊} ≢ 0 mod 𝔭. When our meaning is clear, we often omit the subscript Q to simplify our notation.

For a ∈ 𝔬_{F} we let (\frac{a}{𝔭}) = ±1 denote the usual Legendre symbol given by a^{\frac{q−1}{2}}

mod 𝔭. When F = ℚ, we let (\frac{a}{b}) denote the quadratic residue symbol defined in [Sh, pp442–3], where the sign of a determines the parity of the character (\frac{a}{·}). For a Dirichlet character ϕ, we define its twist ϕ_{(u)}(·) = ϕ(·) (\frac{−u}{·}) by −u.

For any fixed u ∈ 𝔬_{F} we let Φ_{n} denote a Hecke character on F defined for all primes 𝔭 ∤ 2N by

Φ_{n}(𝔭) = Φ_{n}(𝔭, u) = \{ \begin{matrix}(\frac{(−1)^{\frac{n}{2}} D_{𝔭}}{𝔭} ) & if n is even, \\ (\frac{(−1)^{\frac{n−1}{2}} D_{𝔭}u}{𝔭}) & if n is odd. \end{matrix}

When n is odd the extra 2 is unnecessary since N is already even. Also, when the class number of F is 1 (e.g. F = ℚ) we may replace the local D_{𝔭} by D, in which case Φ_{n} is just a quadratic Dirichlet character.

We define the spaces of modular forms M_{k}(N, ϕ) and S_{k}(N, ϕ) of weight k, level N and character ϕ. If k ∈ ℤ + \frac{1}{2} then these are defined as in [Sh, pp443–4] (however his subscript is twice ours). There is a natural (Petersson) inner product on M_{k}(N, ϕ) given by

< f, g >= \frac{1}{Vol(H/Γ_{0}(N))} ∫_{H/Γ_{0}(N)} f(z)\overline{g(z)} y^{k} \frac{dxdy}{y^{2}} ,

assuming that the product fg is a cusp form. (We always assume y^{1/2} > 0.)

Warning: We will often speak about a square class T ℤ^{2}, T 𝔬_{F}^{2}, or T (𝔸_{𝔬_{F}})^{2}, however this is an abuse of notation since our meaning is to allow all non-zero square multiples of T. Therefore, to be precise we should write T (ℤ−{0})^{2}, T (𝔬_{F} −{0})^{2}, or T ∏_{v}(𝔬_{v} − {0})^{2} respectively.

§3 Reduction maps

We begin by giving a recursive procedure to compute the number of representations r_{𝔭^{k},Q}(m) for k >> 1 using Hensel’s Lemma and 3 reduction maps. This will be useful later for understanding the behavior of the local factors β_{𝔭}(m) appearing in Siegel’s product formula, and will allow us to obtain explicit lower bounds for the growth of the Fourier coefficients of the Eisenstein series E(z) within a square class. A simpler version of our approach can be found in [M-H, pp51–3].

Throughout this section, we fix a prime 𝔭 of F and assume Q = Q_{𝔭} is of the form (2.3). We also implicitly use the letter j to index the forms π_{𝔭}^{ν_{j}}Q_{j} and the vectors x⃗_{j} appearing there.

Definition 3.1. We say that x⃗ ∈ R_{𝔭^{k},Q}(m) is of Zero-type if x⃗ ≡ 0 mod 𝔭, of Good-type if π_{𝔭}^{ν_{j}} x⃗_{j} ≢ 0 mod 𝔭 for some j, and of Bad-type otherwise. The set of all such x⃗ are denoted respectively by R_{𝔭^{k},Q}^{Zero}(m), R_{𝔭^{k},Q}^{Good}(m), and R_{𝔭^{k},Q}^{Bad}(m) (which have sizes r_{𝔭^{k},Q}^{Zero}(m), r_{𝔭^{k},Q}^{Good}(m), and r_{𝔭^{k},Q}^{Bad}(m)).

Our definition of Good-type solutions was motivated by the following property:

Lemma 3.2. We have

(3.1)  r_{𝔭^{k+l},Q}^{Good}(m) = q^{(n−1)l}r_{𝔭^{k},Q}^{Good}(m),

for all k ≥ 2 ord_{𝔭}(2) + 1.

Proof. Suppose x⃗ is a Good-type solution to Q(x⃗) ≡ m (mod 𝔭^{k}), so π_{𝔭}^{ν_{j}} x⃗_{j} ≢ 0 mod 𝔭 for some j. We choose an arbitrary lift (mod 𝔭^{k+l}) for the variables outside x⃗_{j} and collect all terms outside Q_{j} together, writing them as m′.

If dim(Q_{j}) = 1 then we wish to solve ux^{2} ≡ m′ mod 𝔭^{k+l} with x ≡ x_{j} mod 𝔭^{k}. By Hensel’s Lemma such a solution exists when k ≥ 2 ord_{𝔭}(2) + 1, and it is clearly unique.

If dim(Q_{j}) = 2 then we wish to apply Hensel’s Lemma as above to f = ax^{2} + bxy + cy^{2} after choosing an arbitrary lift of one of the coordinates, say y. To meet the criterion of Hensel’s Lemma, we must find some coordinates in which \frac{∂f}{∂x}(x⃗_{j}) ≢ 0, which is to say that f is non-singular at x⃗_{j} (mod 𝔭). Checking when both partials vanish, we see that the only possible singular point is x⃗_{j} ≡ (0, 0), which contradicts our assumption that x⃗ is of Good-type. Since b is a unit, we may even lift these solutions when k = 1. □

Let x⃗ denote a general solution of a given type. We now describe several reduction maps useful for understanding the number of solutions of each type, allowing the possibility that x⃗ satisfies additional congruence conditions of the form x⃗_{j} ≡ 0 or x⃗_{j} ≢ 0 (mod 𝔭) for each j so long as these extra conditions are not implied or contradicted by the reduction-type congruence conditions on x⃗. If such conditions on x⃗_{j} are allowed for all j ∈ 𝕊, we denote them by x⃗_{𝕊} ∈ C.

Good-type Solutions: For these we have the map

R_{𝔭^{k}}^{Good, x⃗∈C}(m) ⟶^{π_{G}} R_{𝔭^{k−1}}^{Good, x⃗∈C}(m)

defined by reducing x⃗ mod 𝔭^{k−1}. By Lemma 2.1 this is surjective with multiplicity q^{n−1}, so the number of Good-type solutions can be computed explicitly either from the mod 𝔭 solutions (if 𝔭 ∤ 2) or from the solutions mod 4𝔭 (if 𝔭 | 2).

Zero-type Solutions: These solutions are characterized by the congruence x⃗ ≡ 0 mod 𝔭, thus arise only when 𝔭^{2} | m. Reduction of these solutions depends on the map

R_{𝔭^{k}}^{Zero}(m) ⟶^{π_{Z}} R_{𝔭^{k−2}}(m/π_{𝔭}^{2})

defined by x⃗ ↦ x′′ = \frac{1}{π_{𝔭}} x⃗ mod 𝔭^{k−2}. This is well-defined since \frac{1}{π_{𝔭}} x⃗ is defined modulo 𝔭^{k−1}.

We observe that π_{Z} is surjective with multiplicity q^{n} since the elements x′ mod 𝔭^{k−1} which reduce to a fixed x′′ are in 1-to-1 correspondence with π_{Z}^{−1}(x′′) under x⃗ = π_{𝔭}x′, and there are q^{n} such x′.

Bad-type Solutions: These arise only when 𝔭 | m. To describe their reduction we define

𝕊_{0} = {j | ν_{j} = 0} ,  𝕊_{1} = {j | ν_{j} = 1} ,  𝕊_{2} = {j | ν_{j} ≥ 2} ,

and let s_{i} = ∑_{j∈𝕊_{i}} dim(Q_{j}). Then the Bad-type solutions are characterized by the (mod 𝔭) congruences x⃗ ≢ 0 and x⃗_{𝕊_{0}} ≡ 0. We will have two reduction maps π_{B′} and π_{B′} which respectively correspond to division by π_{𝔭} and division by π_{𝔭}^{2}. In the process, we introduce two auxiliary forms Q′ and Q′′ whose data is denoted with a ′ or ′′ accordingly. For these we have Q_{j} = Q′_{j} = Q′′_{j} for all j.

Bad-type I: Division by π_{𝔭} is appropriate for the case when 𝕊_{1} ≠ ∅ and x⃗_{𝕊_{1}} ≢ 0. Then we have the map

R_{𝔭^{k},Q}^{Bad, x⃗_{𝕊_{1}} ≢0, x⃗_{𝕊_{1}∪𝕊_{2}} ∈C}(m) ⟶^{π_{B′}} R_{𝔭^{k−1},Q′}^{Good, x⃗_{𝕊_{1}∪𝕊_{2}} ∈C}(m/π_{𝔭})

defined for each index j by

x⃗_{j} ↦ \frac{1}{π_{𝔭}} x⃗_{j}, ν′_{j} = ν_{j} + 1  if j ∈ 𝕊_{0},
x⃗_{j} ↦ x⃗_{j}, ν′_{j} = ν_{j} − 1  if j ∉ 𝕊_{0},

which is surjective with multiplicity q^{s_{1}+s_{2}} since we are free to choose lifts of the components of the image at 𝕊_{1} ∪ 𝕊_{2}.

Bad-type II: Division by π_{𝔭}^{2} is appropriate for the remaining case where either 𝕊_{1} = ∅ or x⃗_{𝕊_{1}} ≡ 0, and can occur only when 𝕊_{2} ≠ ∅. In this case, we define the map

R_{𝔭^{k},Q}^{Bad, x⃗_{𝕊_{1}} ≡0, x⃗_{𝕊_{2}} ∈C}(m) ⟶^{π_{B′′}} R_{𝔭^{k−2},Q′′}^{x⃗_{𝕊_{2}} ≢0, x⃗_{𝕊_{2}} ∈C}(m/π_{𝔭}^{2})

given componentwise by

x⃗_{j} ↦ \frac{1}{π_{𝔭}} x⃗_{j}, ν′′_{j} = ν_{j}  if j ∈ 𝕊_{0} ∪ 𝕊_{1},
x⃗_{j} ↦ x⃗_{j}, ν′′_{j} = ν_{j} − 2  if j ∈ 𝕊_{2},

which is surjective and has multiplicity q^{2n−s_{0}−s_{1}}. To see this, notice that the map is q-to-1 over the 𝕊_{0} ∪𝕊_{1} components by the same reasoning as for π_{Z}, and is q^{2}-to-1 over the 𝕊_{2} components since the inverse map there corresponds to multiplication by π_{𝔭}^{2}.

Definition 3.3. We define the depth of each type of solution (i.e., Good, Zero, Bad) of R_{𝔭^{k},Q}(m) to be the maximal difference k − k′ for any x⃗ ∈ R_{𝔭^{k},Q}(m) to be mapped into R_{𝔭^{k′},Q̂}(m̂) under consecutive application of the respective type of maps π_{G}, π_{Z}, and π_{B^{∗}} ∈ {π_{B′}, π_{B′′}} (for some Q̂ and m̂).

Lemma 3.4. Suppose Q is an integral quadratic form over F, 𝔭 is prime ideal in 𝔬_{F}, and m ∈ 𝔬_{F}. Then for k >> 1 we can compute r_{𝔭^{k},Q}(m) recursively in terms of solutions mod 𝔭 (or 4𝔭 if 𝔭 | 2) using the maps π_{G}, π_{Z}, π_{B′}, and π_{B′′}. In fact, the Good, Zero, and Bad-type depths of R_{𝔭^{k},Q}(m) are bounded by k, ord_{𝔭}(m), and ord_{𝔭}(N) + 1 respectively, where N is the level of Q.

