On the number of representations of integers by quadratic forms with congruence conditions

Accepted Manuscript On the number of representations of integers by quadratic forms with congruence conditions

Bumkyu Cho

PII: DOI: Reference:

S0022-247X(17)31138-1 https://doi.org/10.1016/j.jmaa.2017.12.060 YJMAA 21921

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Journal of Mathematical Analysis and Applications

Received date:

21 October 2017

Please cite this article in press as: B. Cho, On the number of representations of integers by quadratic forms with congruence conditions, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2017.12.060

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ON THE NUMBER OF REPRESENTATIONS OF INTEGERS BY QUADRATIC FORMS WITH CONGRUENCE CONDITIONS BUMKYU CHO Abstract. The generating functions of the number of representations of integers by quadratic forms under congruence conditions on the variables will be characterized in terms of modular forms. According to this characterization we present several formulas for the representation numbers as examples.

1. Introduction As is well known, there are many results about representations of integers by quadratic forms. In contrast, it seems that there are only a few results about representations with extra congruence conditions on the variables. In the previous work [1, 2, 3], the author exploited class field theory to characterize the representations of integers that can be expressed as x2 + ny 2 or x2 + xy + ny 2 (n ∈ N) under the congruence conditions x ≡ 1 (mod m) and y ≡ 0 (mod m). For example, the author obtained that a positive integer n relatively prime to 6 is of the form x2 + y 2 with x ≡ 1 (mod 3), y ≡ 0 (mod 3) if and only if n = p1 ⋯pk q1 ⋯ql a2 , where k ≥ 0 is even, l ≥ 0, a ≥ 1, (a, 6) = 1, and the pi ’s and qj ’s are mutually distinct primes such that pi ≡ 5 (mod 12) and qj ≡ 1 (mod 12) (see [2, Example 2]). There are also a few results about the number of representations under congruence conditions on the variables of the quadratic forms. In [10, Vorlesung 11] Hurwitz obtained that a formula for the number of representations of 4n (n odd) as the sum x2 + y 2 + z 2 + w2 of four odd squares is given as 16σ(n), where σ(n) ∶= ∑d∣n d. He made use of the arithmetic of quaternions to prove his result. Deutsch also gave a quaternionic proof of a formula for the representation numbers by the quadratic form x2 + y 2 + 2z 2 + 2w2 with some restrictions on the parity of the variables (see [6, Theorem 58]). Both the results of Hurwitz and Deutsch can be seen as the ones under congruence conditions modulo 2. The author and Park [4] 2010 Mathematics Subject Classification. Primary 11E25; Secondary 11F11. The author was supported by NRF-2015R1C1A1A02037526 and the Dongguk University Research Fund of 2017. 1

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BUMKYU CHO

could also obtain some results for the same and other quadratic forms under congruence conditions modulo 2 by making use of the so-called Liouville-type identities. Recently, Tsuyumine considered the ternary quadratic form x2 + y 2 + z 2 under congruence conditions modulo 8 or modulo a prime, and proved its equidistribution property (see [15]). The purpose of this article is to obtain formulas for the representation numbers with congruence conditions. This will be accomplished by means of modular forms. More precisely, one sees in Section 2 that the generating functions of the representation numbers with congruence conditions turn out to be modular forms of certain level and weight, and as its corollaries we present several formulas for the representation numbers in Section 3. For example, we present a formula for the representation number ∣{(x, y) ∈ Z2 ∣ n = x2 + y 2 , x ≡ 1 (mod 3), y ≡ 0 (mod 3)}∣ ) ) ∑d∣n ( −4 ) if n ≡ 1 (mod 3) and as 0 otherwise (see Example 3.1). Moreover, as 12 (1 + ( −3 n d the formula for the representation number ∣{(x, y, z, w) ∈ Z4 ∣ x2 + y 2 + z 2 + w2 = n, x ≡ y ≡ z ≡ 1 (mod 3), w ≡ 0 (mod 3)}∣ turns out to be σ(n/3) − σ(n/9) − 4σ(n/12) + 4σ(n/36) (see Example 3.5). 2. Main theorems and their proofs We begin with some basic definitions and facts about modular forms and theta functions. Let H = {τ ∈ C ∣ Im(τ ) > 0} be the complex upper half-plane and H∗ = H∪Q∪{∞}. Here the elements of Q ∪ {∞} are called cusps. The group SL2 (Z) acts on H∗ by linear fractional transformation γ(τ ) =

aτ +b cτ +d

for γ = (ac db) ∈ SL2 (Z). The principal congruence subgroup

Γ(N ) with N ∈ N is defined to be Γ(N ) = {( ac db ) ∈ SL2 (Z) ∣ a ≡ d ≡ 1 (mod N ), b ≡ c ≡ 0 (mod N )} and any subgroup of SL2 (Z) containing Γ(N ) is called a congruence subgroup of level N . In this article we mainly employ the congruence subgroups Γ0 (N ) and Γ1 (N ) that are defined as follows: Γ0 (N ) (respectively, Γ1 (N )) consists of all (ac db) ∈ SL2 (Z) such that c ≡ 0 (mod N ) (respectively, c ≡ 0 (mod N ) and a ≡ d ≡ 1 (mod N )). Let k, N be positive integers and ψ a Dirichlet character modulo N . Then the slash operator ∣k,ψ is defined as ¯ (f ∣k,ψ γ)(τ ) = ψ(d)(cτ + d)−k f (γτ )

