1Binary Theta Series andCM Modular Forms
1. Introduction
Let rn(q) = {(x, y) ∈ Z2 : q(x, y) = n} denote the numberof representations of n ∈ Z by the positive definite binary
quadratic form
q(x, y) = ax2 + bxy + cy 2,a, b, c ∈ Z, a > 0.

Fermat, Euler, Lagrange, Gauss: When is rn(q) > 0?
Dirichlet(1839), Weber(1882): If gcd(a, b, c) = 1, i.e. if q isprimitive, then ∃∞ primes p : rp(q) > 0. — Study:XZq (s) =rn(q)n−s.
n≥1Following Jacobi, Hermite, Kronecker, Weber, consider theclosely related binary theta seriesXX
2πiq(x,y)zϑq (z) =e=rn(q)e2πinz .
x,y∈Z

n≥0Theorem 0 (a) Weber(1893): Let D = ∆(q) = b2 − 4ac
denote the discriminant of q and ψD = D· . Then az + ba b
ϑq= ψD (d)(cz + d)ϑq (z), ∀∈ Γ0(|D|).
cz + dc d(b) Hecke(1926), Schoeneberg(1939): ϑq is holomorphic at the cusps, so ϑq ∈ M1(|D|, ψD ).

2Fix: a discriminant D < 0. Thus:√2
D = fD dK , where K = Q( D), dK = disc(K), and fD ≥ 1
is some integer. Let
ΘD := hϑq : q ∈ QD iC and Θ(D) := hϑq : q ∈ Q(D)iCbe the C-subspaces generated by the theta-series, whereQ(D) = {q = (a, b, c) ∈ Z2 : a > 0, ∆(q) = D/t2},
QD = {q = (a, b, c) ∈ Q(D) : gcd(a, b, c) = 1, ∆(q) = D}.
ThusΘD ⊂ Θ(D) ⊂ M1(|D|, ψD ).
Questions: 1) How large are the spaces ΘD and Θ(D)?
What is the dimension of the subspaces of cusp forms, i.e. of
ΘSD = ΘD ∩ S1(|D|, ψD ) and Θ(D)S = Θ(D) ∩ S1(|D|, ψD )?
Hecke (1926): Θ(D) 6= M1(|D|, ψD ), for many D’s.
2) How can a binary theta series ϑq be expressed in terms ofthe (extended) Atkin-Lehner basis of M1(|D|, ψD )?
3) Is there an intrinsic characterization of these spaces?

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2. Some Observations
1) The group GL2(Z) acts on the sets QD and Q(D), andϑq 0 = ϑq ,for all q 0 ∈ q GL2(Z).

By using the Dirichlet/Weber result, one can show that the set{ϑq : q ∈ QD / GL2(Z)} is a basis of ΘD . In particular,dim ΘD = hD := |QD / GL2(D)|2) By Gauss’s theory of composition of forms, the set
Cl(D) = QD / SL2(Z)has the structure of an abelian group. If hD := | Cl(D)|, then1hD = (gD + hD ), where gD = [Cl(D) : Cl(D)2]2
denotes the number of genera of forms of discriminant D.
3) For a character χ ∈ Cl(D)∗ on Cl(D), putX1 Xϑχ(z) :=χ(q)ϑq (z) =an(χ)e2πinz ∈ ΘD ,wDn≥0q∈Cl(D)where wD = 2 for D < −4 and w−3 = 6, m−4 = 4.
It is immediate that {ϑχ}χ∈Cl(D)∗ generates ΘD and hence by1) forms a basis of ΘD (subject to the identification ϑχ = ϑχ).
Note: It turns out (cf. Theorem 1) that the coefficients an(χ) are
multiplicative in n, and that hence ϑχ is a Hecke eigenfunction w.r.t. to the Hecke algebra T(D) generated by the Hecke
operators Tp with (p, D) = 1.