Proof. By definition, the maps π_{G} and π_{B′} give Good-type solutions. The map π_{Z} gives all types of solutions, which may be broken down into Good, Zero, and Bad-type solutions. The image of the map π_{B′′} is less clear, but can be written as

R_{𝔭^{k−2},Q′′}^{x⃗_{𝕊_{2}} ≢0, x⃗_{𝕊_{2}} ∈C}(m/π_{𝔭}^{2}) = R_{𝔭^{k−2},Q′′}^{x⃗_{𝕊\begin{matrix}′ & ′ \\ 2 \end{matrix}} ≢0, x⃗_{𝕊_{2}} ∈C}(m/π_{𝔭}^{2}) ∪ R_{𝔭^{k−2},Q′′}^{x⃗_{𝕊\begin{matrix}′ & ′ \\ 2 \end{matrix}} ≡0,x⃗_{𝕊_{2}−𝕊\begin{matrix}′ & ′ \\ 2 \end{matrix}} ≢0, x⃗_{𝕊_{2}} ∈C}(m/π_{𝔭}^{2}),

where 𝕊′′_{2} is the set 𝕊_{2} with respect to the form Q′′ defined above. Each of these terms can be handled recursively by considering Good-type and Bad-type solutions with extra congruence conditions mod 𝔭 of the kind we allow. The Bad-type solutions of the first term can be handled by the maps π_{B′} and π_{B′′} (since the condition x_{𝕊′_{2}} ≢ 0 is trivial in the setting of π_{B′′}), while the Bad-type solutions of the second term only require π_{B′}. Therefore we are reduced to counting certain types of solutions mod 𝔭 (or mod 4𝔭 if 𝔭 | 2).

From Lemma 3.2 the Good-type depth is bounded by k − 1, and the Zero-type depth is clearly bounded by ord_{𝔭}(m) since it involves division by π_{𝔭}^{2} and if x⃗ is of Zero-type then ord_{𝔭}(m) ≥ 2. The Bad-type depth is controlled by largest ν_{j} in (1.4) since we may have at most ⌊\frac{ν_{j}}{2} ⌋ consecutive maps π_{B′′} (each with depth 2), and then possibly an additional π_{B′} (with depth 1). From (2.5), we see that this is ≤ max_{j}{ν_{j}} + 1 ≤ ord_{𝔭}(N) + 1. □

Remark 3.4.1.

a) When π_{𝔭} ∤ m then all solutions are of Good-type.

b) When π_{𝔭} ∤ N then all solutions are of Good-type or Zero-type.

c) From Lemma 3.4, the Bad-type term r_{𝔭^{k},Q}^{Bad}(m) has depth ≤ ord_{𝔭}(N) + 1. To guarantee the constancy of the Bad-type term as m becomes more 𝔭divisible, we must assume divisibility of m by an additional π_{𝔭} so that all of our Good-type solutions count representations of 0 mod 𝔭. Thus our condition for constancy becomes ord_{𝔭}(m) ≥ ord_{𝔭}(N) + 2.

d) Some authors like to speak about primitive representations of m, meaning those representations which are not a multiple of a representation of some proper divisor of m. In our terminology, primitive representations are those representations not of Zero-type (see Definition 5.3).

e) For k >> 1, we may use Lemma 3.2 together with the reduction maps above to obtain recursion formulas for the number of solutions mod 𝔭^{k} in terms of other solutions mod 𝔭^{k}. The factors associated to the maps π_{Z}, π_{B′}, and π_{B′′} are q^{2−n}, q^{1−s_{0}}, and q^{2−s_{0}−s_{1}} respectively.

Definition 3.5. We say that a number m is locally represented at 𝔭 (by Q) if r_{𝔭^{k}}(m) > 0 for all k >> ν. Also, we say that m is locally represented if it is locally represented at p for all primes 𝔭.

Remark 3.5.1. When n ≥ 3 and 𝔭 ∤ N, every number m is locally represented at 𝔭. In general, to check whether m is locally represented at 𝔭, Lemma 3.4 tells us that it suffices to check locally whether any of the quotients of m by a square factor are represented mod 𝔭^{ord_{𝔭}(4N)+2} when 𝔭 | N, and mod 𝔭 when 𝔭 ∤ N.

Definition 3.6. We define a number m to be 𝔭-stable if m is locally represented ay 𝔭, and for all k >> ν the quantity

r_{𝔭^{k}}^{Good}(π_{𝔭}^{2ν}m) + r_{𝔭^{k}}^{Bad}(π_{𝔭}^{2ν}m)

is constant for all ν ≥ 1. We further define m to be stable if it is 𝔭-stable for all primes 𝔭, and 𝕊-stable if it is 𝔭-stable for all 𝔭 ∈ 𝕊.

Remark 3.6.1. From Remark 3.4.1(b) we know that all m are 𝔭-stable when 𝔭 ∤ N, and using Lemma 3.4 and Remark 3.4.1(c) we see that all m ∈ 𝔰_{𝔭} = 𝔭^{ord_{𝔭}(N)} locally represented at 𝔭 are 𝔭-stable when 𝔭 | N. Together, these imply that the ideal 𝔰 = ∏_{𝔭|N} 𝔰_{𝔭} = N𝔬_{F} is a stable ideal in the sense that m is stable for all locally represented m ∈ 𝔰.

To do explicit calculations with Lemma 3.2 when n = 3 or 4, it is useful to have on hand the number of solutions mod 𝔭 ∤ 2 of quadratic forms in ≤ 4 variables provided in Table 1 below. These can be verified by computing the appropriate Gauss sums (see Appendix). If 𝔭 | 2 then we are interested in the number of solutions mod 4𝔭 which is not as straightforward and so must be computed on a case-by-case basis.

Table 1 - Number of solutions mod 𝔭 when 𝔭 ∤ 2N, u ∈ (𝔬_{F} /𝔭)^{×}, and n ≤ 4:

n = 1  n = 2  n = 3  n = 4
r_{𝔭}(0)  1  q + (q − 1) (\frac{−D_{𝔭}}{𝔭} )  q^{2}  q^{3} + q (q − 1) (\frac{D_{𝔭}}{𝔭} )
r_{𝔭}(u) 1 + (\frac{D_{𝔭}u}{𝔭})  q − (\frac{−D_{𝔭}}{𝔭} )  q^{2} + q (\frac{−D_{𝔭}u}{𝔭})  q^{3} − q (\frac{D_{𝔭}}{𝔭} )

Definition 3.7. We say that a prime 𝔭 is anisotropic for Q (or alternatively, that Q is anisotropic at 𝔭) if for every vector x⃗ ∈ (F_{𝔭})^{n}, we have

Q(x⃗) = 0 ⟹ x⃗ = 0.

If this is not so, we say that a prime 𝔭 is isotropic for Q (or alternatively, that Q is isotropic at 𝔭). We denote the set of anisotropic (resp. isotropic) primes by Aniso (resp. Iso), and often omit Q when our meaning is clear.

Lemma 3.8. Suppose Q is a non-degenerate integral quadratic form over F, 𝔭 is a prime ideal in 𝔬_{F}, k >> 1, and m is 𝔭-stable with 𝔭 | m. Then

Q is anisotropic at 𝔭 ⟺ r_{𝔭^{k},Q}(m) = r_{𝔭^{k},Q}^{Zero}(m).

Proof. Since m is 𝔭-stable and 𝔭 | m, it suffices to prove the Lemma under the assumption that m is sufficiently 𝔭-divisible, and we assume this in what follows.

(⇒) Suppose Q is anisotropic at 𝔭. Then we know r_{𝔭^{k},Q}^{Good}(m) = 0 since by Lemma 3.2 any such solution would lift to a non-zero representation of zero in 𝔬_{𝔭}, hence in F_{𝔭}, which do not exist by assumption.

From [OM, pp153, 167–171] (or more clearly [Ca, pp58–9] when F = ℚ) we know

(3.2) Q is anisotropic at 𝔭 ⟺ \{ \begin{matrix}−D_{𝔭} ∉ (F_{𝔭}^{×})^{2} & if n = 2, \\ c ≠ (\frac{−1,−D_{𝔭}}{𝔭} ) & if n = 3, \\ c = (\frac{−1,−1}{𝔭}) and D_{𝔭} ∈ (F_{𝔭}^{×})^{2} & if n = 4, \end{matrix}

where c is the Hasse invariant of Q at 𝔭. Since the norm residue symbol (\frac{a,b}{𝔭}) depends only on the square classes of a and b over F_{𝔭}, we know Q is anisotropic exactly when the form Q′′ from the map π_{B′′} is anisotropic. The form Q′ from π_{B′} is obtained by multiplying Q by π_{𝔭}, then changing the 𝕊_{1} ∪ 𝕊_{2} coefficients within a square class by dividing by π_{𝔭}^{2}. Since multiplication by π_{𝔭} doesn’t change the anisotropicity of Q, we see that if Q is anisotropic, then so are the Q′, Q′′ arising from it.

Using π_{B′} and π_{B′′}, the Bad-type solutions can be computed recursively in terms of (finitely many) r_{p^{k},Q̃}^{Good}(m̃) for some Q̃ and m̃. However m is assumed to be highly 𝔭-divisible and the Bad-type depth is bounded independently of m, so each of the r_{p^{k},Q̃}^{Good}(m̃) must be zero which guarantees there are no Bad-type solutions. Hence r_{𝔭^{k},Q}(m) = r_{𝔭^{k},Q}^{Zero}(m).

(⇐) Suppose Q is isotropic at 𝔭, so there is some non-zero vector over F_{𝔭} which represents zero. By clearing denominators appropriately we obtain a primitive vector x⃗_{0} over 𝔬_{𝔭}. Then x⃗_{0} mod 𝔭^{k} gives a non-Zero-type solution x⃗_{1} in R_{𝔭^{k},Q}(0). By the reduction procedure mentioned above, we may reduce x⃗_{1} to a Good-type representation x_{1}^{∗} ∈ R_{𝔭^{k^{∗}} ,Q^{∗}}^{Good}(0) for some auxilliary form Q^{∗}.

Since m is assumed highly 𝔭-divisible we have R_{𝔭^{k^{∗}} ,Q^{∗}}^{Good}(0) = R_{𝔭^{k^{∗}} ,Q^{∗}}^{Good}(m^{∗}) where m^{∗} is obtained by applying the same reduction to m, Lemma 3.2 guarantees the existence of a Good-type representation x⃗_{2} over F_{𝔭} of m^{∗} for k >> 1 with x⃗_{2} ≡ x⃗_{1} ≡ x⃗_{0} mod 𝔭. By reversing the reduction procedure with x⃗_{2}, we obtain a non-Zero-type representation of m by Q. □

Remark 2.8.1. From (3.2) we see that if Q is anisotropic at 𝔭, then 𝔭 | D. Since Q is anisotropic at 𝔭 ⟺ 2Q is anisotropic at 𝔭, Lemma 2.2 tells us that all anisotropic primes divide N. Therefore if 𝔭 ∤ N then 𝔭 is isotropic.

Remark 3.9. Due to the very simple nature of our approach, one could equally well use these methods to understand the number of solutions r_{P,𝔭^{k}}(m) of a diagonalizable homogeneous polynomial P (x⃗) of degree d in n variables, and the associated local factors β_{P,𝔭}(m) as in (5.2). In particular, one can describe r_{P,𝔭^{k}}(m) in terms of Good-type, Zero-type, and Bad-type solutions, however the reduction procedure for Bad-type solutions will involve d reduction maps π_{B}^{(l)} corresponding to the d possible divisions of x⃗ by powers π_{𝔭}^{l} with 1 ≤ l ≤ d. However, the definitions of stability can be made exactly as in the quadratic case, and they lead to a simple description of the growth of the local factor β_{P,𝔭}(mπ_{𝔭}^{dν}) when m is 𝔭-stable. Generalizations of Siegel-type results to forms of higher degree have been studied by Igusa [Ig].

§4 Modular forms

Throughout this section we take F = ℚ and assume that n ≥ 3.