REPRESENTATION NUMBERS WITH CONGRUENCE CONDITIONS

3

where f is a meromorphic function on H and γ = ( ac db ) ∈ Γ0 (N ). It is tedious to verify that (f ∣k,ψ γ)∣k,ψ γ ′ = f ∣k,ψ (γγ ′ ) for γ, γ ′ ∈ Γ0 (N ). Let k, N , ψ be as above and let Γ be a congruence subgroup of level N contained in Γ0 (N ). By definition a modular form of weight k for Γ with Nebentypus ψ is a function f satisfying (1) f is holomorphic on H (2) f ∣k,ψ γ = f for all γ ∈ Γ (3) f is holomorphic at all cusps. If a modular form f vanishes at all cusps, then it is called a cusp form. The C-vector space of modular forms of weight k for Γ with Nebentypus ψ is denoted by Mk (Γ, ψ). The subspace consisting of Eisenstein series (respectively, cusp forms) is denoted by Ek (Γ, ψ) (respectively, Sk (Γ, ψ)). In the case that ψ is the trivial character, we simply denote them by Mk (Γ), Ek (Γ), and Sk (Γ). We also employ the standard notation q = e2πiτ for τ ∈ H. Let A ∈ Mn (Z) (n even) be a positive-definite matrix whose entries satisfy the following conditions: Aij = Aji ,

Aii ≡ 0 (mod 2).

Such matrices are called even. Then the quadratic form Q(x) ∶=

1t xAx = ∑ bij xi xj , 2 1≤i≤j≤n

x = t (x1 . . . xn )

is positive-definite and 1 bii = Aii . 2 Let D = (−1)n/2 det(A) be the discriminant of the quadratic form Q. For example, bij = Aij

for i < j,

Q(x) = ax21 + bx1 x2 + cx22 has discriminant b2 − 4ac. The level of the quadratic form Q is defined to be the smallest integer N ∈ N such that N A−1 is an even matrix. Proposition 2.1. [13, Theorem 1 in Chapter IX] The level of the quadratic form Q is given as N = ∣D∣/ gcd(cij , cii /2 ∣ 1 ≤ i, j ≤ n) where cij denotes the (i, j) cofactor of A. We further remark that N and D have the same prime divisors. Let P (x) be a spherical function of degree ν with respect to A. It is a homogeneous polynomial of degree ν in

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BUMKYU CHO

variables x1 , . . . , xn (see [12, p.186]) given as ⎧ ⎪ a constant if ν = 0, ⎪ ⎪ ⎪ ⎪ t P (x) = ⎨ lAx (l ∈ Cn ) if ν = 1, ⎪ ⎪ ⎪ t ν n t ⎪ ⎪ ⎩ a linear combination of ( lAx) (l ∈ C , lAl = 0) if ν > 1. Let h ∈ Zn be a column vector satisfying Ah ≡ 0 (mod N ). We now define the theta function θ(τ ; h, A, N, P ) by θ(τ ; h, A, N, P ) = ∑ P (x)e(A[x]τ /2N 2 ) x∈Zn x≡h (N )

where A[x] = t xAx and e(z) = e2πiz . Then the theta functions are holomorphic in H and θ(τ ; h1 , A, N, P ) = θ(τ ; h2 , A, N, P )

if h1 ≡ h2 (mod N ),

θ(τ ; −h, A, N, P ) = (−1)ν θ(τ ; h, A, N, P ). Proposition 2.2. [13, Theorem 2 in Chapter IX] We have the transformation formulas θ(τ + 1; h, A, N, P ) = e(A[h]/2N 2 )θ(τ ; h, A, N, P ), θ(−1/τ ; h, A, N, P ) = (−i)n/2+2ν ∣D∣−1/2 τ n/2+ν

∑

l∈Zn /N Zn Al≡0 (N )

e(t lAh/N 2 )θ(τ ; l, A, N, P ).

The theta functions behave in a relatively simple way under certain modular transformations. Proposition 2.3. [13, Theorem 5 in Chapter IX] or [14, Proposition 2.1] For any γ = ( ac db ) ∈ Γ0 (N ) we have θ(γτ ; h, A, N, P ) = (

D ) e(abA[h]/2N 2 )(cτ + d)n/2+ν θ(τ ; ah, A, N, P ). d

Now we are ready to state and prove our main theorems. Theorem 2.4. Let Q(x) = ∑1≤i≤j≤n aij xi xj (n even) be a positive-definite integral quadratic form of discriminant DQ and level NQ , and let AQ be the coefficient matrix of the quadratic form Q so that Q(x) = AQ [x]/2. Let P (x) be a spherical function of degree ν with respect to AQ . Then for every m ∈ N and u ∈ Zn , we see that u;m (τ ) ∶= fQ,P

is a modular form of weight

n 2

∑

x∈Zn x≡u (mod m)

P (x)q Q(x)

+ ν for Γ0 (m2 NQ ) ∩ Γ1 (m) with Nebentypus (

more, if ν ≥ 1, then it is a cusp form. Here

D ( ⋅Q )

DQ ⋅ ).

denotes the Kronecker symbol.

Further-

REPRESENTATION NUMBERS WITH CONGRUENCE CONDITIONS

5

Proof. Appealing to Proposition 2.1 we easily see that the level of the quadratic form associated with the coefficient matrix mAQ is mNQ , whence we may take A = mAQ , N = mNQ , h = uNQ in Proposition 2.3. Since θ(mτ ; uNQ , mAQ , mNQ , P ) =

∑

x∈Zn x≡uNQ (mNQ )

P (x)e(AQ [x]τ /2NQ2 )

= NQν ∑ P (x)q Q(x) , x∈Zn x≡u (m)

we need to show that θ(mτ ; uNQ , mAQ , mNQ , P ) ∈ Mn/2+ν (Γ0 (m2 NQ ) ∩ Γ1 (m), (

DQ )). ⋅

For every γ = ( ac db ) ∈ Γ0 (m2 NQ ) ∩ Γ1 (m), one can deduce by Proposition 2.3 that θ(m(γτ ); uNQ , mAQ , mNQ , P ) mn DQ = ( ) e(abAQ [u]/2)(cτ + d)n/2+ν θ(mτ ; auNQ , mAQ , mNQ , P ) d DQ = ( ) (cτ + d)n/2+ν θ(mτ ; uNQ , mAQ , mNQ , P ) d because n is even and auNQ ≡ uNQ (mod mNQ ). Proposition 2.2 implies that the theta functions are holomorphic (respectively, have zeros if ν ≥ 1) at all cusps. This completes 

the proof.