44) The L-function associated to the form ϑχ isXL(s, χ) = L(s, ϑχ) =an(χ)n−s.
n≥1This function is frequently found in the literature (e.g., in Lang,Elliptic Functions, 1st ed.), and was recently studied in detailby Z.-H. Sun and K. S. Williams (2006).
5) If D is a fundamental discriminant, i.e. if D = dK , then it iswell-known that each ϑχ is a primitive form (newform) andhence in this case the ϑχ’s are part of the canonical AtkinLehner basis of M1(|D|, ψD ).
However, in the general case this is no longer true for everyχ ∈ Cl(D)∗ because some of the characters χ ∈ Cl(D)∗ arenot primitive, i.e. they are liftsχ = χ0 ◦ π of characters χ0 ∈ Cl(D0)∗of some “lower level” D0|D (where DD0 = t2 > 1) via the canonical map
π = πD,D0 : Cl(D) → Cl(D0).

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3. Main Results
Theorem 1: The space ΘD is a T(D)-submodule of M1(|D|, ψD )of multiplicity one, and has a canonical basis {ϑχ} consisting ofnormalized T(D)-eigenforms. Furthermore, ϑχ is a cusp form
if and only if χ is not a quadratic character.
STheorem 2: We have ΘD = ΘED ⊕ ΘD , whereΘED = ΘD ∩E1 (|D|, ψD ) denotes the Eisenstein space part andΘSD = ΘD ∩ S1(|D|, ψ) denotes the cusp space part of ΘD , and(1)

S1
dim ΘED = gD and dim ΘD = 2 (hD − gD ).

def
SRemark: Thus ΘD = 0 ⇔ hD = gD ⇔ D is an idoneal discrim-inant. (This implies a result of Kitaoka (1971).)
Theorem 3: Let χ ∈ Cl(D)∗, where D = fD2 dK .
(a) ∃! divisor fχ|fD and a unique primitive character χpr ∈Cl(Dχ), where Dχ = fχ2dK , such that χ = χpr ◦ π̄D,Dχ .
(b) The form ϑχpr ∈ ΘDχ is a primitive form (newform) of level|Dχ|. Moreover, there exist constants cn(χ) ∈ R such thatX(2)ϑχ(z) =cn(χ)ϑχpr (nz),n|f¯χ2where f¯χ = fD /fχ. Furthermore, the function n 7→ cn(χ) is
multiplicative and has generating function
X(3)C(s, χ) :=cn(χ)n−s = L(s, ϑχ)/L(s, ϑχpr ).
n|f¯χ2

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Remark: While L(s, ϑχpr ) is a classical Hecke L-function associated to a Hecke character and hence is well-understood, theL-function L(s, ϑχ) is more complicated and is, in fact, unknown in general.
Thus, (3) does not help in determining the constants cn(χ).
However, C(s, χ) can be computed directly by using facts
about ideals in quadratic orders.
As a consequence, we thus obtain an explicit expression for theL-function L(s, χ) = L(s, ϑχ) :Corollary: If χ ∈ Cl(D)∗, then L(s, χ) has the Euler productY(4)L(s, χ) =Lp(s, χ)pwhere for p - f¯χ the p-Euler factor Lp(s, χ) is given by−s−2s −1Lp(s, χ) = 1 − ap(χ)p + ψD (p)p−s−2s −1= 1 − ap(χpr )p + ψDχ (p)p,whereas for p | f¯χ (and pēp ||f¯χ), it is given by1(1−2s)ēp1−
ψ(p)p(1−2s)ēpDχp1−pLp(s, χ) =+.
1−2s−s−2s1−p1 − ap(χpr )p + ψDχ (p)pRemark: This generalizes the work of Sun and Williams (2006)(for D < 0), who obtained a formula for the p-Euler factors ofL(s, χ) in the case that the class group Cl(D) is cyclic.

7Definition: Let f ∈ Mk (N, ψ) be a T(N )-eigenfunction witheigencharacter λf : T(D) → C. We say that f has CM (complex multiplication) by a Dirichlet character θ if
λf (Tp)θ(p) = λf (Tp),

for all p - N cond(θ),

or, equivalently, if
λf (Tp) = 0 for all p - N cond(θ) with θ(p) 6= 1.
We let MkCM (N, ψ; θ) denote the space generated by all T (N )eigenfunctions f ∈ Mk (N, ψ) which have CM by θ.
Theorem 4: For every discriminant D < 0 we have that(5)

Θ(D) = M1CM (|D|, ψD ) := M1CM (|D|, ψD ; ψD ).