Our interest in modular forms stems from the fact that the theta function Θ_{Q}(z) is known to be a modular form of weight n/2 on Γ_{0}(N) for some quadratic character χ (see [An-Zh, Thrm 2.2, p61]). When n = 3 the Shimura lift will play a key role in our analysis, so we make some related definitions before proceeding.

Definition 4.1. Suppose n ≥ 3 is odd. For f(z) = ∑_{m=1}^{∞} a(m)e(mz) ∈ S_{n/2}(N, χ) and some fixed square free integer t > 0 we define its Shimura lift Shi(f) = Shi(f, t) following [Sh, p441] to be the modular form g(z) = ∑_{m=0}^{∞} b(m)e(mz) ∈ M_{n−1}(N/2, χ^{2}) satisfying

(4.1)  ∑_{m_{0}=1}^{∞} b(m_{0})m_{0}^{−s} = L(s − \frac{n−3}{2}, Φ_{n}) ∑_{m_{0}=1}^{∞} a(tm_{0}^{2})m_{0}^{−s}.

Notice that since n is odd the character Φ_{n} depends on t, and that χ^{2} is the trivial character since χ is quadratic. Additionally, Shimura showed that Shi(f) is actually a cusp form when n ≥ 5.

Remark. This definition agrees with Shimura’s since for odd n we have χ(·) = (\frac{D}{·} ).

We now describe the extent to which the Shimura lift fails to be cuspidal when n = 3. Let U(N, χ) be the subspace of S_{3/2}(N, χ) spanned by

(4.2)  {u(z) = ∑_{m∈ℤ} ψ(m)me(hm^{2}z)}

where ψ is a primitive Dirichlet character of conductor R with ψ(−1) = −1, ψ = Φ_{3} = χ_{(h)} on (ℤ/Nℤ)^{×}, 4hR^{2} | N and h > 0. Since the character ψ depends only on the square class tℤ^{2} containing h, we henceforth denote it by ψ_{T} for any T ∈ tℤ^{2}. Notice that there are only finitely many tℤ^{2} on which the Fourier coefficients of U(N, χ) are non-zero.

It is known (see [Ci, Cor 4.10, p108]) that U(N, χ) is the subspace of S_{3/2}(N, χ) whose Shimura lift (for any t) is not cuspidal. We denote by U^{⊥}(N, χ) the subspace of S_{3/2}(N, χ) perpendicular to U(N, χ) under the Petersson inner product.

Our approach to understanding Θ_{Q}(z) is as follows. When n ≥ 4 we write

(4.3)  Θ_{Q}(z) = E(z) + f(z)

as the sum of an Eisenstein series E(z) = ∑_{m≥0} a_{E}(m)e(mz) and a cusp form f(z), and analyze the growth of the Fourier coefficients separately. However when n = 3, the situation is complicated by the existence of cusp forms with non-cuspidal Shimura lift, and we write

(4.4)  Θ_{Q}(z) = E(z) + H(z) + f(z)

where H(z) = ∑_{m>0} a_{H}(m)e(mz) ∈ U(N, χ) and f(z) ∈ U^{⊥}(N, χ). In this case, we will also be interested in the form g = Shi(f) ∈ S_{2}(N/2).

Definition 4.2. When n = 3 we say that m ∈ tℤ^{2} is spinor p-stable at some prime p if a_{H}(mp^{2ν}) = a_{H}(m)ψ_{t}(p^{ν})p^{ν} for all ν ≥ 0 with ψ_{t} as in (4.2), and spinor stable if m is spinor p-stable for all primes p. Notice that lcm {h_{j} | h_{j} ∈ tℤ^{2}} is spinor stable. We also say that m is very p-stable if it is both p-stable (in the sense of Definition 3.6) and spinor p-stable, and likewise say that m is very stable if it is very p-stable at all primes p.

From the work of [Si], [KuRa1-2], [Ra] and [SP2, Satz 2, p291], we may realize the following Fourier coefficients as weighted averages over Gen(Q) and Spn(Q) respectively:

(4.5)  a_{E}(m) = ∑_{Q′∈Gen(Q)} \frac{r_{Q′}(m)}{|Aut(Q′)|} ; ∑_{Q′∈Gen(Q)} \frac{1}{|Aut(Q′)|} ≥ 0,

(4.6)  a_{E}(m) + a_{H}(m) = ∑_{Q′∈Spn(Q)} \frac{r_{Q′}(m)}{|Aut(Q′)|} ; ∑_{Q′∈Spn(Q)} \frac{1}{|Aut(Q′)|} ≥ 0.

In fact, Kneser [Kn] (when Q is indefinite) and Hsia [Hs] (more generally) have shown that Gen(Q) splits into two half genera consisting of equal numbers of spinor genera, and that a_{E}(m) + a_{H}(m) is the same for all spinor genera in a given half genus. By comparing (4.5) and (4.6), we see that H(z) changes sign as we switch between these half genera.

Definition 4.3. For convenience, if t is square free and there is some m ∈ tℤ^{2} with a_{H}(m) ≠ 0, then we say that tℤ^{2} is a spinor square class because of its close connection with the spinor genera in the genus of Q.

We also say that a locally represented number m is of (spinor) exceptionaltype if a_{E}(m) = |a_{H}(m)|, and non-exceptional otherwise. We see that m is of exceptional-type exactly when it is extremal in the sense of Lemma 5.8(a).

Remark. Schulze-Pillot [SP1] has given a complete local characterization of numbers of exceptional-type, extending the sufficient conditions given by Hsia in [Hs].

We will need effective upper bounds for the Fourier coefficients of a normalized newform of weight \frac{n}{2}. When n is odd this comes in two parts. Within a square class the Shimura lift gives a good bound, but for square free numbers additional analytic information is needed. Since the state-of-the-art for these square free estimates is constantly changing, we include their precise statement as an assumpton in what follows. When n is even, no assumptions are required.

Assumption 4.4. Suppose n ≥ 3 is odd and f ∈ S_{n/2}(N, χ) is an eigenform for all Hecke operators T_{p^{2}} for all p ∤ N and normalized so that < f, f >= 1. Then for all square free t > 0 we have

|a(t)| < B_{ε′}t^{\frac{n−1}{4} −η+ε′}

for some ε′ > 0 and 0 ≤ η ≤ \frac{1}{4}, and some effective constant B_{ε′} > 0.

As of this writing, the best known exponent is η = \frac{1}{12} in [Co-Iw], however the implied constant is probably quite large and has not been made explicit. Even for smaller η, where the constant is probably smaller, the author knows of no results with an explicit constant.

For simplicty, we adopt the following convention for decomposing a cusp form for Γ_{0}(N) as a linear combination of Hecke eigenforms.

If g(z) = ∑_{m≥0} b(m)e(mz) has integral weight k, then we write

(4.7)  g(z) = ∑_{i=1}^{r} γ_{i} g_{i}(z)

where g_{i}(z) = ∑_{m=1}^{∞} b_{i}(m)e(mz) and the g_{i}(z) are eigenforms for all T_{p} with p ∤ N and normalized so that their first non-zero Fourier coefficient is 1. By the theory of newforms [At-Le] and Deligne’s bound on Hecke eigenvalues [De] we have |b_{i}(m)| ≤ τ(m)m^{\frac{k−1}{2}} , therefore

(4.8)  |b(m)| ≤ τ(m)m^{\frac{k−1}{2}} ∑_{i=1}^{r} |γ_{i}|.

If f(z) has half-integral weight \frac{n}{2} with n ≥ 3 odd, and f ∈ U^{⊥}(N, χ) when n = 3, then we write f(z) = ∑_{k} δ_{k}f_{k}(z) where f_{k}(z) = ∑_{m=1}^{∞} a_{k}(m)e(mz) are Hecke eigenforms for all T_{p^{2}} with p ∤ N and normalized so that < f_{k}, f_{k} >= 1. Then δ_{k}Shi(f_{k}, t) = γ_{i}g_{i}(z) for some i with g_{i} as in the integral weight case above, so γ_{i} = δ_{k}a_{k}(t). When t is fixed, by Möebuis inversion, this together with Deligne’s bound on Hecke eigenvalues and τ(m/d) ≤ τ(m) gives the estimate

(4.9)  |a(m)| ≤ τ(m_{0})^{2}m0^{\frac{n−2}{2}} ∑_{i=1}^{r} |γ_{i}|

where m = tm_{0}^{2}. However, as t varies we can use Assumption 4.4 to obtain

(4.10)  |a(m)| ≤ (B_{ε′} ∑_{k} |δ_{k}|) t^{\frac{n−1}{4} −η+ε′}τ(m_{0})^{2}m0^{\frac{n−2}{2}} .

§5 The main term

In this section we establish some exact formulas and precise lower bounds for a_{E}(m) and a_{E}(m) + a_{H}(m) by combining our reduction procedure with Siegel’s product formula and results from §4. While we will only need information over ℚ for our main results, we initially state and prove a bound for a_{E}(m) more generally over a totally real number field F, obtaining the lower bound over ℚ as a corollary. Precise statements about a_{E}(m) + a_{H}(m) over F could be proved similarly by replacing §4 with the analogous facts for Hilbert modular forms.

Following Siegel, for m ∈ 𝔬_{F} and a totally definite integral quadratic form Q defined over F we define the local representation density β_{v}(m) at a place v of F by

(5.1)  β_{v}(m) = lim_{U→\{m\}} \frac{Vol (Q^{−1}(U))}{Vol(U)},

where U is an open neighborhood of m ∈ F_{v}. (Here we use the usual measure on ℝ, and the Haar measure on F_{𝔭} normalized so that Vol(𝔬_{𝔭}) = 1.) The β_{v}(m) give a measure of the number of local solutions of Q(x) = m over F_{v}. If v is a finite place corresponding to the prime 𝔭 of F, then by reduction mod 𝔭^{ν} we may rewrite β_{v}(m) as

(5.2)  β_{𝔭}(m) = lim_{ν→∞} \frac{r_{𝔭^{ν}} (m)}{q^{(n−1)(ν−1)}} ,

where q = N_{F/ℚ}(𝔭). When n ≥ 3, Siegel’s product formula ([Si1], [KuRa1-2], [Ra]) states that

(5.3)  a_{E}(m) = ∏_{v} β_{v}(m)

where the product runs over all places v of F.

Definition 5.1. For our purposes, it will also be convenient to consider β_{𝔭}(m) which satisfy certain congruence conditions at 𝔭 (i.e. β_{𝔭}^{Good}(m), β_{𝔭}^{Bad}(m), etc.), which are defined by imposing these conditions on r_{𝔭^{ν}} (m) in (5.2).

Definition 5.2. We say that m = (m_{v})_{v} ∈ 𝔸_{𝔬_{F}} is locally represented by Q when β_{v}(m_{v}) ≠ 0 for all places v of F. (This agrees with Definition 3.5 when m ∈ 𝔬_{F}.) It is equivalent to saying a_{E}(m) = 0 where a_{E}(m) is defined by (5.3), and we understand β_{v}(m) to mean β_{v}(m_{v}). We similarly say that m is 𝔭-stable when m_{𝔭} is 𝔭-stable, and that m is stable if m is 𝔭-stable for all primes 𝔭. We likewise extend our definition for m to be supported on some set 𝕊 of primes.

Definition 5.3. We say that a representation x⃗ of m by Q is 𝔭-primitive if its image in R_{Q, 𝔭^{k}}(m) is not of Zero-type, and we say that x⃗ is primitive if it is 𝔭-primitive for all primes 𝔭. We denote the number of primitive representations of m by Q as r_{Q}^{∗}(m).

We define a_{E}^{∗}(m) and a_{E}^{∗}(m)+a_{H}^{∗}(m) analogously to (4.5) and (4.6) as weighted averages of r_{Q}^{∗}(m) over Gen(Q) and Spn(Q) respectively. One can similarly prove a primitive product formula

(5.4)  a_{E}^{∗}(m) = ∏_{v} β_{v}^{Good ∪ Bad}(m)

using (5.3).