Remark. Restricting m to a multiple of NQ , we see that this theorem is the same as [11, Corollary 10.7]. u;m (τ ) if P (x) = 1. We set We simply write fQu;m (τ ) instead of fQ,P u;m (k) ∶= ∣{(x1 , . . . , xn ) ∈ Zn ∣ Q(x1 , . . . , xn ) = k, x ≡ u (mod m)}∣. rQ

Then we easily see that

∞

u;m (k)q k , fQu;m (τ ) = ∑ rQ k=0

so the above theorem says that the generating function fQu;m (τ ) of the representation numu;m bers rQ (k) is actually a modular form in Mn/2 (Γ0 (m2 NQ ) ∩ Γ1 (m), (

DQ ⋅ )).

The next theorem allows us to decompose the space Mn/2 (Γ0 (m2 NQ )∩Γ1 (m), (

DQ ⋅ ))

into

χ-eigenspaces for the congruence subgroup Γ0 (m2 NQ ), and according to this decomposition we can explicitly find a basis for the space Mn/2 (Γ0 (m2 NQ )∩Γ1 (m), ( fQu;m (τ )

DQ ⋅ )).

Finally, writing

as a linear combination of the basis enables us to obtain the desired formulas for

u;m (k). the representation numbers rQ

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BUMKYU CHO

Theorem 2.5. Let k, M, N be positive integers and ψ a Dirichlet character modulo M N . Then we have Mk (Γ0 (M N ) ∩ Γ1 (M ), ψ) ≅ ⊕ Mk (Γ0 (M N ), χψ) χ

where χ runs over all Dirichlet characters modulo M such that χ(−1) = (−1)k ψ(−1). Proof. For f ∈ Mk (Γ0 (M N ) ∩ Γ1 (M ), ψ) and χ a Dirichlet character modulo M , we define fχ (τ ) =

1 ¯ ∣k,ψ γd )(τ ) ∑ χ(d)(f ϕ(M ) d∈(Z/M Z)×

where γd = ( ∗∗ d∗0 ) ∈ Γ0 (M N ) is any fixed matrix with d0 ≡ d (mod M ). We then have for any γd′ ∈ Γ0 (M N ), (fχ ∣k,ψ γd′ )(τ ) =

1 ¯ ∣k,ψ γdd′ )(τ ) ∑ χ(d)(f ϕ(M ) d∈(Z/M Z)×

= χ(d′ )fχ (τ ), whence fχ is contained in Mk (Γ0 (M N ), χψ). One has ∑ fχ (τ ) = χ

1 ¯ ∑ (f ∣k,ψ γd )(τ ) ∑ χ(d) ϕ(M ) d∈(Z/M Z)× χ

= f (τ ) where the summation is taken over all Dirichlet characters χ modulo M . This is because ⎧ ⎪ ⎪ ϕ(M ) if d = 1, ¯ =⎨ ∑ χ(d) ⎪ otherwise. ⎪ χ ⎩ 0 Observe that Mk (Γ0 (M N ), χ1 ψ) ∩ Mk (Γ0 (M N ), χ2 ψ) = {0} if χ1 ≠ χ2 . We thus have the decomposition Mk (Γ0 (M N ) ∩ Γ1 (M ), ψ) ≅ ⊕ Mk (Γ0 (M N ), χψ) χ

where χ runs over all Dirichlet characters modulo M . By definition one has Mk (Γ0 (M N ), χψ) = {0} unless (χψ)(−1) = (−1)k . This completes the proof.



REPRESENTATION NUMBERS WITH CONGRUENCE CONDITIONS

7

3. Examples In the case when the space Sn/2 (Γ0 (m2 NQ ) ∩ Γ1 (m), (

DQ ⋅ ))

of cusp forms is trivial, we

can get an explicit formula for the number of representations of integers by the quadratic form Q(x) with congruence condition x ≡ u (mod m) on the variables. This is because the space Mn/2 (Γ0 (m2 NQ ) ∩ Γ1 (m), (

DQ ⋅ ))

is then spanned by Eisenstein series and their

Fourier coefficients are explicitly known. Let AN,1 be the set of triples ({ψ, ϕ}, t) such that ψ and ϕ, taken as an unordered pair, are primitive Dirichlet characters satisfying (ψϕ)(−1) = −1, and t is a positive integer such that tuv∣N , where u and v denote the conductors of ψ and ϕ, respectively. For ({ψ, ϕ}, t) ∈ AN,1 let E1ψ,ϕ,t (τ ) =

∞ 1 (δ(ϕ)L(0, ψ) + δ(ψ)L(0, ϕ)) + ∑ σ0ψ,ϕ (n)q tn , 2 n=1

where δ(ψ) is 1 if ψ is the trivial character and is 0 otherwise, and σ0ψ,ϕ (n) = ∑ ψ(n/d)ϕ(d). d∣n

Proposition 3.1. [7, Theorem 4.8.1] For any Dirichlet character χ modulo N , the set {E1ψ,ϕ,t ∣ ({ψ, ϕ}, t) ∈ AN,1 , ψϕ = χ} represents a basis for the space E1 (Γ0 (N ), χ) of Eisenstein series of weight one with Nebentypus χ. Hereafter χn (⋅) denotes the quadratic character ( n⋅ ) for n ∈ Z. Example 3.1. For Q(x) = x21 + x22 and ui ∈ {0, 1, 2}, let (u ,u2 );3

fQ 1

(τ ) =

∑

xi ≡ui (3)

∞

(u ,u2 );3

q x1 +x2 = ∑ rQ 1 2

2

(n)q n .

n=0

(u ,u2 );3

Appealing to Theorems 2.4 and 2.5 we see that fQ 1

(τ ) is contained in M1 (Γ0 (36), χ−4 ).

