Central values of $L$-functions of cubic twists

We are interested in finding for which positive integers $D$ we have rational solutions for the equation $x^3+y^3=D.$ The aim of this paper is to compute the value of the $L$-function $L(E_D, 1)$ for the elliptic curves $E_D: x^3+y^3=D$. For the case of $p$ prime $p\equiv 1\mod 9$, two formulas have been computed by Rodriguez-Villegas and Zagier. We have computed formulas that relate $L(E_D, 1)$ to the square of a trace of a modular function at a CM point. This offers a criterion for when the integer $D$ is the sum of two rational cubes. Furthermore, when $L(E_D, 1)$ is nonzero we get a formula for the number of elements in the Tate-Shafarevich group and we show that this number is a square when $D$ is a norm in $\mathbb{Q}[\sqrt{-3}]$.


Introduction
In the current paper we are interested in finding which positive integers D can be written as the sum of two rational cubes: x 3`y3 " D, x, y P Q.
Despite the simplicity of the problem, an elementary approach to solving the Diophantine equation fails. However, we can restate the problem in the language of elliptic curves. After making the equation homogeneous, we get the equation x 3`y3 " Dz 3 that has a rational point at 8 " r1 :´1 : 0s. Moreover, after a change of coordinates X " 12D z x`y , Y " 36D x´y x`y the equation becomes: which defines an elliptic curve over Q written in its Weierstrass affine form. Thus the problem reduces to finding if the group of rational points E D pQq of the elliptic curve E D is non-trivial. We assume D cube free and D ‰ 1, 2 throughout the paper. In this case E D pQq has trivial torsion (see [23]), thus (1) has a solution iff E D pQq has positive rank. From the Birch and Swinnerton-Dyer(BSD) conjecture, this is conjecturally equivalent to the vanishing of LpE D , 1q.
Without assuming BSD, from the work of Coates-Wiles [2] (or more generally Gross-Zagier [8] and Kolyvagin [14]), when LpE D , 1q ‰ 0 the rank of E D pQq is 0, thus we have no rational solutions in p1q.
In the case of prime numbers, Sylvester conjectured that we have solutions in (1) in the case of D " 4, 7, 8 mod 9. In the cases of D prime with D " 2, 5 mod 9, D is not the sum of two cubes. This follows from a 3-descent argument (given in the 19th century by Sylvester, Lucas and Pepin).
We define the invariant where Ω D " is the real period and c 3D " ś p|3D c p is the product of the Tamagawa numbers c p corresponding to the elliptic curve E D at the unramified places p|3D. The definition is made such that in the case of LpE D , 1q ‰ 0 we expect to get from the full BSD conjecture: where #X is the order of the Tate-Shafarevich group.
From the work of Rubin [21], LpE D , 1q ‰ 0 implies the order of XpE D q is finite. Furthermore, using the Cassels-Tate pairing, Cassels proved in [1] that when X is finite the order #X is a square. We actually show that, when D is a norm in Qr ?´3 s, S D is an integer square up to an even power of 3. Current work in Iwasawa theory shows that for semistable elliptic curves at the good primes p we have ord p p#Xrp 8 sq " ord p pS D q, where Xrp 8 s is the p 8 -torsion part of X (see [12]). However, this cannot be applied at the place 3 in our case.
By computing the value of S D , one can determine when we have solutions in (1) and, assuming the full BSD conjecture, one can find in certain cases the order of X: The goal of the current paper is to compute several formulas for S D . In [20], Rodriguez-Villegas and Zagier computed formulas for LpE p , 1q in the case of primes p " 1 mod 9. In the current paper we are extending on their results and compute similar formulas for all integers D.