Theorem 5.4. Let Q be a totally definite integral quadratic form of dimension n ≥ 3 defined over a totally real number field F. Suppose that m = T′(m′)^{2} ∈ T′𝔬_{F}^{2} where T′ is 𝔭-stable at all primes 𝔭 | m′. Then

a_{E}(m) = a_{E}(T′) ∏_{\begin{matrix}𝔭|m′ \\ 𝔭 Iso \end{matrix}} [C_{𝔭}(T′) + \frac{1 − C_{𝔭}(T′)}{q^{(n−2)ν_{𝔭}}}] N_{F/ℚ}((m′)_{Iso})^{n−2},

where ν_{𝔭} = ord_{𝔭}(m′) and C_{𝔭}(T′) is defined after (5.8) below.

Proof. We proceed by computing the growth of each of the local factors β_{v}(m) separately, and use Siegel’s product formula (5.3) to assemble our results.

At a real valuation v | ∞, Siegel [Si, Hilfssatz 26, pp554–5] has computed

(5.5)  β_{v}(m) = \frac{n ω_{n} D^{− \frac{1}{2}}}{2} m^{\frac{n−2}{2}}

where ω_{n} is the volume of the n-sphere ∑_{i=1}^{n} x_{i}^{2} ≤ 1 in ℝ^{n}.

At a prime valuation v associated to 𝔭, we give a lower bound for the ratios of local factors within a square class using our reduction formula. In estimating these ratios, we assume m ∈ T′𝔬_{𝔭}^{2} for some fixed T′, and consider several cases:

m is not 𝔭-stable: In this case, applying the map π_{Z} for k >> 1 gives the weaker inequality

(5.6)  \frac{β_{𝔭}(m π_{𝔭}^{2})}{β_{𝔭}(m)} = \frac{r_{𝔭^{k}}(m π_{𝔭}^{2})}{r_{𝔭^{k}}(m)} ≥ \frac{r_{𝔭^{k}}^{Zero}(m π_{𝔭}^{2})}{r_{𝔭^{k}}(m)} = \frac{1}{q^{n−2}} .

𝔭 anisotropic and m is 𝔭-stable: By Lemma 3.8, 𝔭 is anisotropic iff we have equality in (5.6) above.

𝔭 isotropic and T′ is 𝔭-stable: For convenience, we let ν = ord_{𝔭}(m′) and let K = K(T′) = β_{𝔭}^{Good ∪Bad}(π_{𝔭}^{2}T′). By our 𝔭-stability assumption and repeated application of the map π_{Z}, we have

(5.7)  β_{𝔭}(m) = K + \frac{K}{q^{n−2}} + ⋯ + \frac{K}{q^{(n−2)(ν−1)}} + \frac{β_{𝔭}(T′)}{q^{(n−2)ν}} .

Therefore

(5.8)  \begin{aligned}\frac{β_{𝔭}(m)}{β_{𝔭}(T′)} & = \frac{1}{q^{(n−2)ν}} + \frac{K}{β_{𝔭}(T′)} \frac{(\frac{1}{q^{n−2}} )^{ν} − 1}{\frac{1}{q^{n−2}} − 1} \\ & = \frac{(1 − C_{𝔭}) + C_{𝔭} q^{(n−2)ν}}{q^{(n−2)ν}} \end{aligned}

where C_{𝔭} = C_{𝔭}(T′) = \frac{q^{n−2}}{q^{n−2}−1} \frac{K}{β_{𝔭}(T′)}. Lemma 3.8 tells us that C_{𝔭} > 0. □

Remark 5.4.1.

(a) From the proof of Theorem 5.4 and (5.4), we can see that if m is locally represented and 𝔭-stable, then

\begin{matrix}\frac{a_{E}(m π_{𝔭}^{2ν})}{q^{ν}} & ⟶ C_{𝔭}(m) a_{E}(m) \\ \frac{a_{E}^{∗}(m π_{𝔭}^{2ν})}{q^{ν}} & ⟶ a_{E}^{∗}(m) \end{matrix} \} when 𝔭 is isotropic,

monotonically as ν → ∞, and

a_{E}(m π_{𝔭}^{2ν}) = a_{E}(m) when 𝔭 is anisotropic.

(b) When ord_{𝔭}(T′) ≤ 1 we may easily compute C_{𝔭}(T′) since there are no Zerotype or Bad-type solutions. Then

(5.9)  C_{𝔭}(T′) = \frac{q^{n−2}}{q^{n−2} − 1} \frac{β_{𝔭}^{Good}(T′π_{𝔭}^{2ν})}{β_{𝔭}^{Good}(T′)}

and a short computation with Lemma 9.1 gives the table:

Table 2 - Local constants C_{𝔭}(T′) for 𝔭 ∤ N appearing in (5.8)

C_{𝔭} for 𝔭 ∤ N  n odd  n even
𝔭 ∤ T′  \frac{q^{n−2}−q^{\frac{n−3}{2}} Φ_{n}(𝔭)}{q^{n−2}−1}  \frac{q^{\frac{n−2}{2}}}{q^{\frac{n−2}{2}} −Φ_{n}(𝔭)}
ord_{𝔭}(T′) = 1  \frac{q^{n−2}}{q^{n−2}−1}  \frac{q^{n−2}}{q^{n−2}−1}

Theorem 5.5. Let Q be a positive definite integral quadratic form in 3 variables defined over ℚ, and suppose that m = T̃m^{2} ∈ T̃ℤ^{2} where T̃ is very p-stable at all primes p | m̃. Then

a_{E}(m) + a_{H}(m) = [a_{E}(T̃) + ψT̃(m̃)a_{H}(T̃)] (m̃)_{Iso}

+ a_{E}(T̃) (m̃_{+} ∏_{p|m̃_{−}} [p^{ν_{p}} + 2 ∑_{μ=0}^{ν_{p}−1} p^{μ}] ∏_{p|m̃_{2}} [C_{p}(T̃)(p^{ν_{p}} − 1) + 1] − (m̃)_{Iso}) ,

where ν_{p} = ord_{p}(m̃), C_{p}(T̃) is as in (5.8), and (m̃)_{Iso} = m̃_{+}m̃_{−}m̃_{2} with m̃_{+} = (m̃)_{𝕊_{+}}, m̃_{−} = (m̃)_{𝕊_{−}}, m̃_{2} = (m̃)_{𝕊_{2}}, and

𝕊_{+} = {p ∈ VStable(T̃) ∩ Iso | ψT̃(p) = 1 and p ∤ NT̃},
𝕊_{−} = {p ∈ VStable(T̃) ∩ Iso | ψT̃(p) = −1 and p ∤ NT̃},
𝕊_{2} = {p ∈ VStable(T̃) ∩ Iso | p |N or ord_{p}(T̃) ≥ 2}.

Proof. To see VStable(T̃) ∩ Iso = 𝕊_{+} ⊔ 𝕊_{−} ⊔ 𝕊_{2}, notice that ψT̃(p) = 0 ⟹ p | NT̃ and also if ord_{p}(T̃) = 1 then p | t so p | N. Thus (m̃)_{Iso} = m̃_{+}m̃_{−}m̃_{2}. By taking T′ = T̃, Theorem 5.4 gives an exact formula for a_{E}(m). When n = 3, Table 2 yields the explicit formulas (5.10)

C_{p}(T̃) + \frac{1 − C_{p}(T̃)}{p^{ν}} = \frac{1}{p^{ν}} \{ \begin{matrix}p^{ν} & if p ∤ T̃ and ψT̃(p) = 1, \\ p^{ν} + 2 ∑_{μ=0}^{ν−1} p^{μ} & if p ∤ T̃ and ψT̃(p) = −1, \\ ∑_{μ=0}^{ν} p^{μ} & if ord_{p}(T̃) = 1. \end{matrix}

Since T̃ is very p-stable at all p | m̃, we also have a_{H}(m̃) = ψT̃(m̃) a_{H}(T̃)(m̃)_{Iso}. Combining these proves the theorem. □

To obtain precise lower bounds for a_{E}(m), it is useful to notice that

(5.11)  \frac{β_{𝔭}(T′π_{𝔭}^{2ν})}{β_{𝔭}(T′)} ≥ C′_{𝔭}(T′) for all ν ≥ 0,

where C′_{𝔭}(T′) = min{1, C_{𝔭}(T′)}. This follows since the ratio is 1 when ν = 0, and approaches C_{𝔭}(T′) monotonically as ν → ∞ by Remark 5.4.1(a).

From Table 2 we see that for n ≥ 3 and 𝔭 ∤ N, we have

(5.12)  C′_{𝔭}(T′) = 1 ⟺ \{ \begin{matrix}n = 3, or \\ n ≥ 4, 𝔭 ∤ T′, and Φ_{n}(𝔭) = (−1)^{n−1}. \end{matrix}

Definition 5.6. When n ≥ 4, for each T ∈ 𝔬_{F} we let Stable(T) be the set of all prime ideals 𝔭 in 𝔬_{F} such that T is 𝔭-stable. When n = 3 and F = ℚ, for each T ∈ ℤ we let VStable(T) be the set of all primes p ∈ ℤ such that T is very p-stable.

Let T be a finite union of square classes t(𝔸_{𝔬_{F}})^{2} with t ∈ 𝔸_{𝔬_{F}} and ord_{𝔭}(t) ≤ 1 at all primes 𝔭. By Remark 3.6.1 and Definition 4.2, there is some ideal 𝔰 ⊆ 𝔬_{F} such that any locally represented m is stable (resp. very stable when n = 3 and F = ℚ) when m_{𝔭} ∈ 𝔰_{𝔭} for all primes 𝔭. By choosing representatives of T which generate the ideals dividing 𝔰, we can find a minimal (finite) subset B_{T} ⊂ T such that any locally represented m ∈ T can be written as m = T′(m′)^{2} with T′ ∈ B_{T} and m′ ∈ 𝔸_{𝔬_{F}} supported on Stable(T′) (resp. VStable(T′)). Without any difficulty, we may also assume that ∏_{v} |T′|_{v} = 1 for all T′ ∈ B_{T}.

Since the ratios of local factors in (5.8) are all non-zero numbers, we know that m is locally represented ⟺ T′ is locally represented. Therefore all T′ ∈ B_{T} are locally represented.

Theorem 5.7. Let Q be a totally positive definite integral quadratic form of dimension n ≥ 3 defined over a totally real number field F.

Also, let C′_{𝔭}(T′) = min{1, C_{𝔭}(T′)} and let T = ⋃_{t∈𝕋} t(𝔸_{𝔬_{F}})^{2} where 𝕋 ⊂ 𝔸_{𝔬_{F}} is a (finite) set of representatives for the locally represented square classes in ∏_{𝔭|N} 𝔬_{𝔭}.

a) When n = 3, m is locally represented by Q, and m = T′(m′)^{2} for some T′ not contained in a spinor square class with m′ supported on 𝕊 ⊆ Stable(T′), we have the explicit lower bound

a_{E}(m) ≥ Λ_{3}(T′)N_{F/ℚ}((m′)_{Iso}),

where

Λ_{3}(T′) = a_{E}(T′) ∏_{\begin{matrix}𝔭∈𝕊, 𝔭|N \\ 𝔭 Iso \end{matrix}}C′_{𝔭}(T′)

with C′_{𝔭}(T′) defined as in (5.11).