This space has no cusp forms other than 0, whence we see by the preceding proposition that M1 (Γ0 (36), χ−4 ) is decomposed as M1 (Γ0 (36), χ−4 ) = CE11,χ−4 ,1 ⊕ CE11,χ−4 ,3 ⊕ CE11,χ−4 ,9 ⊕ CE1χ12 ,χ−3 ,1 .

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BUMKYU CHO

Their Fourier expansions are explicitly given as E11,χ−4 ,1 (τ ) =

1 ∞ −4 + ∑ ( ∑ ( ) )q n , 4 n=1 d∣n d ∞

E1χ12 ,χ−3 ,1 (τ ) = ∑ ( ∑ ( n=1

(u ,u2 );3

Calculating sufficiently many rQ 1

d∣n

12 −3 ) ( ) )q n . n/d d (u ,u2 );3

(n), we can express fQ 1

(τ ) as a linear combina-

tion of the above Eisenstein series. It turns out that for every n ∈ N we have (1,0);3

rQ

(n) =

1 −4 −4 12 1 1 −3 )( ). ∑( ) − ∑( ) + ∑( 2 d∣n d 2 d∣ n d 2 d∣n n/d d 9

Note that

(1,0);3 (n) rQ

= 0 if n ≡/ 1 (mod 3). Moreover, appealing to the identity ∑( d∣n

we have (1,0);3 (n) rQ

12 −4 −3 −3 )( ) = ( )∑( ), n/d d n d∣n d

⎧ ⎪ ) ) ∑d∣n ( −4 ) if n ≡ 1 (mod 3) ⎪ 1 (1 + ( −3 n d = ⎨ 2 ⎪ otherwise. ⎪ ⎩ 0

Similarly, we deduce that (1,1);3

rQ

(n) =

1 −4 −4 12 1 1 −3 )( ) ∑( ) − ∑( ) − ∑( 2 d∣n d 2 d∣ n d 2 d∣n n/d d 9

⎧ ⎪ ) ) ∑d∣n ( −4 ) if n ≡ 2 (mod 3) ⎪ 12 (1 − ( −3 n d = ⎨ ⎪ otherwise. ⎪ ⎩ 0 Example 3.2. The following results are easily obtained by the same method as above, whence we merely present their formulas without detailed explanation. (1) If Q(x) = x21 + x22 , then for any n ∈ N we have (1,0);2

rQ

(n) = 2 ∑ ( d∣n

−4 −4 )−2∑( ) d d d∣ n 2

⎧ ⎪ ) if n ≡ 1 (mod 4) ⎪ 2 ∑d∣n ( −4 d = ⎨ ⎪ otherwise, ⎪ ⎩ 0 −4 −4 (1,1);2 (n) = 4 ∑ ( ) − 4 ∑ ( ) rQ d d d∣ n d∣ n 2

4

⎧ ⎪ ) if n ≡ 2 (mod 8) ⎪ 4 ∑d∣ n2 ( −4 d = ⎨ ⎪ otherwise. ⎪ ⎩ 0

REPRESENTATION NUMBERS WITH CONGRUENCE CONDITIONS (u ,u2 );2

(2) For Q(x) = x21 + x1 x2 + x22 , we see rQ 1

(u ,u );3 rQ 1 2 (n)

=

(u ,u );3 rQ 2 1 (n)

=

(1,0);2

rQ

(−u ,−u );3 rQ 1 2 (n)

=

(n) = 2 ∑ ( d∣n

(u ,u1 );2

(n) = rQ 2

(u +u ,−u );3 rQ 1 2 2 (n).

4

⎧ ⎪ ) if n ≡ 1 (mod 2) ⎪ 2 ∑d∣n ( −3 d = ⎨ ⎪ otherwise, ⎪ ⎩ 0 −3 −3 (1,0);3 (n) = ∑ ( ) − ∑ ( ) rQ d d d∣n d∣ n 3

⎧ ⎪ ⎪ ∑ ( −3 ) if n ≡ 1 (mod 3) = ⎨ d∣n d ⎪ otherwise, ⎪ ⎩ 0 −3 −3 (1,1);3 (n) = 3 ∑ ( ) − 3 ∑ ( ) rQ d d n n d∣ d∣ 9

⎧ ⎪ ) if n ≡ 3 (mod 9) ⎪ 3 ∑d∣ n3 ( −3 d = ⎨ ⎪ otherwise. ⎪ ⎩ 0

(3) Given Q(x) = x21 + 2x22 , one has for any n ∈ N (1,0);2

rQ

(n) = ∑ ( d∣n

−8 −8 −4 8 ) − ∑( ) + ∑( )( ) d d d d∣ n d∣n n/d 2

⎧ ⎪ ) if n ≡ 1 (mod 8) ⎪ 2 ∑d∣n ( −2 d = ⎨ ⎪ otherwise, ⎪ ⎩ 0 −8 −8 (0,1);2 (n) = 2 ∑ ( ) − 2 ∑ ( ) rQ d d d∣ n d∣ n 2

4

⎧ ⎪ ) if n ≡ 2 (mod 4) ⎪ 2 ∑d∣ n2 ( −2 d = ⎨ ⎪ otherwise, ⎪ ⎩ 0 −8 −8 −4 8 (1,1);2 (n) = ∑ ( ) − ∑ ( ) − ∑ ( rQ )( ) d d d d∣n d∣ n d∣n n/d 2