Our main theorem is the following: p ei i , S D is an integer square up to an even power of 3.
Theorem 1.1 above follows from the formula for S D presented below. Let K " Qr ?´3 s. For D a norm in Qr ?´3 s, we write D " D 1 D 2 2 such that D 0 " D 1 D 2 is the radical of D. Let π 1 , π 2 " 1 mod 3 be elements of norm D 1 and D 2 respectively. Let σpDq the number of distinct primes dividing D and ϕ Euler's totient function.
Theorem 1.2. Using the above notation, let D " ś pi"1p3q p ei i be a positive integer that is a product of split primes in K and D 0 " ś p|D p be its radical. Then S D is an integer square up to an even power of 3 and we have: where the term T D {3 is an integer if σpDq is even and T D { ?´3 is an integer if σpDq is odd. Moreover, we have the formula: where: • θ r pzq " ř nPZ p´1q n e πipn`r D´1 6 q 2 z , for r " 0, 1 are theta functions of weight 1{2, • τ "´b`?´3 2 is a CM point such that b 2 "´3 mod 12D 2 and pπ 1 π 2 q 2 divides pτ q, • H O is the ray class field of modulus 3D 0 , • ω k0 is the unique cube root of unity that makes T D {3 or T D { ?´3 an integer.
This theorem follows from a more general result for all integers D prime to 6 that is proved using automorphic methods: Theorem 1.3. For all integers D prime to 6, 3c 3D S D is an integer and we have the formula: where Θ K pzq " ÿ a,bPZ e 2πizpa 2`b2´a bq is the theta function of weight one associated to the number field K " Qr ?´3 s, ω "´1`?´3 2 is a third root of unity, and and H 3D is the ring class field associated to the order O 3D " Z`3DO K .
Note that each of the elliptic curves E D is a cubic twist of E 1 . In the case of quadratic twists of elliptic curves, an important tool in computing the values of the L-functions is the work of Waldspurger [28]. For example, this is used to obtain Tunnell's theorem for congruent numbers in [26]. However, the cubic twist case proves to be significantly more difficult. We instead take advantage of the fact that E D is an elliptic curve with complex multiplication by O K " Zrωs the ring of integers of the number field K " Qr ?´3 s. Then from CM theory there is a Hecke character χ ED : AK{KˆÑ Cˆsuch that LpE D , sq " L f ps, χ ED q and we compute the value of L f ps, χ ED q using automorphic methods.
We present now an outline of the proof of Theorem 1.3. To compute the value of Lps, χ ED q we look at the Hecke character adelically and using Tate's thesis we compute Tate's zeta function Zps, χ ED , Φ K q for Φ K a Schwartz-Bruhat function in SpA K q. After integrating we get a linear combination of Eisenstein series. By evaluation at s " 1, we write LpE D , 1q as a linear combination of theta functions at CM-points. We further show using Shimura's reciprocity law that the terms are all Galois conjugates over K.
The idea of the proof of Theorem 1.2 is based on factoring each weight one theta function Θ K pzq into a product of theta functions of weight 1{2. The method we are using is a factorization lemma of Rodriguez-Villegas and Zagier from [19] applied to the formula in Theorem 1.3. This gives us the square of a linear combination of theta functions evaluated at CM points. Finally, using Shimura reciprocity law, we show that all the factors are Galois conjugates to each other and recover an integer square.
Note that using the formula (4) we can show that an integer D cannot be written as the sum of two cubes by computationally checking whether LpE D , 1q ‰ 0. Furthermore, assuming BSD, S D " #X and thus we can compute the expected order of X explicitly. result in the current form, very helpful discussions and for help with using PARI for computational purposes. We would also like to thank Max Planck Institute in Bonn and Tsinghua University in Beijing for their hospitality.