Moreover, if we have an effective estimate L_{F} (1, ψ_{t}) ≥ C_{ε}N_{F/ℚ}(t)^{−ε} for some ε > 0 with C_{ε} > 0 as t runs over square free t ∈ 𝔬_{F} > 0, then we have

a_{E}(m) ≥ Λ̂_{3} N_{F/ℚ}(t)^{\frac{1}{2} −ε}N_{F/ℚ}((m_{0})_{Iso}) for all m ∈ 𝔬_{F} ,

where m = tm_{0}^{2} and

Λ̂_{3} = \frac{C_{ε} (2πD^{− \frac{1}{2}} )^{[F:ℚ]}}{ζ_{F} (2)} min_{T′∈B_{T}} \{N_{F/ℚ}((T′/t)_{Aniso})^{\frac{1}{2}} ∏_{\begin{matrix}𝔭∈Stable(T′) \\ 𝔭|N, 𝔭 Iso \end{matrix}} C′_{𝔭}(T′) ∏_{𝔭|N} \frac{β_{𝔭}(T′)}{1 + q}\} .

b) When n = 4 and m is locally represented by Q, we have the lower bound:

a_{E}(m) ≥ Λ̂_{4} N_{F/ℚ}((m)_{Iso}) ∏_{\begin{matrix}𝔭|m, 𝔭∤N \\ χ(𝔭)=−1 \end{matrix}} \frac{q − 1}{q + 1} for all m ∈ 𝔬_{F} > 0,

where

Λ̂_{4} = min_{T′∈B_{T}} \{\frac{(2ω_{4}D^{− \frac{1}{2}} )^{[F:ℚ]}}{N_{F/ℚ}((T′)_{Aniso}) L_{F} (2, χ)} ∏_{𝔭|N} \frac{β_{𝔭}(T′)}{(1 − \frac{χ(𝔭)}{q^{2}})} ∏_{\begin{matrix}𝔭∈Stable(T′) \\ 𝔭|N, 𝔭 Iso \end{matrix}} C′_{𝔭}(T′)\} .

c) When n ≥ 5 and m is locally represented by Q, we have the lower bound:

a_{E}(m) ≥ Λ̂_{n} N_{F/ℚ}(m)^{\frac{n−2}{2}} for all m ∈ 𝔬_{F} > 0,

where the constants Λ̂_{n} for n ≥ 5 are given by

Λ̂_{n} = (\frac{n ω_{n} D^{− \frac{1}{2}}}{2} )^{[F:ℚ]} \{ \begin{matrix}\frac{ζ_{F} (n−2)E_{n}(X)}{ζ_{F} (n−1)ζ_{F} ( \frac{n−1}{2} )} min_{T′∈B_{T}} \{B̂_{n}(T′)\} & if n is odd, \\ \frac{ζ_{F} (n−2)}{L_{F} ( \frac{n}{2} ,Φ_{n})ζ_{F} ( \frac{n−2}{2} )^{2}} min_{T′∈B_{T}} \{B̂_{n}(T′)\} & if n is even, \end{matrix}

with

B̂_{n}(T′) = \{ \begin{matrix}∏𝔭|N \frac{β_{v}(T′)}{1+q^{(n−1)/2}} & if n is odd, \\ ∏𝔭|N \frac{β_{v}(T′)}{(1− \frac{Φn(𝔭)}{q^{n/2}} )(1− \frac{1}{q^{(n−2)/2}} )} & if n is even, \end{matrix}

and

E_{n}(X) = \begin{matrix}∏ & p<X & (\frac{1}{1+ \frac{1}{p^{s}}} ) & [F:ℚ] \\ ∏ & N𝔭≤X & (1 + \frac{1}{N𝔭^{s}} ) \end{matrix} (\frac{ζ_{ℚ}(2s)}{ζ_{ℚ}(s)})^{[F:ℚ]} ,

for any X.

Proof. Fix some T′ ∈ 𝔸_{𝔬_{F}} with ord_{𝔭}(T′) ≤ 1 at all 𝔭 ∤ N and satisfying the product formula ∏_{v} |T′|_{v} = 1, and consider m = T′(m′)^{2} ∈ T′(𝔸_{𝔬_{F}})^{2} with m′ supported on Stable(T′). Combining (5.5) and our estimates β_{𝔭}(m) ≥ C′_{𝔭}(T′)β_{𝔭}(T′) from (5.11), we see that

(5.13)  a_{E}(m) ≥ a_{E}(T′) N_{F/ℚ}((m′)_{Iso})^{n−2} ∏_{\begin{matrix}𝔭∈Stable(T′) \\ 𝔭 Iso \end{matrix}} C′_{𝔭}(T′).

The first part of (a) follows from (5.12) since C′_{𝔭}(T′) = 1 for all 𝔭 ∤ N when n = 3.

For the remaining cases, we must establish a bound over all square classes t(𝔸_{𝔬_{F}})^{2}. This is related to the generic behavior of the local factors as t varies, and depends on the parity of n. After accounting for the local factors at 𝔭 ∤ N we will be left with finitely many lower bounds for a_{E}(m), and taking their minimum proves the theorem. This generic behavior was first investigated by Siegel [Si, Hilfssatz 12, pp539–42], who showed that it is related to certain special values of L-functions.

Suppose n is even: When n is even, the quadratic character Φ_{n} associated to Q is independent of m, and the generic part of ∏_{v} β_{v}(m) here is essentially L(\frac{n}{2}, Φ_{n}). Explicitly, since ord_{𝔭}(T′) ≤ 1 at all 𝔭 ∤ N, we can use Table 1 to compute

∏_{𝔭∤N} β_{𝔭}(T′) = ∏_{𝔭∤NT′} (1 − \frac{Φ_{n}(𝔭)}{q^{\frac{n}{2}}}) ∏_{𝔭|T′, 𝔭∤N} \frac{(q^{\frac{n−2}{2}} + Φ_{n}(𝔭)) (q^{\frac{n}{2}} − Φ_{n}(𝔭))}{q^{n−1}}

= ∏_{𝔭∤N} (1 − \frac{Φ_{n}(𝔭)}{q^{\frac{n}{2}}}) ∏_{𝔭|T′, 𝔭∤N} \frac{(q^{\frac{n−2}{2}} + Φ_{n}(𝔭)) (q^{\frac{n}{2}} − Φ_{n}(𝔭))}{q^{\frac{n−2}{2}} (q^{\frac{n}{2}} − Φ_{n}(𝔭))}

≥ ∏_{𝔭∤N} (1 − \frac{Φ_{n}(𝔭)}{q^{\frac{n}{2}}}) ∏_{\begin{matrix}𝔭|T′, 𝔭∤N \\ Φ_{n}(𝔭)=−1 \end{matrix}} (1 − \frac{1}{q^{\frac{n−2}{2}}} ) .

When n ≥ 6 we can estimate this directly, getting

a_{E}(T′) ≥ \frac{N_{F/ℚ}(T′)^{\frac{n−2}{2}}}{L_{F} (\frac{n}{2}, Φ_{n})ζ_{F} (\frac{n−2}{2})} ∏_{𝔭|N} \frac{β_{𝔭}(T′)}{(1 − \frac{Φ_{n}(𝔭)}{q^{\frac{n}{2}}}) (1 − _{\frac{1}{q^{\frac{n−2}{2}}}} )} ∏_{v|∞} β_{v}(1),

while for n = 4 we have

a_{E}(T′) ≥ \frac{N_{F/ℚ}(T′)}{L_{F} (2, χ)} ∏_{𝔭|N} \frac{β_{𝔭}(T′)}{(1 − \frac{χ(𝔭)}{q^{2}})} ∏_{v|∞} β_{v}(1) ∏_{\begin{matrix}𝔭|T′, 𝔭∤N \\ χ(𝔭)=−1 \end{matrix}} (1 − \frac{1}{q}) .

From Table 2 we see that C′_{𝔭}(T′) = 1 unless 𝔭 ∤ T′ and Φ_{n}(𝔭) = −1, in which case C′_{𝔭}(T′) is the local factor at 𝔭 of ζ_{F} (n−2)/ζ_{F} (\frac{n−2}{2}). When n ≥ 6, this together with (5.13) and (5.5) gives

a_{E}(m) ≥ N_{F/ℚ}(m)^{\frac{n−2}{2}} \frac{ζ_{F} (n − 2) (\frac{n}{2}ω_{n}D^{− \frac{1}{2}} )^{[F:ℚ]}}{L_{F} (\frac{n}{2}, Φ_{n})ζ_{F} (\frac{n−2}{2})^{2}} ∏_{𝔭|N} \frac{β_{𝔭}(T′)}{(1 − \frac{Φ_{n}(𝔭)}{q^{\frac{n}{2}}}) (1 − _{\frac{1}{q^{\frac{n−2}{2}}}} )},

and when n = 4 this gives

a_{E}(m) ≥ \frac{N_{F/ℚ}((m)_{Iso}) (2ω_{4}D^{− \frac{1}{2}} )^{[F:ℚ]}}{N_{F/ℚ}((T′)_{Aniso}) L_{F} (2, χ)} ∏_{𝔭|N} \frac{β_{𝔭}(T′)}{(1 − \frac{χ(𝔭)}{q^{2}})} ∏_{\begin{matrix}𝔭∈Stable(T′) \\ 𝔭|N, 𝔭 Iso \end{matrix}}C′_{𝔭}(T′)

∏_{\begin{matrix}𝔭|T′, 𝔭∤N \\ χ(𝔭)=−1 \end{matrix}} (1 − \frac{1}{q}) ∏_{\begin{matrix}𝔭|m′, 𝔭∤N \\ χ(𝔭)=−1 \end{matrix}}(1 + \frac{1}{q})^{−1} .

Taking the minimum over all T′ ∈ B_{T} as above gives (b) for n = 4 and (c) when n ≥ 6 and even.

Suppose n is odd: When n is odd, then the generic factor is more complicated since the quadratic character Φ_{n} depends on m (or more precisely, its square free part t). Since ord_{𝔭}(T′) ≤ 1 at all 𝔭 ∤ N, we have

∏_{𝔭∤N} β_{𝔭}(T′) = ∏_{𝔭∤NT′} (1 + \frac{Φ_{n}(𝔭)}{q^{\frac{n−1}{2}}}) ∏_{p∤N, 𝔭|T′} (1 − \frac{1}{q^{n−1}} )

= \frac{L_{F} (\frac{n−1}{2}, Φ_{n})}{ζ_{F} (n − 1)} ∏_{𝔭|N} \frac{1 − \frac{Φ_{n}(𝔭)}{q^{\frac{n−1}{2}}}}{1 − \frac{1}{q^{n−1}}} ,

since Φ_{n}(𝔭) = 0 for all 𝔭 | T′ where 𝔭 ∤ N. From this we have

a_{E}(T′) ≥ \frac{L_{F} (\frac{n−1}{2}, Φ_{n})}{ζ_{F} (n − 1)}N_{F/ℚ}(T′)^{\frac{n−2}{2}} ∏_{𝔭|N} \frac{β_{𝔭}(T′)}{1 + q^{\frac{n−1}{2}}} ∏_{v|∞} β_{v}(1).

From Table 2 we see that when n is odd, C′_{p}(T′) = 1 unless n ≥ 5 and Φ_{n}(𝔭) = 1, in which case C′_{𝔭}(T′) is the local factor at 𝔭 of ζ_{F} (n − 2)/ζ_{F} (\frac{n−1}{2}). Combining this with (5.13) and (5.5) gives

a_{E}(m) ≥ N_{F/ℚ}(m)^{\frac{n−2}{2}} \frac{L_{F} (\frac{n−1}{2}, Φ_{n})ζ_{F} (n − 2)}{ζ_{F} (n − 1)ζ_{F} (\frac{n−1}{2})} (\frac{n}{2}ω_{n}D^{− \frac{1}{2}} )^{[F:ℚ]} ∏_{𝔭|N} \frac{β_{𝔭}(T′)}{1 + q^{\frac{n−1}{2}}}

when n ≥ 5.