(n) and

For every n ∈ N, one has

−3 −3 )−2∑( ) d d d∣ n

3

(u +u2 ,−u2 );2

(n) = rQ 1

9

⎧ ⎪ ) if n ≡ 3 (mod 8) ⎪ 2 ∑d∣n ( −2 d = ⎨ ⎪ otherwise. ⎪ ⎩ 0

10

BUMKYU CHO (u ,u2 );2

(4) When Q(x) = x21 + x1 x2 + 2x22 we see rQ 1

(u +u2 ,−u2 );2

(n) = rQ 1

(n) and obtain for any

n∈N (1,0);2

rQ

(n) = 2 ∑ ( d∣n

−7 −7 −7 )−4∑( )+2∑( ) d d d n n d∣ d∣ 2

4

⎧ ⎪ ) if n ≡ 1 (mod 2) ⎪ 2 ∑d∣n ( −7 d = ⎨ ⎪ otherwise, ⎪ ⎩ 0 −7 −7 (0,1);2 (n) = 2 ∑ ( ) − 2 ∑ ( ) rQ d d d∣ n d∣ n 2

4

⎧ ⎪ ) if n ≡ 2 (mod 4) ⎪ 2 ∑d∣ n2 ( −7 d = ⎨ ⎪ otherwise. ⎪ ⎩ 0 We are now going to deal with quadratic forms of four variables. Let AN,2 be the set of triples (ψ, ϕ, t) such that ψ and ϕ, taken this time as an ordered pair, are primitive Dirichlet characters satisfying (ψϕ)(−1) = 1, and t is an integer such that 1 < tuv∣N , where u and v denote the conductors of ψ and ϕ, respectively. For any triple (ψ, ϕ, t) ∈ AN,2 we define

⎧ ∞ ∞ ⎪ if ψ = ϕ = 1 ⎪ ∑ σ(n)q n − t ∑n=1 σ(n)q tn = ⎨ 1 n=1 ψ,ϕ ∞ tn ⎪ otherwise. ⎪ ⎩ 2 δ(ψ)L(−1, ϕ) + ∑n=1 σ1 (n)q Here δ(ψ) has the same meaning as before, and E2ψ,ϕ,t (τ )

σ1ψ,ϕ (n) = ∑ ψ(n/d)ϕ(d)d. d∣n

Proposition 3.2. [7, Theorem 4.6.2] For any Dirichlet character χ modulo N , the set {E2ψ,ϕ,t ∣ (ψ, ϕ, t) ∈ AN,2 , ψϕ = χ} represents a basis for the space E2 (Γ0 (N ), χ) of Eisenstein series of weight two with Nebentypus χ. Remark. We will not deal with an example of weight greater than 2, whence we merely refer the reader to [7, Theorem 4.5.2] for the basis of Ek (Γ0 (N ), χ) with k ≥ 3. Example 3.3. If we let Q(x) = x21 +x22 +x23 +x24 , then fQu;2 (τ ) is a modular form in M2 (Γ0 (16)) by Theorem 2.4. Appealing to the theory of modular forms we see S2 (Γ0 (16)) = {0}, whence M2 (Γ0 (16)) is spanned by Eisenstein series as follows: M2 (Γ0 (16)) = ( ⊕ CE21,1,t ) ⊕ CE2χ−4 ,χ−4 ,1 . 1
REPRESENTATION NUMBERS WITH CONGRUENCE CONDITIONS

)σ(n), we obtain that for every n ∈ N In terms of the identity σ1χ−4 ,χ−4 (n) = ( −4 n

(1,0,0,0);2

rQ

(n) = (1 + ( =

(1,1,0,0);2

rQ

(n) = =

(1,1,1,0);2

rQ

(n) = =

(1,1,1,1);2

rQ

(n) = =

−4 ) )σ(n) − 3σ(n/2) + 2σ(n/4) n

⎧ ⎪ ⎪ 2σ(n) if n ≡ 1 (mod 4) ⎨ ⎪ otherwise, ⎪ ⎩ 0 4σ(n/2) − 12σ(n/4) + 8σ(n/8) ⎧ ⎪ ⎪ 4σ(n/2) if n ≡ 2 (mod 4) ⎨ ⎪ otherwise, ⎪ ⎩ 0 −4 (1 − ( ) )σ(n) − 3σ(n/2) + 2σ(n/4) n ⎧ ⎪ ⎪ 2σ(n) if n ≡ 3 (mod 4) ⎨ ⎪ otherwise, ⎪ ⎩ 0 16σ(n/4) − 48σ(n/8) + 32σ(n/16) ⎧ ⎪ ⎪ 16σ(n/4) if n ≡ 4 (mod 8) ⎨ ⎪ otherwise. ⎪ ⎩ 0

Example 3.4. By a routine computation as above we can obtain the following results. (1) The modular form fQu;2 (τ ) with Q(x) = x21 + x22 + x23 + 2x24 is contained in

M2 (Γ0 (32), χ2 ) = ( ⊕ CE21,χ2 ,a ) ⊕ ( ⊕ CE2χ2 ,1,b ) ⊕ CE2χ−2 ,χ−4 ,1 ⊕ CE2χ−4 ,χ−2 ,1 . a=1,2,4

b=1,2,4

11

12

BUMKYU CHO

We thus obtain that for every n ∈ N

(1,0,0,0);2

rQ

(n) = ∑ ( d∣n

2 2 2 −2 )d − 2∑( )d + ( )∑( )d n/d n d∣n d d∣ n n/2d 2

⎧ 2 ⎪ ⎪ 2 ∑d∣n ( n/d ) d if n ≡ 1 (mod 4) = ⎨ ⎪ otherwise, ⎪ ⎩ 0 2 −2 (1,1,0,0);2 (n) = 4 ∑ ( rQ )d − 8∑( )d d∣ n n/2d d∣ n n/4d 2

4

⎧ 2 ⎪ ⎪ 4 ∑d∣ n2 ( n/2d ) d if n ≡ 2 (mod 4) = ⎨ ⎪ otherwise, ⎪ ⎩ 0 −2 2 2 2 2 −2 (1,1,1,0);2 (n) = −(1 + ( ) ) ∑ ( ) d + ∑ ( ) d + (1 + ( ) ) ∑ ( rQ )d − 2∑( )d n n d∣n d d∣ n d d∣n n/d d∣ n n/2d 2