Background
Let K " Qr ?´3 s and denote O K " Zrωs its ring of integers, where ω "´1`?´3 2 is a fixed cube root of unity. We will denote by K v the localization of K at the place v, and

The L-function
Our goal is to compute several formulas for the central value of the L-function LpE D , 1q of the elliptic curve E D : x 3`y3 " Dz 3 . The elliptic curve E D has complex multiplication (CM) by O K . Then from CM theory we can find a Hecke character χ : AK{KˆÑ C corresponding to the elliptic curve E D such that LpE D , sq " L f ps, χ D ϕq. We can compute explicitly χ " χ D ϕ (see Ireland and Rosen [11] for more details), where ϕ is the Hecke character corresponding to E 1 and χ D is the Hecke character corresponding to the cubic twist. More precisely, writing the characters classically, we have: • ϕ : Ip3q Ñ Kˆis defined on the set of ideals prime to 3 by taking ϕpAq " α, where α is the unique generator of the ideal A such that α " 1 mod 3.
• χ D : ClpO 3D q Ñ t1, ω, ω 2 u is the cubic character defined below in Section 2.2; it is defined over ClpO 3D q the ring class group corresponding to the order O 3D " Z`3DO K .

The cubic character
We define the cubic character χ D and recall some of its properties following Ireland and Rosen [11]. Let ω "´1`?´3 2 and for α P Zrωs prime to 3, we define the cubic residue character`α˘3 : Ip3αq Ñ t1, ω, ω 2 u, where Ip3αq is the set of fractional ideals of K prime to 3α. For a prime ideal p of K, we define´α p¯3 " ω j , for 0 ď j ď 2 such that It is further defined multiplicatively on the fractional ideals of Ip3αq.
It is easy to check that the definition makes sense, as the group pZrωs{pZrωsqˆhas Nm p´1 elements, thus α Nm p´1 " 1 mod p. As Nm p " 1 mod 3, we can factor out α Nm p´1´1 " pα pNm p´1q{3´1 qpα pNm p´1q{3´ω qpα pNm p´1q{3´ω2 q and as K is an UFD, p divides exactly one of these terms, exactly α pNm p´1q{3´´α p¯3 .
The character χ D is defined following [11] to be: We also define χ π pAq "`π A˘3 where π is a generator of an ideal of norm D. Note that χ π pAq " χ π pAq. An important result is the cubic reciprocity law (see [11] for more details): Theorem 2.1. (Cubic reciprocity law). For π 1 , π 2 " 2 mod 3 generators of the prime ideals It follows immediately from the cubic reciprocity law that for α "˘1 mod 3, we have χ D pαq " χ α pDq. Also from the cubic reciprocity it follows that χ D ppαqq " 1 for α " a mod 3D, where a is an integer prime to 3D. Thus χ D is invariant on the ideals of P Z,3D " tpαq: α P K such that α " a mod 3D for some integer a such that pa, 3Dq " 1}. The ring class group of the order O 3D " Z`3DO K is defined to be ClpO 3D q " Ip3Dq{P Z,3D , where Ip3Dq is the set of fractional ideals prime to 3D, and thus χ D is invariant on ClpO 3D q.
Finally, we can relate the cubic character to the Galois conjugates of D 1{3 : Lemma 2.2. Let D be an integer prime to 3 and π a generator of an ideal of norm D. Then for an ideal A of K prime to 3D, we have: where σ A P GalpC{Kq is the Galois action corresponding to the ideal A in the Artin correspondence. Note that this immediately implies D 1{3 χ D pAq " pD 1{3 q σ´1 A .
Proof. It is enough to show the result for a prime ideal p of K, p prime to 3D. Let σ p "´L {K p¯b e the Frobenius element corresponding to the prime ideal p of O K , where L " Krπ 1{3 s. Then from the definition of the Frobenius, element for π 1{3 P L, we get pπ 1{3 q σp " pπ 1{3 q Nm p mod pO L .

Hecke characters
A classical Hecke character r χ : Ipf q Ñ Cˆof conductor f can be expressed on the set of principal ideals P pf q prime to f in the form r χppαqq " r ǫpαqr χ´1 0 pαq, where r ε : pO K {f O K qˆÑ T is a character taking values in a finite group T and r χ 0 is an infinity type continuous character, meaning that r χ 0 : CˆÑ Cˆis a continuous character.
The idelic Hecke character is a continuous character χ : Aˆ{KˆÑ Cˆ. There is a unique correspondence between the idelic and the classical Hecke characters defined as follows: at 8 for z P C we define χ 8 pzq " r χ´1 0 pzq for z P Cˆand at the places v ∤ f we define χpOv ̟ v q :" r χpp v q, for ̟ v a uniformizer of O Kv and p v the prime corresponding to the place v. At the places v|f , the value of χ v can be determined using the Weak Approximation Theorem.
We are interested in the character χ " χ D ϕ defined before. By abuse of notation, we will use ϕ, χ D both for the classical and the adelic Hecke characters.
Recall ϕ : Ip3q Ñ Cˆis the Hecke character defined by χppαqq " α for α " 1 mod 3. For the place v ∤ 3, denote by ̟ v a uniformizer of O Kv such that ̟ v " 1 mod 3. Then for ϕ : AK Ñ Cˆ, we can define: Recall χ D : Ip3Dq Ñ t1, ω, ω 2 u is the cubic character and we showed that it is well-defined on ClpO 3D q, the ring class group for the order O 3D " Z`3DO K . We define the character χ D adelically over Note that we can rewrite l f P AK ,f in the form l f " kl 1 with k P Kˆand l 1 P ś v∤8 OK v . We can find k 1 P O K such that k 1 " l 1 mod 3DO Kv and we define χ D,f plq " χ D,f pl 1 q " χ D ppk 1 qq. More precisely, we get: The values of χ D and ϕ at the ramified places can be computed using the Weak approximation theorem.
3 LpE D , 1q and Tate's zeta function In this section we will compute the value of LpE D , 1q " Lp1, χ D ϕq, working with χ D , ϕ as automorphic Hecke characters. Let K " Qr ?´3 s and ω "´1`?´3 2 a fixed cube root of unity as before. We will show the following result: where Θ K pzq " ř m,nPZ e 2πipm 2`n2´m nqz , H 3D is the ring class field for the order O 3D " Z`3DO K and c 3D " ś p|3D c p is the product of the Tamagawa numbers c p of E D .
We will compute the formula (5) using Tate's zeta function. We start by recalling some background and notation.

Haar measure
We take V " K as a quadratic vector space over Q with the norm as its quadratic form. We take dx v to be the the self-dual additive Haar measure and dˆα v the multiplicative Haar measure dv x v " dxv |xv|v normalized such that volpOK v q " 1, if v ∤ 8, and dˆz " dz |z|8 where dz is the usual Lebesgue measure, and |z| 8 " |z| 2 C is the square of the usual absolute value over C.