To find a lower bound for L_{F} (1, Φ_{n}) when s ∈ ℝ > 1, we write

L_{F} (s, Φ_{n}) ≥ ∏_{N𝔭≤X} (1 + \frac{1}{N𝔭^{s}} )^{−1} ∏_{N𝔭>X} (1 + \frac{1}{N𝔭^{s}} )^{−1}

for some parameter X, and estimate

∏_{N𝔭>X} (1 + \frac{1}{N𝔭^{s}} ) ≤ ∏_{p>X} (1 + \frac{1}{p^{s}} )^{d} = (\frac{ζ_{ℚ}(s)}{ζ_{ℚ}(2s)})^{d} ∏_{p<X} (\frac{1}{1 + \frac{1}{p^{s}}} )^{d}

where d = [F : ℚ], by taking all primes 𝔭 | p to split completely. This shows

L_{F} (s, Φ_{n}) ≥ \begin{matrix}∏ & p<X & (\frac{1}{1+ \frac{1}{p^{s}}} ) & d \\ ∏ & N𝔭≤X & (1 + \frac{1}{N𝔭^{s}} ) \end{matrix} (\frac{ζ_{ℚ}(2s)}{ζ_{ℚ}(s)})^{d}

which becomes optimal as X → ∞, and gives (c) by taking the minimum of the β_{𝔭}(T′) at 𝔭 | N over all T′ in the lower bound for a_{E}(m) above.

When n = 3 we have C′_{𝔭}(T′) = 1 for all 𝔭 ∤ N, so

a_{E}(m) ≥ \frac{N_{F/ℚ}(t(m_{0})_{Iso}^{2})^{\frac{1}{2}}}{N_{F/ℚ}((T′/t)_{Aniso})^{− \frac{1}{2}}} L_{F} (1, ψ)\frac{(2πD^{− \frac{1}{2}} )^{[F:ℚ]}}{ζ_{F} (2)} ∏_{𝔭|N} \frac{β_{𝔭}(T′)}{1 + q} ∏_{\begin{matrix}𝔭∈Stable(T′) \\ 𝔭|N, 𝔭 Iso \end{matrix}} C′_{𝔭}(T′).

If we know L_{F} (1, ψ) = L_{F} (1, χ_{(t)}) ≥ C_{ε}N_{F/ℚ}(t)^{−ε}, then we have

a_{E}(m) ≥ \frac{N_{F/ℚ}(t(m_{0})_{Iso}^{2})^{\frac{1}{2}}}{N_{F/ℚ}(t)^{ε}} \frac{C_{ε} (2πD^{− \frac{1}{2}} )^{[F:ℚ]}}{ζ_{F} (2) N_{F/ℚ}((T′/t)_{Aniso})^{− \frac{1}{2}}} ∏_{𝔭|N} \frac{β_{𝔭}(T′)}{1 + q} ∏_{\begin{matrix}𝔭∈Stable(T′) \\ 𝔭|N, 𝔭 Iso \end{matrix}} C′_{𝔭}(T′),

which similarly gives the second part of (a). □

Remark 5.7.1. When the class number of F is 1, it is unnecessary to consider the adelic square classes t(𝔸_{𝔬_{F}})^{2} in Theorem 5.7, and we can freely replace them by the more conventional square classes t𝔬_{F}^{2}. In particular, when F = ℚ we are dealing with the square classes tℤ^{2}.

Lemma 5.8. Suppose n = 3 and F = ℚ.

a) We have the general inequality

|a_{H}(m)| ≤ a_{E}(m).

b) Given any set 𝕋 ⊆ VStable(m), we have the refined inequality

|a_{H}(m)| ≤ ( ∏_{\begin{matrix}p∈𝕋, \\ ψ(p)≠0 \end{matrix}} C′_{p}(m)) a_{E}(m),

with equality iff m is of exceptional-type (in which case |a_{H}(m)| = a_{E}(m)). Suppose H(z) ≠ 0, tℤ^{2} is a spinor square class, and ψ_{t} is its associated quadratic character. Then t | N and

c) ψ_{t}(p) = 0 for all anisotropic primes p.

d) If m is of exceptional-type (so m ∈ tℤ^{2}), then C′_{p}(m) = 1 for all primes p ∈ VStable(m) with ψ_{t}(p) ≠ 0.

Proof. To see (a), by the discussion after (4.6), if |a_{H}(m)| > a_{E}(m) then a_{H}(m) > a_{E}(m) for some spinor genus in Gen(Q), contradicting (4.6) ≥ 0.

For (b), we let 𝕊 = {p ∈ 𝕋 | ψ(p) ≠ 0 and C_{p}(m) < 1} and let m_{1} → ∞_{𝕊} mean that m_{1} ∈ ℕ is divisible only by primes p ∈ 𝕊 and that m_{1} → ∞ in such a way that ord_{p}(m_{1}) → ∞ for all p ∈ 𝕊.

From Remark 5.4.1(a), we see that

lim_{m_{1}→∞_{𝕊}} \frac{a_{E}(mm_{1}^{2})}{m_{1}} = a_{E}(m) ∏_{𝔭∈𝕊} C_{p}(m)

and

lim_{m_{1}→∞_{𝕊}} \frac{|a_{H}(mm_{1}^{2})|}{m_{1}} = |a_{H}(m)|,

hence the inequality in (b) follows from (a).

Now suppose we have equality in (b). Then we wish to show

lim_{m_{1}→∞_{𝕊}} \frac{a_{E}^{∗}(mm_{1}^{2}) + a_{H}^{∗}(mm_{1}^{2})}{a_{E}^{∗}(mm_{1}^{2})} = 0

which contradicts [SP3, Korollar 2, pp130-131] unless m is of exceptional-type. By Remark 5.4.1(a) we see that lim_{m_{1}→∞_{𝕊}} \frac{a_{E}^{∗}(mm_{1}^{2})}{m_{1}} ≠ 0 since all p ∈ 𝕊 are isotropic, thus we are reduced to showing

lim_{m_{1}→∞_{𝕊}} \frac{a_{E}^{∗}(mm_{1}^{2}) + a_{H}^{∗}(mm_{1}^{2})}{m_{1}} = 0.

However, this follows from our assumption by evaluating the limit inside the defining divisor sum (which has finitely many terms since 𝕊 is a finite set).

Finally, if m is of exceptional-type then we have equality in (a), hence (a) and (b) coincide and C_{p}(m) ≥ 1 for all p ∈ VStable(m) with ψ(p) ≠ 0, proving (d). □

Remark 5.8.1. Since the character ψ_{t}(·) = Φ_{3}(·) = (\frac{−tD}{·} ) is associated to the pair (Q, tℤ^{2}) while the anisotropic primes are related only to Q, we do not expect a general relationship between the anisotropic primes and those primes with ψ_{t}(p) = 0. However, for a spinor square class tℤ^{2}, Lemma 5.8(c) and (d) gives that {p ∈ VStable(T′) | C′_{p}(T′) = 1} ⊆ {p | ψ_{t}(p) ≠ 0} ⊆ Iso when T′ ∈ tℤ^{2} is of exceptionaltype.

Theorem 5.9. Let Q be a positive definite integral quadratic form in 3 variables defined over ℚ, and suppose that m = T̃m^{2} is locally represented by Q and contained in a spinor square class, and that m̃ is supported on VStable(T̃). Then we have the following lower bounds for a_{E}(m) + a_{H}(m):

a) If T̃ is non-exceptional, then

(i) When a_{H}(m) = 0, we may apply Theorem 8.3(a) to obtain

a_{E}(m) + a_{H}(m) ≥ Λ_{3}(T̃)(m̃)_{Iso}.

(ii) When a_{H}(m) ≠ 0, we have the lower bound

a_{E}(m) + a_{H}(m) ≥ K_{0}(T̃)(m̃)_{Iso}

where

K_{0}(T̃) = a_{E}(T̃) ∏_{\begin{matrix}p∈VStable(T̃) \\ p|N, ψ_{T}(p)≠0 \end{matrix}} C′_{p}(T̃) − |a_{H}(T̃)| > 0.

b) If T̃ is of exceptional-type and m is non-exceptional, then

(i) When a_{H}(m) ≥ 0, we may apply Theorem 5.7(a) to obtain

a_{E}(m) + a_{H}(m) ≥ Λ_{3}(T̃)(m̃)_{Iso}.

(ii) When a_{H}(m) < 0, we have the lower bound

a_{E}(m) + a_{H}(m) ≥ a_{E}(T̃) m̃_{+} 2^{ϵ} (σ(m̃_{−}) − m̃_{−}) m̃_{2} ∏_{p|m̃_{2}} [\frac{(C_{p}(T̃) − 1)(p − 1)}{p}] ,

where (m̃)_{Iso} = m̃_{+}m̃_{−}m̃_{2} is as in Theorem 5.5 and ϵ is defined to be 0 if m̃_{−} = 1 and 1 if m̃_{−} > 1. Moreover, this lower bound is an equality when for some prime p either m̃_{−} = p^{ν} and m̃_{2} = 1, or m̃_{−} = 1 and m̃_{2} = p.

Proof. Part (i) of (a) and (b) follows directly from Theorem 5.7(a).

Part (ii) of (a): Since a_{H}(m) ≠ 0, both a_{H}(T̃) and ψT̃(m̃) ≠ 0. By Lemma 5.8(c), this implies that all p | m̃ are isotropic. Using Theorem 5.7(a) with T′ = T̃ and 𝕊 = {p ∈ VStable(T̃) | ψT̃(p) ≠ 0}, we see

a_{E}(m) + a_{H}(m) ≥ [a_{E}(T̃) ∏_{p∈𝕊} C′_{p}(T̃) − |a_{H}(T̃)|] (m̃)_{Iso},

which is > 0 by Lemma 5.8(b) since T̃ is not of exceptional-type.

Part (ii) of (b): Theorem 5.5 gives an exact formula for a_{E}(m) + a_{H}(m), and our conditions imply that its first term vanishes. Since m is non-exceptional, from the remaining term we see that m̃_{−}m̃_{2} > 1.

We obtain our estimate by applying Theorem 5.5 in stages (going from T̃_{0} = T̃m_{+}^{2}, to T̃_{1} = T̃_{0}m_{−}^{2}, to m = T̃_{1}m_{2}^{2}), and rounding off by assuming |a_{H}(T̃_{i})| = a_{E}(T̃_{i}) at each stage.

For each p | m̃_{2}, we have

a_{E}(T̃_{i}p^{2ν}) + a_{H}(T̃_{i}p^{2ν}) ≥ a_{E}(T̃_{i})(1 + C_{p}(T̃_{i})(p^{ν} − 1)) − |a_{H}(T̃_{i})|p^{ν}

= a_{E}(T̃_{i})(C_{p}(T̃_{i}) − 1)(p^{ν} − 1)

≥ a_{E}(T̃_{i}) [\frac{(C_{p}(T̃_{i}) − 1)(p − 1)}{p}] p^{ν},

where T̃_{i} ∈ T̃_{1}ℤ^{2} and p ∤ T̃_{i}. By choosing T̃_{i+1} = T̃_{i}p^{2ν_{p}} and inducting on all p | m̃_{2}, we have

a_{E}(m) + a_{H}(m) ≥ [a_{E}(T̃_{1}) − |a_{H}(T̃_{1})|] ∏_{p|m̃_{2}} [\frac{(C_{p}(T̃) − 1)(p − 1)}{p}] ,

(Since as we vary T̃_{i} the ord_{p}(T̃_{i}) are the same whenever p | T̃_{i}, we see that C_{p}(T̃_{i}) = C_{p}(T̃).) From Theorem 5.5, we also have that

a_{E}(T̃_{1}) − a_{H}(T̃_{1}) ≥ [a_{E}(T̃_{0}) − |a_{H}(T̃_{0})|] 2^{ϵ} [σ(m̃_{−}) − m̃_{−}] ,

and

a_{E}(T̃_{0}) − |a_{H}(T̃_{0})| = a_{E}(T̃) m̃_{+},

which combine to prove the theorem. □

§6 Main Results

We now state some effective lower bounds which are sufficient to ensure that a number m is represented by a positive definite quadratic form Q in n ≥ 3 variables.

Theorem 6.1. Suppose Q is a positive definite quadratic form in 3 variables and t is square free. Then any sufficiently large m = T̃m^{2} ∈ tℤ^{2} locally represented by Spn(Q) with m̃ supported on VStable(T̃) and having a priori bounded divisibility at the anisotropic primes is represented by Q, except possibly when T̃ is anti-exceptional and m = T̃p^{2} for some prime p ∤ N with ψT̃(p) = −1.