⎧ 2 ⎪ ⎪ 4 ∑d∣n ( n/d ) d if n ≡ 3 (mod 8) = ⎨ ⎪ otherwise, ⎪ ⎩ 0 2 2 2 2 (0,0,0,1);2 (n) = −2 ∑ ( ) d + 2 ∑ ( ) d + 4 ∑ ( rQ )d − 8∑( )d d d n/2d n/4d n n n n d∣ d∣ d∣ d∣ 2

4

2

2

4

⎧ 2 2 ⎪ ⎪ −2 ∑d∣ n2 ( d ) d + 4 ∑d∣ n2 ( n/2d ) d if n ≡ 2 (mod 4) = ⎨ ⎪ otherwise, ⎪ ⎩ 0 2 2 2 −2 (1,0,0,1);2 (n) = ∑ ( rQ )d − 2∑( )d − ( )∑( )d n d∣n d d∣n n/d d∣ n n/2d 2

⎧ 2 ⎪ ⎪ 2 ∑d∣n ( n/d ) d if n ≡ 3 (mod 4) = ⎨ ⎪ otherwise, ⎪ ⎩ 0 2 (1,1,0,1);2 (n) = 8 ∑ ( rQ ) d, d∣ n n/4d 4

(1,1,1,1);2

rQ

(n) = −(1 − (

−2 2 2 2 2 −2 ) ) ∑ ( ) d + ∑ ( ) d + (1 − ( ) ) ∑ ( )d − 2∑( )d n n d∣n d d∣ n d d∣n n/d d∣ n n/2d 2

⎧ 2 ⎪ ⎪ 4 ∑d∣n ( n/d ) d if n ≡ 5 (mod 8) = ⎨ ⎪ otherwise. ⎪ ⎩ 0

2

REPRESENTATION NUMBERS WITH CONGRUENCE CONDITIONS

13

(2) Let Q(x) = x21 + x1 x2 + x22 + x23 + x3 x4 + x24 . Because Q(x) is preserved by the change of variables x1 ↦ x1 + x2 and x2 ↦ −x2 , we see that (u ,u2 ,u3 ,u4 );2

rQ 1

(u ,u1 ,u3 ,u4 );2

(n) = rQ 2

(u ,u4 ,u1 ,u2 );2

(n) = rQ 3

(u +u2 ,−u2 ,u3 ,u4 );2

(n) = rQ 1

(n)

and obtain that for every n ∈ N (1,0,0,0);2

(n) = 2σ(n) − 6σ(n/2) − 6σ(n/3) + 4σ(n/4) + 18σ(n/6) − 12σ(n/12) ⎧ ⎪ ⎪ 2σ(n) − 6σ(n/3) if n ≡ 1 (mod 2) = ⎨ ⎪ otherwise, ⎪ ⎩ 0 (1,0,1,0);2 (n) = 4σ(n/2) − 4σ(n/4) − 12σ(n/6) + 12σ(n/12). rQ rQ

u;2 (n) (3) If Q(x) = x21 + x22 + x23 + x24 + x1 x2 + x1 x3 + x1 x4 , then a computation shows that rQ

with u = (u1 , . . . , u4 ) ∈ {0, 1}4 and n ∈ N is given as ⎧ ⎪ 8σ(n/2) − 24σ(n/4) + 16σ(n/8) if u1 = 0 and exactly one of u2 , u3 , u4 is 0 ⎪ ⎪ ⎪ ⎪ ⎨ 24σ(n/4) − 48σ(n/8) if u1 = u2 = u3 = u4 = 0 ⎪ ⎪ ⎪ ⎪ otherwise. ⎪ ⎩ 2σ(n) − 6σ(n/2) + 4σ(n/4) Example 3.5. We deal with Q(x) = x21 + x22 + x23 + x24 . The modular form fQu;3 (τ ) is then contained in M2 (Γ0 (36)) by Theorems 2.4 and 2.5. Unlike the previous examples, it turns out that this space contains a cusp form η(6τ )4 , where η(τ ) denotes the Dedekind eta function defined by ∞

η(τ ) = q 1/24 ∏(1 − q n ). n=1

In fact, we have M2 (Γ0 (36)) = ( ⊕ CE21,1,a ) ⊕ ( ⊕ CE2χ−3 ,χ−3 ,b ) ⊕ Cη(6τ )4 . 1
b=1,2,4

If we denote by c(n) the nth coefficient of the cusp form η(6τ )4 , i.e. ∞

η(6τ )4 = ∑ c(n)q n = q − 4q 7 + 2q 13 + 8q 19 − 5q 25 − 4q 31 − 10q 37 + ⋯, n=1

14

BUMKYU CHO

then we deduce that for every n ∈ N 1 −3 2 2 −3 (1 + ( ) )σ(n) − σ(n/3) − (1 + ( ) )σ(n/4) 6 n 3 3 n/4 8 2 1 + σ(n/9) + σ(n/12) − 2σ(n/36) + c(n) 2 3 3 ⎧ 1 −3 2 −3 2 ⎪ ⎪ 6 (1 + ( n ) )σ(n) − 3 (1 + ( n/4 ) )σ(n/4) + 3 c(n) if n ≡ 1 (mod 3) = ⎨ ⎪ otherwise, ⎪ ⎩ 0 1 −3 2 2 −3 (1,1,0,0);3 (1 − ( ) )σ(n) − σ(n/3) − (1 − ( (n) = rQ ) )σ(n/4) 6 n 3 3 n/4 8 1 + σ(n/9) + σ(n/12) − 2σ(n/36) 2 3 ⎧ 1 −3 2 −3 ⎪ ⎪ (1 − ( n ) )σ(n) − 3 (1 − ( n/4 ) )σ(n/4) if n ≡ 2 (mod 3) = ⎨ 6 ⎪ otherwise, ⎪ ⎩ 0 (1,0,0,0);3