Schwartz-Bruhat functions
We choose the Schwartz-Bruhat function Φ f P SpA K,f q such that Tate's zeta function Zps, Φ, χ D ϕq defined below to be nonzero. More precisely, char pa`DZprωsq for p|D,

Tate's zeta function
We recall Tate's zeta function. For a Hecke character χ : AK{KˆÑ Cˆand a Schwartz-Bruhat and globally as Zps, As a global integral this is It has meromorphic continuation to all s P C and in our case it is entire. We will compute Z f ps, χ f , Φ f q for χ " χ D ϕ and the Schwartz-Bruhat function Φ f chosen above. From Tate's thesis, we have the equality of local factors L v ps, χ D ϕq " Z v ps, χ D ϕq at all the unramified places, and thus L f ps, χ D ϕq " Z f ps, χ D ϕq ź p|3D L p ps, χ D,p ϕ p q Z p ps, χ D,p ϕ p , Φ p q . As ϕ, χ D and |¨| are trivial when Φ p is nonzero for p|3D, we can compute easily ś p|3D Z p ps, χ D,p ϕ p , Φ p q " ś p|D vol pZ`3DZ p rωsqˆvol p1`3Z 3 rωsqˆand this equals 1 6 ś p|D pp´`p 3˘q´1 . The terms L p ps, χ D ϕq " 1 for p|3D by definition. Thus for all s and for Φ the Schwartz-Bruhat function chosen above, we have: where V 3D " 1 6 ź p|D pp´´p 3¯q´1 .
Next we compute the value of Z f ps, χ D,f ϕ f , Φ f q as a linear combination of Hecke characters and use (6) to get the value of L f ps, χ D,f ϕ f q: Lemma 3.2. For all s P C and the Schwartz-Bruhat function Φ f P SpA K,f q chosen above, we have: where Proof. We first take the quotient by Kˆin the integral defining Z f ps, χ D ϕ, Φ f q and get: where |¨| C is the usual absolute value over C. Then the integral reduces to: Similarly, |¨| f is trivial on units, thus on U p3Dq, while χ D is invariant on U p3Dq by definition. Moreover, ϕ is trivial on all the units at all the unramified places, while, at 3, ϕ is invariant under 1`3Z 3 rωs, thus it is trivial on all of U p3Dq. Thus we can take the quotient by U p3Dq as well. Note that the integral is now a finite sum: We compute volpU p3Dqq " volp1`3Z 3 ωq ś p|D volpZ`DZ p rωsq " V 3D and, changing notation, we get: Finally together with (6) we get the result of the lemma.

Representative classes of ClpO 3D q
We will use the following lemma (see [18]) that is easy to show: Lemma 3.3. Any primitive ideal of O K can be be written in the form A " ra,´b`?´3 2 s Z as a Z-module, where b is an integer (determined only modulo 2a) such that b 2 "´3 mod 4a and Nm A " a.
Conversely, given an integer satisfying the above congruence and A defined as above, we get that A is an ideal in O K of norm a. We will use the notation k A for the generator k A " 1 mod 3 of a primitive ideal A in O K . If we choose a lattice such that A " ra,´b`?´3 2 s Z , we denote the corresponding CM point τ A "´b`?´3 2a .
We can write adelically ClpO 3D q » U p3DqzAK ,f {Kˆ. This follows from the Strong approximation theorem, as K is a PID and thus we have U p3DqzAK ,f {Kˆ-p ź p|3D pZ p rωsqˆ{pZ3 DZ p rωsqˆq{ ´ω . Then we can define the map ź p|3D pZ p rωsqˆ{pZ`3DZ p rωsqˆ{ ´ω Ñ Ip3Dq{P Z,3D given by where we choose the representative˘αω k , such that α 3 " 1 mod 3, and k α is an element of O K such that k α " α v mod 3D. Note that this is well defined as pk α q gives us a unique class in ClpO 3D q, and two elements α 1 , α 2 get sent to the same class in ClpO 3D q only if α 1 " α 2 mod 3D.
Thus for α f P p O K we can choose a class rA α s in ClpO 3D q by taking a representative A α f " pk α q, for k α P O K such that k α " α p mod 3DZ p rωs for p|3D. Note that this choice is not unique. However, we can pick the representatives A α to be primitive ideals.
Thus we can further write A α as a Z-lattice A α " ra,´b`?´3 2 s Z , where a " Nm A α and b is chosen (not uniquely) such that b 2 "´3 mod 4a. We define the corresponding CM point τ Aα "´b`?´3 2a .