If either T̃ is non-exceptional or a_{H}(m) ≥ 0, we see that m is represented when

\frac{\sqrt{(m_{0})_{Iso}}}{τ((m_{0})_{Iso})^{2}} > M_{3} τ((m_{0})_{Aniso})^{2}\sqrt{(m_{0})_{Aniso}},

where m = t(m_{0})^{2}, M_{3} = M_{3}(T̃) = ∑_{i} |γ_{i}|/[(T̃/t)Iso^{\frac{1}{2}} min\{Λ_{3}(T̃), K_{0}(T̃)\}] > 0 with

Λ_{3} and K_{0} as in Theorem 5.7(a) and Theorem 5.9(a)(ii) respectively, and the γ_{i} are as in (4.7) with g = Shi(f, t).

If T̃ is of exceptional-type and m is non-exceptional, then m is represented when

2^{ϵ} \frac{\sqrt{m̃_{+}m̃_{2}}}{τ((m_{0})_{Iso})^{2}} \frac{σ(m̃_{−}) − m̃_{−}}{\sqrt{m̃_{−}}} > \frac{∑_{i=1}^{r} |γ_{i}|}{a_{E}(T̃)} \frac{τ((m_{0})_{Aniso})^{2}\sqrt{(m_{0})_{Aniso}}}{∏p|m̃_{2} [\frac{(C_{p}(T̃)−1)(p−1)}{p}]} ,

where m = t(m_{0})^{2}, (m̃)_{Iso} = m̃_{+}m̃_{−}m̃_{2} with m̃_{±,2} as defined in Theorem 5.5, ϵ = 1 when m̃_{2} > 1 and zero otherwise, and the γ_{i} are as in (4.7) with g = Shi(f, t).

Proof. The general statement follows from the following two cases, noting that when m = T̃p^{2} as above our bound is no better than for m = T̃. Thus, if T̃ is not represented, we can say nothing about the representability of T̃p^{2}.

Case 1: If either T̃ is non-exceptional or a_{H}(m) > 0, then by Theorem 5.9 we have

a_{E}(m) ≥ min{Λ_{3}(T̃), K_{0}(T̃)}(m̃)_{Iso} > 0.

Combining this with (4.9) and solving to ensure a_{E}(m) > |a_{H}(m)| gives the desired bound.

Case 2: If T̃ is of exceptional-type and m is non-exceptional, then the result follows from combining Theorem 5.9(b)(ii) and (4.9) to ensure a_{E}(m) > |a_{H}(m)|. □

Theorem 6.2. Let Q be a positive definite integral quadratic form in 3 variables, and suppose L(1, ψ_{t}) ≥ C_{ε}t^{−ε} for some ε > 0 with C_{ε} > 0 as t runs over all positive square free numbers. Then any sufficiently large locally represented m ∈ ℕ not contained in a spinor square class is represented by Q. In fact, m is represented when

\frac{t^{η−(ε+ε′)}(m_{0})Iso^{\frac{1}{2}}}{τ((m_{0})_{Iso})^{2}} ≥ \frac{B_{ε′} ∑_{k} |δ_{k}|}{Λ̂_{3}}τ((m_{0})_{Aniso})^{2}(m_{0})Aniso^{\frac{1}{2}},

where m = tm_{0}^{2}, Λ̂_{3} is defined in Theorem 5.7(a), the δ_{k} are defined after (4.8), and η, ε′, and B_{ε′} are as in Assumption 4.4.

Proof. This follows by using Theorem 5.7(a) and (4.10) with n = 3, and solving to ensure a_{E}(m) > |a(m)|. □

From Theorem 6.1, we have a good understanding of which numbers are represented by Q within any given square class tℤ^{2}, though our information is only complete if tℤ^{2} is not one of the finitely many (spinor) exceptional-type square classes. Assuming the Riemann hypothesis for Dirichlet L-functions (or at least that they have no Siegel zeros) gives the effective lower bound L(1, χ_{(t)}) ≥ C_{ε}t^{−ε} necessary for Theorem 6.2, and allows us to uniformly understand the representation behavior across all but finitely many square classes. Obtaining complete information about the representation behavior within the (spinor) exceptional-type square classes is more subtle and depends on understanding the behavior of the Fourier coefficients of the cusp form f(z) in (4.4). We take up this issue in [Ha2].

Theorem 6.3. Let Q be a positive definite integral quadratic form in 4 variables. Then any sufficiently large locally represented m with a priori bounded divisibility at the anisotropic primes is represented by Q. In fact, m is represented when

\frac{\sqrt{(m)_{Iso}}}{τ((m)_{Iso})} ∏_{\begin{matrix}p∤N, p|m \\ χ(p)=−1 \end{matrix}} \frac{p − 1}{p + 1} > \frac{∑_{i=1}^{r} |γ_{i}|}{Λ̂_{4}}τ((m)_{Aniso})\sqrt{(m)_{Aniso}}

where Λ̂_{4} > 0 is as in Theorem 5.7(b), and the γ_{i} are as in (4.7).

Proof. This follows by combining Theorem 5.7(b) and (4.8) with k = \frac{n}{2} = 4 to ensure a_{E}(m) > |a(m)|. □

Remark. Schulze-Pillot [SP3] has given an explicit lower bound for representability when n = 4, though his bound is a bit coarser since it depends only on the level N of Q, not Q itself, thus is not very practical for computations.

For completeness, we state an explicit bound for the case n ≥ 5 as well, though there are already many such bounds available in the literature. (See [Wa] and [Hs-Ic] for example.)

Theorem 6.4. Suppose Q be a positive definite integral quadratic form in n ≥ 5 variables. Then any sufficiently large locally represented number m is represented by Q. In fact, m is represented when

\frac{m^{\frac{n−2}{4}}}{τ(m)} ≥ \frac{∑_{i=1}^{r} |γ_{i}|}{Λ̂_{n}} if n is even,

and under Assumption 1 when

\frac{t^{\frac{n−3}{4} +η−ε′}(m_{0})^{\frac{n−2}{2}}}{τ(m_{0})^{2}} ≥ \frac{B_{ε′} ∑_{k} |δ_{k}|}{Λ̂_{n}} if n is odd,

with m = tm_{0}^{2}, the γ_{i} and δ_{k} as defined at the end of §3, and Λ̂_{n} as in Theorem 5.7(c).

Proof. This follows by combining Theorem 5.7(c), (4.8), and (4.10) with k = \frac{n}{2} to ensure a_{E}(m) > |a(m)|. □

Theorem 6.5. Suppose p is anisotropic and m is p-stable. Then

r_{Q}(mp^{2ν}) = r_{Q}(m) for all ν ≥ 0.

Proof. It suffices to prove this for ν = 1, since the theorem follows by repeated application of this result. Notice that since every representation Q(x⃗) = m gives rise to a representation Q(px⃗) = mp^{2}, we have the general inequality r_{Q}(mp^{2}) ≥ r_{Q}(m).

Suppose p is anisotropic. Then by Remark 5.4.1(a) we have a_{E}(mp^{2}) = a_{E}(m). However (4.5) expresses a_{E}(m) as an average of r_{Q′}(m) over all Q′ ∈ Gen(Q), so a_{E}(mp^{2}) = a_{E}(m) implies r_{Q′}(mp^{2}) = r_{Q′}(m) for all Q′ ∈ Gen(Q). □

Theorems 6.3 and 6.4 allow us to determine a finite set of numbers to check for representability, assuming bounded growth at anisotropic primes. Once this is done, for each m which is not represented we may use Theorem 6.5 to determine the representation behavior allowing anisotropic factors. While this does not guarantee the existence of only finitely many numbers which are not represented (given local representability), it does provide a practical procedure for determining the numbers represented by a positive definite integral quadratic form in 4 variables, and also in 3 variables aside from the sequences T̃p^{2} mentioned in Theorem 6.1.

For computational purposes, we now state some useful inequalities.

Lemma 6.6. For some fixed N ∈ ℕ and quadratic Dirichlet character χ, let

F_{4}(m) = \frac{\sqrt{m}}{τ(m)} ∏_{\begin{matrix}p∤N, p|m \\ χ(p)=−1 \end{matrix}} \frac{p − 1}{p + 1}.

Then F_{4}(m) is a multiplicative function and for any prime p, we have

F_{4}(mp^{ν}) ≥ F_{4}(m)

when either p ≥ 11 and ν ≥ 1, p = 7 or 5 and ν ≥ 2, p = 3 and ν ≥ 5, or p = 2 and ν ≥ 11.

Proof. Clearly F_{4}(ab) = F_{4}(a)F_{4}(b) when gcd (a, b) = 1, so F_{4}(m) is multiplicative. For the second part, we write m = m_{1}p^{ν_{1}} where p ∤ m_{1}. Then

F_{4}(mp^{ν}) = \frac{p^{ν/2}\sqrt{m_{1}p^{ν_{1}}}}{(1 + ν_{1} + ν)τ(m_{1})} ∏_{\begin{matrix}p′∤N, p′|mp \\ χ(p′)=−1 \end{matrix}} \frac{p′ − 1}{p′ + 1}

≥ \frac{p^{ν/2}(p − 1)}{p + 1} \frac{1 + ν_{1}}{1 + ν_{1} + ν} F_{4}(m),

and we are interested in when \frac{p^{ν/2}(p−1)}{p+1} \frac{1+ν_{1}}{1+ν_{1}+ν} ≥ 1. The most restrictive case is when ν_{1} = 0, and we can see that this is true when either p or ν is sufficiently large. For p ≥ 11, we see that any ν ≥ 1 will do, while for p = 2, 3, 5, or 7 we must have ν ≥ 11, 5, 2, or 2 respectively. □

Remark 6.6.1. The function F_{4}(m) in Lemma 6.6 is just the right side of the expression in Theorem 6.3 when m = (m)_{Iso} and N is the level of Q.

§7.1 Asymptotics

While it is not our main goal, the methods of the previous sections allow us to describe the asymptotic behavior of r_{Q}(m) as m → ∞. This is especially interesting when n = 3, though the exact nature of the main term depends on how m → ∞.

Suppose n = 3 and F = ℚ, fix a square class tℤ^{2}, and suppose m → ∞ is evenutally in T̃ℤ^{2} for some 𝕊-stable T̃, where m = T̃m^{2} and m̃ is supported on VStable(T̃). (If m → ∞ by becoming increasingly divisible by square factors, for example, then Remark 3.6.1 guarantees that m is eventually in some such T̃ℤ^{2}.) To make sensible estimates we require that sign(a_{H}(m)) → {0, ±1}, and for simplicity we also assume m̃ = (m̃)_{Iso}. Then using (4.9), Theorem 5.5, and estimating as in Theorem 5.9(b)(ii), we have

r_{Q}(m) = a_{E}(m) + a_{H}(m) + O(m^{\frac{1}{2} +ε})

where a_{H}(m) = ±′|a_{H}(m)| and

LL(m̃) ≤ [a_{E}(m) ±′ |a_{H}(m)|] − [a_{E}(T̃) ±′ |a_{H}(T̃)|] m̃ ≤ UU(m̃),

with

LL(m̃) = m̃_{+}m̃_{2}2^{ϵ}(σ(m̃_{−}) − m̃_{−}) ∏_{p|m̃_{2}} \frac{(C_{p}(T̃) ±′ 1)(p − 1)}{p}

and

UU(m̃) = m̃_{+}m̃_{2}2^{#{p|m̃_{−}}}(σ(m̃_{−}) − m̃_{−}) ∏_{p|m̃_{2}}(C_{p}(T̃) ±′ 1).