(n) =

(1,1,1,0);3

(n) = σ(n/3) − σ(n/9) − 4σ(n/12) + 4σ(n/36),

(1,1,1,1);3

(n) =

rQ

rQ rQ

1 −3 2 2 −3 (1 + ( ) )σ(n) − σ(n/3) − (1 + ( ) )σ(n/4) 6 n 3 3 n/4 8 1 1 + σ(n/9) + σ(n/12) − 2σ(n/36) − c(n) 2 3 3 ⎧ 1 −3 2 −3 ⎪ (1 + ( n ) )σ(n) − 3 (1 + ( n/4 ) )σ(n/4) − 13 c(n) if n ≡ 1 (mod 3) ⎪ = ⎨ 6 ⎪ otherwise. ⎪ ⎩ 0

Example 3.6. For Q(x) = x21 + x22 + 2x23 + 2x24 , we see that fQu;2 (τ ) is contained in the space M2 (Γ0 (32)). Its basis is given as M2 (Γ0 (32)) = ( ⊕ CE21,1,a ) ⊕ ( ⊕ CE2χ−4 ,χ−4 ,b ) ⊕ Cη(4τ )2 η(8τ )2 . 1
b=1,2

Let c(n) denote the nth coefficient of the Fourier expansion of the cusp form η(4τ )2 η(8τ )2 , namely, ∞

η(4τ )2 η(8τ )2 = ∑ c(n)q n = q − 2q 5 − 3q 9 + 6q 13 + 2q 17 − q 25 + ⋯. n=1

REPRESENTATION NUMBERS WITH CONGRUENCE CONDITIONS

We then find that for every n ∈ N

(1,0,0,0);2

rQ

(n) = =

(1,1,0,0);2

rQ

(n) = =

(0,0,1,0);2

rQ

(n) = =

(1,0,1,0);2

rQ

(n) = =

(1,1,1,0);2

rQ

(n) = =

(0,0,1,1);2

rQ

(n) = =

(1,0,1,1);2

rQ

(n) = =

(1,1,1,1);2

rQ

(n) = =

1 −4 3 (1 + ( ) )σ(n) − σ(n/2) + σ(n/4) + c(n) 2 n 2 ⎧ ⎪ σ(n) + c(n) if n ≡ 1 (mod 4) ⎪ ⎨ ⎪ otherwise, ⎪ ⎩ 0 −4 2(1 + ( ) )σ(n/2) − 6σ(n/4) + 4σ(n/8) n/2 ⎧ ⎪ ⎪ 4σ(n/2) if n ≡ 2 (mod 8) ⎨ ⎪ otherwise, ⎪ ⎩ 0 2σ(n/2) − 6σ(n/4) + 4σ(n/8) ⎧ ⎪ ⎪ 2σ(n/2) if n ≡ 2 (mod 4) ⎨ ⎪ otherwise, ⎪ ⎩ 0 1 −4 3 (1 − ( ) )σ(n) − σ(n/2) + σ(n/4) 2 n 2 ⎧ ⎪ ⎪ σ(n) if n ≡ 3 (mod 4) ⎨ ⎪ otherwise, ⎪ ⎩ 0 8σ(n/4) − 24σ(n/8) + 16σ(n/16) ⎧ ⎪ ⎪ 8σ(n/4) if n ≡ 4 (mod 8) ⎨ ⎪ otherwise, ⎪ ⎩ 0 4σ(n/4) + 4σ(n/8) − 40σ(n/16) + 32σ(n/32) ⎧ ⎪ ⎪ 4σ(n/4) + 4σ(n/8) if n ≡ 4, 8, 12 (mod 16) ⎨ ⎪ otherwise, ⎪ ⎩ 0 1 −4 3 (1 + ( ) )σ(n) − σ(n/2) + σ(n/4) − c(n) 2 n 2 ⎧ ⎪ ⎪ σ(n) − c(n) if n ≡ 1 (mod 4) ⎨ ⎪ otherwise, ⎪ ⎩ 0 −4 2(1 − ( ) )σ(n/2) − 6σ(n/4) + 4σ(n/8) n/2 ⎧ ⎪ ⎪ 4σ(n/2) if n ≡ 6 (mod 8) ⎨ ⎪ otherwise. ⎪ ⎩ 0

15

16

BUMKYU CHO

Example 3.7. Consider the quadratic form Q(x) = x21 +x1 x2 +x22 +x23 +x3 x4 +x24 . By definition we see that (u ,u2 ,u3 ,u4 );3

rQ 1

(±u2 ,±u1 ,u3 ,u4 );3

(n) = rQ

(u ,u4 ,u1 ,u2 );3

(n) = rQ 3

(u +u2 ,−u2 ,u3 ,u4 );3

(n) = rQ 1

and we infer that fQu;3 (τ ) is contained in M2 (Γ0 (27)) whose basis is given as M2 (Γ0 (27)) = ( ⊕ CE21,1,a ) ⊕ ( ⊕ CE2χ−3 ,χ−3 ,b ) ⊕ Cη(3τ )2 η(9τ )2 . a=3,9,27

b=1,3

If we set ∞

η(3τ )2 η(9τ )2 = ∑ c(n)q n = q − 2q 4 − q 7 + 5q 13 + 4q 16 − 7q 19 − 5q 25 + ⋯, n=1