Eisenstein series of weight 1
We will now connect Ips, α f , Φ f q " ř kPKˆk |k| 2s C Φ f pkα f q to an Eisenstein series. We define the following classical Eisenstein series of weight 1: Here the sum is taken over all m, n P Z except for the pair p0, 0q, and ε "`3˘is the quadratic character associated to the field extension K{Q. The Eisenstein series E ε ps, zq does not converge absolutely for s " 0, but we can still compute its value using the Hecke trick (see [9]). We compute its Fourier expansion at s " 0 in the following section.
Using this notation, we have the following equality: OK v , let A α f " pk α q be a choice of an ideal in the corresponding class of ClpO 3D q. We write A α " ra α ,´b`?´3 2 s Z and take τ Aα "´b`?´3 2aα the corresponding CM point. Then we have: Remark 3.5. Note that the variable τ Aα on the left hand side is not uniquely defined. However, the function is going to be invariant on the class rA α s in ClpO 3D q.
Recall that we defined k α such that k α " α p mod 3DZ p rωs for all p|3D. Then kk α P a`3DZ p rωs for pa, pq " 1 and kk α P 1`3Z 3 rωs. Furthermore, for We can rewrite this further: Finally, we will make this explicit. Note that we must have kk α P A α , where A α " pk α q, as well as kk α P a p`D Z p rωs for some integer a p , pa p , pq " 1, and kk α P 1`3Z 3 rωs. By the Chinese remainder theorem, we can find an integer a such that a " a p mod D and a " 1 mod 3.
Here P Z,3D " tk P K : k " a mod 3DO K for some integer a, pa, 3Dq " 1u and P 1,3 " tk P K : k " 1 mod 3u. We rewrite: Finally, we want to write the elements of A α X P Z,D X P 1,3 explicitly. Recall that we can write A α as a Z-lattice A α " ra,´b`?´3 2 s Z . Then all of the elements of A are of the form ma`n´b`?´3 2 for some integers m, n P Z. Moreover, note that the intersection of A and P Z,3D " tk P O K : k " n mod 3D, for some integer n, pn, 3Dq " 1u is tma`3Dn´b`?´3 2 : m, n P Zu. Further taking the intersection with P 1,3 , we must have ma " 1, thus, as a is norm in O K , m " 1 mod 3, and we can rewrite Ips, α f , Φ f q in the form: By changing n Ñ´n and taking out a factor of a 1´2s , we have: Note that for Repsq ą 1 the integral converges absolutely, and we can rewrite the sum as: where εpmq "`m 3˘i s the usual quadratic character. On the right hand side we recognize the Eisenstein series E ε p2s´2, τ Aα q and we get Ips, α f , Φ f q " 1 2 a 1´s kα E ε p2s´2, Dτ Aα q. By analytic continuation, we can extend the equality to all s P C. Now we can rewrite the linear combination in Lemma 3.2 by taking representatives A for the classes of ClpO 3D q. Note that for α P p OK with α " 1 mod 3 we have ϕ f pα f q " 1 and Using the lemma above and after inverting each class A Ñ A in ClpO 3D q, we get: For all s, taking representative ideals A " ra,´b`?´3 2 s Z for the classes in the ring class group ClpO 3D q, we have: where A " pk A q with k A " 1 mod 3 and τ A "´b`?´3 2a the associated CM points.
3.6 Fourier expansion of E ε ps, zq at s " 0 We want to connect the Eisenstein series E ε ps, zq " ÿ c,d 1 εpdq p3cz`dq|3cz`d| 2s to the theta function associated to the number field K. It is a modular form of weight 1 for the congruence group Γ 1 p3q. Note that this differs from the theta function Θ K chosen by Rodriguez-Villegas and Zagier in [20] by a factor of 1{2.
More precisely, we are going to show the following version of the Siegel-Weil theorem: Proof. We will show this by computing the Fourier expansion of E ε ps, zq at s " 0 using the Hecke trick and comparing it to the Fourier expansion of Θ K pzq. We will follow closely the exposition of Pacetti [16]. This is also done by Hecke in [9]. We first rewrite E ε ps, zq in the form: We define for z in the upper-half plane Hpz, sq " ř mPZ 1 pz`mq|z`m| 2s and then we can rewrite the form above as: Pacetti ([16]), following Shimura (Lemma 1, p. 84, [24]), computed the Fourier expansion of Hpz, sq when s Ñ 0 to be lim sÑ0 Hps, zq "´πi´2πi 8 ř n"1 q n . It gives us in the relation above: We compute separately the inner sum and get: where Gpεq " is the quadratic Gauss sum corresponding to ε. Then we can rewrite: Since ε is a quadratic character, we compute Lp1, εq " π ?
3 9 (see Kowalski [15]) and this gives us the Fourier expansion: It is actually easy to show that ÿ m|n εpmq represents the number of ideals of norm n in O K and we can recognize the sum in the bracket on the RHS of (7) to equal the theta function Θ K pzq " 1`6 ř A e 2πipNm Aqz , which finishes the proof.