To produce a well-defined constant, we fix a set of isotropic primes 𝕊 such that ∑_{𝕊′} \frac{1}{p} < ∞ where 𝕊′ = \{p ∈ 𝕊 | ψ_{t}(p) = −1\}, and suppose m̃ → ∞_{𝕊}. In this case, we have the simpler estimate

a_{E}(m) ±′ |a_{H}(m)| ∼ (C_{3,𝕊} ±′ |a_{H}(T̃)|) m̃,

where C_{3,𝕊} is defined by the (finite) product C_{3,𝕊} = a_{E}(T̃) ∏_{p∈𝕊} C_{p}(T̃).

§7.2 Quadratic Polynomials

While our original intention was to describe (compute) the numbers represented by a positive definite integral quadratic form Q, these results can be easily modified to describe the numbers represented by an integer-valued quadratic polynomial P whose homogeneous degree 2 part is positive definite.

By completing the square, this is equivalent to asking which numbers are represented by Q(x⃗ + b⃗) + c for some b⃗ ∈ ℚ^{n} and c ∈ ℚ, where Q is the degree 2 part of P. Now clearing deminators by multiplying through by some d ∈ ℕ, the question becomes which m ∈ d𝔬_{F} are represented by Q(x⃗ + b⃗′) + c′ with b⃗′ ∈ ℤ^{n} and c′ ∈ ℤ. However this question can be answered by considering the congruence theta function, whose m^{th} coefficient counts x⃗ ∈ R_{Q}(m) with x⃗ ≡ b⃗′ (mod d). The same results for the usual theta function apply to the congruence theta function, which gives the same results for them as for Θ_{Q}(z), though the constants involved will be different. The modifications necessary to include congruence conditions are described in §2 of [Du-SP]. Roughly speaking, there is an analogue of the Siegel product formula due to Van der Blij [VB] whose local factors count solutions satisfying the desired congruence, and the theta functions are modular forms with the same weight (though they may have a different level N′ and be modular only for the group Γ_{1}(N′)). Since our reduction procedure will respect these congruences, and our error term bound depends only on general properites of cusp forms, the only change is to modify the local constants C_{p} at those primes p | d to account for the additional congruence condition.

§8 An Example

We now describe the numbers represented by the form Q = x^{2} +3y^{2} +5z^{2} +7w^{2}. This form has level N = 420, χ = (\frac{105}{·}), and its only anisotropic place is v = ∞. Checking locally at p = 2, 3, 5, 7, we see that there are no congruence obstructions since all congruence classes (mod 8,3,5,7 respectively) have Good-type solutions. It was conjectured in 1982 by Kneser that Q represents all natural numbers except for 2 and 22, and he checked that this was true for all numbers < 40, 000. This conjecture was later popularized by Kaplansky, and has been the positive definite form in 4 variables with smallest discriminant whose representation behavior was unknown. (The smaller forms in 4 variables can be handled by various arithmetic tricks.)

Using the QFlib package for Pari/GP [Ha1], we see that the relevant constants for Theorem 6.3 are ∑_{i=1}^{r} |γ_{i}| ≈ 39.34 and Λ̂_{4} = 2/9. Since Q has no anisotropic primes, it suffices to simply check the representability of all m with

F_{4}(m) < 177.03,

where F_{4} is as in Lemma 6.6.

To implement this, we first check all square free numbers less than this bound. We do this by first computing the primes involved in such a computation, and then forming the list of square free numbers divisible only by these primes which again satisfy our bound. With the bound 177.03 above, we find that the first 11,765 primes are involved in our search. This leads to 68,776 square free numbers to check, and the largest of these is 2 · 3 · 5 · 7 · 11 ⋯ 31 ≈ 2 × 10^{11}. Using Magma on a Celeron (Coppermine) 1GHz, it takes ≈ 17 hours to compute the first 10^{6} Fourier coefficients of Θ_{Q}(z), which makes a direct computation up to 2 × 10^{11} infeasible.

For quadratic forms in n ≥ 3 variables, one approach is to reconstruct the Θfunction Fourier coefficients for each number m we’re interested in by computing a_{E}(m) locally using Siegel’s product formula, and reconstructing a(m) form the constants γ_{i} and the Hecke eigenvalues ω_{i}(p^{2}) (or ω_{i}(p) for p | N) of f_{i} for all p | m.

For quadratic forms Q in n ≥ 4 variables, there is a convenient trick due to Bhargava we can use. Suppose we can write Q = Q_{1} + cw^{2} for some form Q_{1} in ≥ 3 variables, and suppose m is represented by Q for some large w = w_{0}. Then m − cw_{0}^{2} is much smaller than m (at best it’s ≤ 2c\sqrt{m}) and is represented by Q_{1}. However Q_{1} has fewer variables than Q, so one can compute its theta coefficients more quickly and look for these representations. (In fact, to compute all coefficients ≤ X of Θ_{Q}, it takes ≈ cX^{\frac{n}{2}} time.) By repeating this for several large values of w_{0}, one can quickly reduce the size of the list of possibly non-represented numbers.

Applying this strategy of Q = x^{2} +3y^{2} +5z^{2} +7w^{2} with Q_{1} = x^{2} +3y^{2} +7z^{2}, we can compute the first 10^{7} coefficients of Θ_{Q_{1}}(z) in ≈ 65 minutes, and easily check that the only possibly not represented numbers are 2 and 22. Since they are not represented, we see that Q represents all integers ≥ 0 except for 2 and 22.

§9 Appendix

It will be useful to have on hand the following straighforward Gauss sum computations (originally due to Siegel) to compute the local densities β_{𝔭}(m) when 𝔭 ∤ 2.

Lemma 9.1. Let Q(x⃗) = ∑_{i=1}^{n} a_{i}x_{i}^{2} be an integral quadratic form defined over a number field F with 𝔭 ∤ a_{i}, and let r_{𝔭}(m) denote the number of solutions Q(x⃗) ≡ m (mod 𝔭). Then

r_{𝔭}(0) = \{ \begin{matrix}q^{n−1} & if n is odd, \\ q^{n−1} + (q − 1)q^{\frac{n−2}{2}} (^{\frac{(−1)^{\frac{n}{2}} a_{1}···a_{n}}{𝔭}} ) & if n is even, \end{matrix}

r_{𝔭}(u) = \{ \begin{matrix}q^{n−1} + q^{\frac{n−1}{2}} (^{\frac{(−1)^{\frac{n−1}{2}} a_{1}···a_{n} u}{𝔭}}) & if n is odd, \\ q^{n−1} − q^{\frac{n−2}{2}} (^{\frac{(−1)^{\frac{n}{2}} a_{1}···a_{n}}{𝔭}} ) & if n is even, \end{matrix}

where u ∈ (𝔬_{F} /𝔭)^{×} = 𝔽_{q}^{×}.

Proof. We let r_{𝔭}^{n}(m) = r_{𝔭}(m) above and proceed by induction on n.

When n = 1 we check by hand that

r_{𝔭}^{1}(0) = 1, r_{𝔭}^{1}(u) = 1 + (\frac{au}{𝔭}) .

When n ≥ 1 we assume by induction that

(9.1)  r_{𝔭}^{n+1}(u) = C_{n+1} + D_{n+1} (\frac{u}{𝔭})

for all u ∈ 𝔽_{q}^{×} where C_{n+1}, D_{n+1} are independent of u, and that r_{𝔭}^{n+1}(0) = C_{n+1} when D_{n} = 0.

We begin with the solutions Q(x⃗) ≡ 0. By allowing all but the last variable to be freely chosen, we have

r_{𝔭}^{n+1}(0) = ∑_{\begin{matrix}m,z∈𝔽_{q} \\ m+a_{n+1}z=0 \end{matrix}} r_{𝔭}^{n}(m) (1 + (\frac{z}{𝔭})) .

Rewriting the relationship between m and z as m = −a_{n+1}z gives

∑_{z∈𝔽_{q}} r_{𝔭}^{n}(−a_{n+1}z) + ∑_{z∈𝔽_{q}} r_{𝔭}^{n}(−a_{n+1}z) (\frac{z}{𝔭}) .

Since z ↦ −a_{n+1}z is a permutation of 𝔽_{q} we are summing over all possible values of Q, so the first term is just q^{n}. To analyze the other term, we substitute (9.1) which gives the preliminary formula

r_{𝔭}^{n+1}(0) = q^{n} + (q − 1)D_{n} (\frac{−a_{n+1}}{𝔭} ) .

We now compute r_{𝔭}^{n}(u), and justify our inductive assumption (9.1). Suppose r_{𝔭}^{n}(u) is independent of u (i.e., D_{n} = 0), then by a similar argument to the previous computation, we have

r_{𝔭}^{n+1}(u) = ∑_{\begin{matrix}m,z∈𝔽_{q} \\ m+a_{n+1}z=u \end{matrix}} r_{𝔭}^{n}(m) (1 + (\frac{z}{𝔭}))

= ∑_{m∈𝔽_{q}} r_{𝔭}^{n}(m) + ∑_{m∈𝔽_{q}} r_{𝔭}^{n}(m) (\frac{(u − m) a_{n+1}}{𝔭} )

= q^{n} + (r_{𝔭}^{n}(0) − C_{n}) (\frac{u a_{n+1}}{𝔭} ) + ∑_{m≠u} C_{n} (\frac{(u − m) a_{n+1}}{𝔭} )

= q^{n} + (r_{𝔭}^{n}(0) − C_{n}) (\frac{u a_{n+1}}{𝔭} ) .

Therefore r_{𝔭}^{n+1}(u) has the form C_{n+1} + D_{n+1} (\frac{u}{𝔭}) where

(9.2)  C_{n+1} = q^{n},  D_{n+1} = (r_{𝔭}^{n}(0) − C_{n}) (\frac{a_{n+1}}{𝔭} ) .

Comparing this with our previous result about r_{𝔭}^{n}(0) we see that in fact (9.1) holds for all m ∈ 𝔽_{q} when D_{n} = 0.

Now suppose r_{𝔭}^{n}(u) depends on u (i.e., D_{n} ≠ 0). By induction we assume

(9.3)  r_{𝔭}^{n}(m) = C_{n} + D_{n} (\frac{m}{𝔭})

for all m ∈ 𝔽_{q}. Therefore

r_{𝔭}^{n+1}(αu) = ∑_{\begin{matrix}m,z∈𝔽_{q} \\ m+a_{n+1}z=αu \end{matrix}} (C_{n} + D_{n} (\frac{m}{𝔭})) (1 + (\frac{z}{𝔭}))

= ∑_{\begin{matrix}m,z∈𝔽_{q} \\ αm+αa_{n+1}z=αu \end{matrix}} (C_{n} + D_{n} (\frac{αm}{𝔭})) (1 + (\frac{αz}{𝔭} ))

= ∑_{z∈𝔽_{q}} C_{n} + ∑_{z∈𝔽_{q}} C_{n} (\frac{αz}{𝔭} ) + ∑_{m∈𝔽_{q}} D_{n} (\frac{αm}{𝔭}) + ∑_{\begin{matrix}m,z∈𝔽_{q} \\ m+a_{n+1}z=u \end{matrix}} D_{n} (\frac{αm}{𝔭})(\frac{αz}{𝔭} ) .

By reparameterizing, we see that each of the terms is independent of α. So when D_{n} ≠ 0, (9.1) holds and D_{n+1} = 0. To compute C_{n} in this case, we use the formula

(9.4)  r_{𝔭}^{n+1}(u) = \frac{q^{n+1} − r_{𝔭}^{n+1}(0)}{q − 1} = q^{n} − D_{n} (\frac{−a_{n+1}}{𝔭} ) .

All that remains is to determine D_{n}. Since D_{1} = 1, we see that D_{2n} = 0. When n is odd, substituting (9.4) into (9.2) we obtain the relation D_{n+1} = q (\frac{−a_{n}a_{n+1}}{𝔭} ) D_{n−1}, therefore

D_{2n+1} = q^{n} (\frac{(−1)^{n}a_{1} ⋯ a_{n}}{𝔭} ) . □

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