Corollary:(6)

dim Θ(D) = dim M1CM (|D|, ψD ) =X

2ω(f )hD/f 2 ,f |fD

where ω(f ) denotes the number of distinct prime divisors off . Moreover, the dimensions of the Eisenstein part and of thecuspidal part of M1CM (|D|, ψD ) are given byXCM
dim E1 (|D|, ψD ) =2ω(f )gD/f 2 ,f |fD

dim S1CM (|D|, ψD ) =X

2ω(f )(fD/f 2 − gD/f 2 ).

f |fD

Remark: There is no (known) formula for dim M1(|D|, ψD ).

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4. Ingredients
1) Dedekind’s Isomorphism:∼

λD : Cl(D) → Pic(OD ),√D+ D
where OD = Z + Z 2 ⊂ OK is the order of discriminant
D (and/or of conductor fD in K).

2) A classification of the invertible ideals of OD :⇒ the multiplicativity of an(χ),
the value of cn(χ) for n|D, etc.
3) A study of the conductor of χ ∈ Cl(D)∗ : via the isomorphism∼IK (fD OK )/PK,Z(fD ) → Pic(OD ),one can identify each χ ∈ Cl(D)∗ with a Hecke character χ̃
on the group IK (fD OK ) of fractional ideals prime to the ideal
fD OK . A key fact is:
χ is primitive on Cl(D) ⇔ χ̃ is primitive mod fD OK .
4) Genus theory (Gauss/Kronecker/Weber): this identifies quadraticcharacters χ ∈ Cl(D)∗ with certain Dirichlet characters.
5) Extended Atkin-Lehner theory: this describes:
1) the characters λ ∈ T(N )∗ = Hom(T(N ), C) of the Hecke
algebra T(N ) ⊂ End(Mk (N, ψ)) in terms of primitive eigenfunctions (newforms);2) the structure of the T(N )-eigenspace associated to λ:Mk (N, ψ)[λ] = {f ∈ Mk (N, ψ) : f |k Tn = λ(Tn)f, ∀(n, N ) = 1}

9For Theorem 4, we also need:6) (a) The Deligne/Serre theory of Galois representations
ρf : GQ = Gal(Q/Q) → GL2(C)attached to T(N )-eigenfunctions f ∈ M1(N, ψ).
(b) A characterization of characters of ring class fields via(strongly) dihedral Galois representations of GQ (= reinterpretation of a result of Bruckner (1966)).
(c) A characterization of CM forms via their associated Galois
representations (→ Theorem 5 below).

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5. Galois representations
Deligne/Serre (1974): If f ∈ M1(N, ψ) is a normalized T(N )eigenfunction, then ∃! Galois representation
ρf : GQ → GL2(C)such that for all primes p - N
tr(ρf (F rp)) = λf (Tp) = ap(f ),det(ρf (F rp)) = ψ(p).
Furthermore, ρf is irreducible ⇔ f is a cusp form.
Definition: An Galois representation ρ : GQ → GL2(C) is calledstrongly dihedral if Im(ρ) ' Dn is a dihedral group (n ≥ 3).
Moreover, ρ is said to be of dihedral type if Im(ρ)/Z(Im(ρ)) 'Dn is a dihedral group (n ≥ 2).
Theorem 5: Let f ∈ S1(N, ψ) be a newform.
(a) f has CM by some character θ ⇔ ρf is of dihedral type.
(b) f has CM by ψ ⇔ ρf is strongly dihedral.
Theorem 6: Let ρ : G → GL2(C) be Galois representation.
(a) (Hecke, Weil, Deligne/Serre) If ρ is of dihedral type and isodd, then ρ = ρf for some f ∈ S1(N, ψ).
(b) (Bruckner, 1966) ρ is strongly dihedral if and only if thefield Fix(Ker(ρ)) is contained in some ring class field.
Remark: Theorems 3, 5, 6 ⇒ Theorem 4.