Formula for Lp1, χ D ϕq
Applying Corollary 3.6 for s " 1 we get L f p1, Furthermore, from Theorem 3.7, this is the same as: We need one more step before rewriting the formula as a trace. We will use the following lemma: s Z a primitive ideal of norm Nm A " a, with generator A " pk A q, where k A " 1 mod 3 and τ A "´b`?´3 2 , we have: Proof. Since A " ra,´b`?´3 2 s Z as a Z-lattice, we can write its generator k A in the form k A " ma`3n´b`?´3 2 for some integers m, n such that m " 1p3q and gcdpm, 3nq " 1. Then we can find integers A, B such that mA`3nB " 1, and thus p A B 3n m q is a matrix in Γ 1 p3q. Since Θpzq is a modular form of weight 1 for we can compute the term on the LHS to be Θ K ppAτ AB qk A q and, after expanding, we are evaluating 4a`a bB`b p´mA`3nBq`1 2 P Z and thus using the period 1 of Θ K we get This finishes the proof.
Using the Lemma above we can rewrite (8) as: Now we will rewrite the formula (9) as a trace. We can define f pzq " Θ K pDzq Θ K pzq and this is a modular function for Γ 0 p3Dq. We will prove in Section 5 in Proposition 5.5 that f pωq P H 3D , the ring class field of corresponding to the order O 3D . Moreover, we show in the same proposition that, for A " ra,´b`?´3 2 s Z a primitive ideal in O K , we have the Galois conjugate: where σ A is the Galois action corresponding to the ideal A via the Artin map. Furthermore, from Corollary 2.2 we have pD 1{3 q σ´1 A " D 1{3 χ D pAq and then formula (9) becomes: Moreover, D 1{3 P H 3D (see for example Cohn [3]). Thus we can rewrite the sum on the left hand side as Tr H3D {K´D 1{3 ΘK pDωq ΘK pωq¯. We can compute the extra terms as well. Rodriguez-Villegas and Zagier in [20] cite Θ K´´9`?´3 18¯"´6 Γ`1 3˘3 {p2πq 2 . Using several of the properties of Θ K proved in the Appendix, we can compute Θ pωq " Γ`1 3˘3 {p2π 2 q.
, we get the formula of Theorem 3.1: Note that this implies S D P K. Moreover, as D 1{3 ΘpDωq{Θpωq is invariant under complex conjugation, we get S D P R which furthermore implies S D P Q. We will show in Section 4.5 that actually 3c 3D S D P Z. Remark 3.9. If we take D " D 1 D 2 2 such that D 0 " D 1 D 2 is square-free, note that the character χ D " χ D1 χ D2 is well defined on the class group ClpO 3D0 q. Then the above computations work for D 0 and the character χ D " χ D1 χ D2 instead of χ D0 " χ D1 χ D2 and we get: Note that, for D " D 1 D 2 2 , LpE D , sq " Lps, χ D1 χ D2 ϕq and thus we have: As before from Corollary 2.2 we have pD 2 χ D2 pAq and finally we can write the expression above as: 4 Second formula for S D For r P Z, µ P t1{2, 1{6u, we define the theta functions of weight 1{2: Throughout the paper we will use the notation r P Z{DZ to mean any family of representatives for the residues r mod D. We denote In this section we will use a Factorization formula of Rodriguez-Villegas and Zagier from [19] to show the following theorem: where T D {3 P Z if σpDq is even and T D { ?´3 P Z if σpDq is odd. We have the exact formula: Here τ "´b`?´3 2 is a CM-point, with b 2 "´3 mod 12D 2 , π 1 , π 2 are elements in O K such that π 1 , π 2 " 1 mod 3, π 1 π 2 has norm D 0 and π 1 π 2 2 has norm D, and such that pπ 1 π 2 q 2 divides the ideal´´b`?´3 2¯, H O is the ray class field of modulus 3D 0 and ω k0 is the unique cube root of unity that makes T D real or purely imaginary.
Below we discuss the details of D square-free. All definitions and proofs can be easily extended to all D. We do that in Section 4.6.
Take τ "´b`?´3 2 a CM point such that b 2 "´3 mod 12D 2 and an element π " 1 mod 3 of norm D in O K such that π 2 divides the ideal p´b`?´3 2 q.
We will use the notation: We are actually going to show in Corollary 4.10 that S 1{2 " 0, thus it is enough to compute the formula (14) for S 1{6 .
Using a Factorization formula of Rodriguez-Villegas and Zagier from [19] we will write the theta functions Θ µ of weight 1 as linear combinations of products of theta functions of weight 1{2 in Proposition 4.4. We define: for a cubic root of unity ω k0 , then S 1{6 " |T D,1{6 pτ q| 2 .
We show in Lemma 4.17 that T D " 1 ϕpDq Tr HO{K θ 1,1{6 pτ q θ 0 pτ q π´2 {3 ω k0 and that T D P K. Furthermore, we show in proposition 4.12 that T D " p´1q σpDq T D and thus T D P Q or T D { ?´3 P Q and thus Moreover, in Section 4.5 we show that 3c 3D S D is an integer, hence T D {3 P Z for σpDq even and T D { ?´3 P Z for σpDq odd. Finally, for D a product of split primes, we have S D ‰ 0 only for D " 1 mod 9. In this case the Tamagawa numbers equal c 3D " 3 1`σpDq , thus we have: Hence S D is an integer square up to an even power of 3 and this finishes the proof of Theorem 4.1.