then we find for any n ∈ N (1,0,0,0);3

rQ

(n) = =

(1,1,0,0);3

rQ

(n) = =

(1,1,1,0);3

rQ

(n) = =

(1,1,1,1);3

rQ

(n) = =

(1,0,1,0);3

rQ

(n) = =

1 −3 2 1 2 (1 + ( ) )σ(n) − σ(n/3) + σ(n/9) + c(n) 6 n 3 2 3 ⎧ ⎪ ) )σ(n) + 23 c(n) if n ≡ 1 (mod 3) ⎪ 16 (1 + ( −3 n ⎨ ⎪ 0 otherwise, ⎪ ⎩ 3 −3 9 (1 + ( ) )σ(n/3) − 6σ(n/9) + σ(n/27) 2 n/3 2 ⎧ −3 ⎪ 32 (1 + ( n/3 ) )σ(n/3) if n ≡ 3 (mod 9) ⎪ ⎨ ⎪ otherwise, ⎪ ⎩ 0 1 −3 2 1 1 (1 + ( ) )σ(n) − σ(n/3) + σ(n/9) − c(n) 6 n 3 2 3 ⎧ 1 −3 1 ⎪ ⎪ 6 (1 + ( n ) )σ(n) − 3 c(n) if n ≡ 1 (mod 3) ⎨ ⎪ otherwise, ⎪ ⎩ 0 3 −3 9 (1 − ( ) )σ(n/3) − 6σ(n/9) + σ(n/27) 2 n/3 2 ⎧ 3 −3 ⎪ ⎪ 2 (1 − ( n/3 ) )σ(n/3) if n ≡ 6 (mod 9) ⎨ ⎪ otherwise, ⎪ ⎩ 0 1 −3 2 1 (1 − ( ) )σ(n) − σ(n/3) + σ(n/9) 6 n 3 2 ⎧ 1 −3 ⎪ 6 (1 − ( n ) )σ(n) if n ≡ 2 (mod 3) ⎪ ⎨ ⎪ otherwise. ⎪ ⎩ 0

(n)

REPRESENTATION NUMBERS WITH CONGRUENCE CONDITIONS

17

Example 3.8. Let Q(x) = x21 + x1 x2 + 2x22 + x23 + x3 x4 + 2x24 . We easily infer from the definition u;2 (n) that of rQ (u ,u2 ,u3 ,u4 );2

rQ 1

(u ,u4 ,u1 ,u2 );2

(n) = rQ 3

(u +u2 ,−u2 ,u3 ,u4 );2

(n) = rQ 1

(n).

It follows that fQu;2 (τ ) is contained in the space M2 (Γ0 (28)) whose basis is given as M2 (Γ0 (28)) = ( ⊕ CE21,1,t ) ⊕ Cη(τ )η(2τ )η(7τ )η(14τ ) ⊕ Cη(2τ )η(4τ )η(14τ )η(28τ ). 1
If we set ∞

η(τ )η(2τ )η(7τ )η(14τ ) = ∑ c(n)q n = q − q 2 − 2q 3 + q 4 + 2q 6 + q 7 − q 8 + ⋯, n=1

then one has that for every n ∈ N (1,0,0,0);2

rQ

(n) =

= (1,0,1,0);2

(n) =

(1,0,0,1);2

(n) =

rQ rQ

= (0,1,0,0);2

(n) =

(0,1,0,1);2

(n) =

rQ rQ

2 4 14 σ(n) − 2σ(n/2) + σ(n/4) − σ(n/7) + 14σ(n/14) 3 3 3 28 4 4 − σ(n/28) + c(n) + c(n/2) 3 3 3 ⎧ 2 4 ⎪ ⎪ 3 σ(n) + 3 c(n) if n ≡ 1 (mod 2) ⎨ ⎪ otherwise, ⎪ ⎩ 0 4 4 28 28 8 σ(n/2) − σ(n/4) − σ(n/14) + σ(n/28) + c(n/2), 3 3 3 3 3 2 4 14 σ(n) − 2σ(n/2) + σ(n/4) − σ(n/7) + 14σ(n/14) 3 3 3 28 2 2 − σ(n/28) − c(n) − c(n/2) 3 3 3 ⎧ 2 14 2 ⎪ ⎪ 3 σ(n) − 3 σ(n/7) − 3 c(n) if n ≡ 1 (mod 2) ⎨ ⎪ otherwise, ⎪ ⎩ 0 4 4 28 28 2 σ(n/2) − σ(n/4) − σ(n/14) + σ(n/28) + c(n/2), 3 3 3 3 3 4 4 28 28 4 σ(n/2) − σ(n/4) − σ(n/14) + σ(n/28) − c(n/2). 3 3 3 3 3 References

1. B. Cho, Primes of the form x2 + ny 2 with conditions x ≡ 1 mod N , y ≡ 0 mod N , J. Number Theory, 130 (2010), 852-861. 2. B. Cho, Integers of the form x2 + ny 2 , Monatsh. Math., 174 (2014), 195-204. 3. B. Cho, Representations of integers by the binary quadratic form x2 + xy + ny 2 , J. Aust. Math. Soc., 100 (2016), 182-191.

18

BUMKYU CHO

4. B. Cho and H. Park, On some results of Hurwitz and Deutsch about certain quadratic forms, J. Number Theory, 144 (2014), 1-14. 5. D. Cox, Primes of the Form x2 + ny 2 , second edition, Wiley, 2013. 6. J. I. Deutsch, A quaternionic proof of the representation formula of a quaternary quadratic form, J. Number Theory, 113 (2005), 149-174. 7. F. Diamond and J. Shurman, A First Course in Modular Forms, Springer-Verlag, Graduate Texts in Mathematics, Vol. 228 (2005). 8. L. E. Dickson, History of the Theory of Numbers, Volume III: Quadratic and Higher Forms, Dover Publications, 2005. 9. E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, 1985. 10. A. Hurwitz, Vorlesungen u ¨ber die Zahlentheorie der Quaternionen, Julius Springer, Berlin, 1919. 11. H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics (Volume 17), Providence RI, American Mathematical Society. 12. T. Miyake, Modular Forms, Springer-Verlag, 1989. 13. B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, 1974. 14. G. Shimura, On modular forms of half integral weight, Ann. Math., 97 (1973), 440-481. 15. S. Tsuyumine, Sums of three squares under congruence condition modulo a prime, J. Number Theory, 159 (2016), 123-159. Department of Mathematics, Dongguk University-Seoul, 30 Pildong-ro 1-gil, Jung-gu, Seoul, 04620, Republic of Korea E-mail address: [email protected]