Factorization lemma
As in the previous section, we write a primitive ideal A as a lattice A " ra,´b`?´3 2 s Z for a " NmpAq and b 2 "´3 mod 4a. We also define the CM point τ A "´b`?´3 2a corresponding to the Z-lattice. We also denote by k A the generator of A such that k A " 1p3q and we write the generator in the form k A " n a a`m a τ A .
A direct application of this is the following: Lemma 4.3. With notation as above, we have: Using the lemma above and the notation θ r,µ pzq " ÿ nPZ p´1q n e πipn`r D´µ q 2 z , we show: Proposition 4.4. For ideals A " ra,´b`?´3 2 s Z , A 1 " ra 1 ,´b`?´3 2 s Z and b such that b 2 "´3 mod 4D 2 a 2 a 1 , we have: Proof. We apply Lemma 4.3 for µ "´1{6 and ν " 1{2, D odd, z "´b`?´3 2Daa1 . It is easy to see on the LHS of the equation that we have θ
? a 1 . We get: Now we only have to show that the RHS equals Θ µ pDτ A q. We claim: Since the absolute values of the two sides already agree, we only need to show that the arguments agree as well, meaning 2πi`m 2`n 2`D mn 2a˘"´2 πi 6a mod 2πiZ. This is equivalent to showing that´m  z " E˚, 0 pzq`ωE˚, 1 pzq`ω 2 E˚, 2 pzq.
A particular case of Lemma 4.4 is for D " 1. As Θ pτ A {3q " 0 from Lemma 6.3 from the Appendix, we get: Corollary 4.5. For b 2 "´3 mod 12a 2 a 1 and A, A 1 as above, we have Let f r,µ pzq " θ r,µ pzq θ 0 pzq . Taking the ratios of the theta functions in Proposition 4.4 and Corollary

we get:
Corollary 4.6. Under the same conditions as above, we have: We are interested in the Galois conjugates of f r,µ pτ q for τ "´b`?´3 2 such that b 2 "´3 mod 12D 2 . For A " ra,´b`?´3 2 s Z a primitive ideal and k A " 1 mod 3 its generator, we write k A in the form k A " n a a`m a´b`?´3 2 with 3|m a and n a " 1p3q. In Section 5.2 we will show in Proposition 5.6 that f r,µ pτ q σ´1 A " f n 1 a r,µ pτ q, where σ A is the Galois action corresponding to the ideal A via the Artin map and n 1 a " n a p3Dq with n 1 a odd. We also show in Lemma 5.4 in the same section that we have we have pf r,µ pτ qq σ´1 A " f r,µ pτ A q, thus we get the following lemma: Lemma 4.7. For an ideal A " ra,´b`?´3 2 s generated by n a a`m a´b`?´3 2 such that m a " 0 mod 3, n a " 1 mod 3 and b 2 "´3 mod 12aD 2 , we have: a r,µ pτ q for n 1 a " n a p3Dq with n 1 a odd.
Using the lemma above, we can rewrite Corollary 4.6: Corollary 4.8. Under the same conditions as above, for A " pn a a`m a´b`?´3 2 q, A 1 " pn a1 a 1m a1´b`?´3 2 q with b 2 "´3 mod 12a 2 a 1 D 2 , we have: where n 1 a " n a p3Dq, n 1 a1 " n a1 p3Dq and n 1 a , n 1 a1 odd.

S D as an absolute value
In the following we will use Corollary 4.8 for a choice of representative ideals for the classes of the ring class group ClpO 3D q. We show first: Proposition 4.9. For τ "´b`?´3 2 such that b 2 "´3 mod 12D 2 and π " 1 mod 3 an element of norm of D such that pπq 2 divides p´b`?´3 2 q, we have: Proof. The structure of the ring class group of conductor 3D for D " ś pi"1 mod 3 p i is given by ClpO 3D q -pZ{DZqˆ(see for example Cox [4]). We will choose as representatives for the classes of ClpO 3D q ideals A s such that Nm A s " s mod D. For b fixed, b 2 "´3 mod 12D, we take: where a s " NmpA s q " s mod D, n s " 1 mod 3D, m s " 0 mod 3. Note that this gives us m " b´1ps´1q mod 3D. Moreover, it is easy to check that the ideals A s for s P pZ{DZqˆare in different classes in ClpO 3D q. We take as before π the element of norm D such that pπq 2 divides the ideal pτ q " p´b`?´3 2 q. Then note that χ D pαq " χ π p α α q " χ π p α 2 |α| 2 q. As b " ?´3 mod π, we get α s " n s a s`ms´b`?´3 2 " s mod π and thus χ π ppα s qq " χ π ps 2 {sq " χ π psq.
Taking representatives s P Z{DZ, s " 1 mod 6, we get m s " 0 mod 6 and n s " 1 mod 6. Summing up over r P Z{DZ with r " 1 mod 6 and taking A 1 " p1q in Corollary 4.8, we get: Summing up for all ts P pZ{DZqˆ, s " 1p6qu and rearranging the terms, we get: f sr,µ pτ q χ π prsq¨f r,µ pτ q χ π prq.
Finally, we will further modify the sums on the RHS in order to sum up over r P pZ{DZqâ s well. In order to emphasize the dependence of θ r,µ on D we will use the notation f r{D pzq " θr,µpzq θ0pzq " ř nPZ e πipn`r D´µ q 2 z p´1q n θ0pzq . Moreover, for p i1 . . . p i k |D, denote: We claim that for k ě 1 we have S pi 1 ...pi k " 0. Note that we can rewrite thus showing S pi 1 ...pi k " 0 for k ě 1 proves our result. To see that S pi 1 ...pi k " 0, let D 1 " D{pp i1 . . . p i k q. We recognize each of the inner sums ÿ r 1 PZ{D 1 Z r"1pmod 6q f sr 1 {D 1 pτ qf r 1 {D 1 pτ q of S pi 1 ...pi k to be equal to D 1 3{2 Θ µ pD 1 τ As q Θ pτ As q from (18) for D :" D 1 .
Denote m " D{D 1 . From the properties of the cubic character, we have χ D " χ m χ D 1 . Moreover, from our choice of ideals, we have Θ µ pD 1 τ As q Θ pτ As q " or s " s 1 mod 3D 1 , as A s and A s 1 are in the same class in ClpO 3D 1 q. Then we can rewrite the sum as: In the inner sum we are summing over s modulo m for all s in pZ{mZqˆ. Moreover, χ m pA s q is a nontrivial character as a function of s, as m 1{3 χ m pA s q " pm 1{3 q σA s " m 1{3 for all A s iff m 1{3 P Qr ?´3 s. As we are summing a non-trivial character over a group, the sum is 0. This finishes the proof.
Using the above proposition now it is easy to see: Corollary 4.10. S D " 1 3c3D S 1{6 and S 1{2 " 0.

S D as a square
In the following we will rewrite Proposition 4.11 so that we get a square. Define F r,µ pzq " f r,µ p3zq and take: R D,µ pzq " ÿ rPpZ{DZqr "1p6q F r,µ pzqχ π prq.
With this notation, we have showed in Proposition 4.9 that S 1{6 " |R D,1{6 pτ {3qD´1 {3 | 2 . One can show that R 3 D,1{6 pτ {3q P K and actually R D,1{6 pτ {3q is really close to being an integer. We will show in this section the following: where T D " R D,1{6 pτ {3qπ´2 {3 ω k0 and T D " p´1q σpDq T D and thus T D is real or purely imaginary.
Here ω k0 is the unique cube root that makes T D real or purely imaginary.
Proof. Let b 1 " b mod 4D 2 , and b 1 ı 0 mod 3. Without loss of generality we can actually pick b, b 1 such that pb 2`3 q{12D 2 and pb 12`3 q{4D 2 are prime to 3D. Let π " 1p3q be an element of norm D such tht pπq 2 divides pτ q. Then we can find ideals A, A 1 prime to 3D such that:

2˙.
We can write the generators k A , k A 1 " 1 mod 3 of A and A 1 , respectively, in the form k a " an a`ma´b`?´3 2 , k a 1 " a 1 n a 1`m a 1´b`?´3 2 , where m a , m 1 a " 0 mod 3, and n a , n 1 a " 1 mod 3. Let τ A "´b`?´3 2a , τ A 1 "´b We are going to show that: These two relations will imply: (iii) R D,1{6 pτ q " χ π pn a qχ π pn a 1 qR D,1{6 pτ {3q.
To show (iii), note that we are in the conditions of Lemma 5.4 from Section 5, as F r,1{6 is a modular function of level 18D 2 . Then F r,1{6 pτ q σ´1 A 1 " F r,1{6 pτ A 1 q and thus we get R D,1{6 pτ A 1 q " We can rewrite this as pR D,1{6 pτ A 1 qq σ A 1 " R D,1{6 pτ q and using piq, we get R D,1{6 pτ q " Finally this implies R D,1{6 pτ q " χ π pn a qχ π pn a 1 qR D,1{6 pτ {3q and we take ω k " χ π pn a qχ π pn a 1 q to get the result.
Using the notation T D " R D,1{6 pτ qπ´2 {3 ω k 1 this is equivalent to T D " p´1q σpDq T D .
Note that we can think of ω k 1 as the unique root of unity which makes R D,1{6 pτ qω k 1 π´2 {3 either real or purely imaginary. We actually give a formula for ω k 1 in the proof of Proposition 4.14.
We show first that, for r odd, we can rewrite the terms θ prq,µ p3z{D 2 q as: From the definition, we have θ prq,µ p3zq " ÿ nPZ e πipn´Dµq 2 3z p´1q n e 2πinr{D . Choosing as before s P Z{DZ such that s " 1 mod 6, we sum over all n modulo D: ÿ nPZ e πipDn`s´Dµq 2 3z p´1q Dn`s e 2πipDn`sqr{D .
To actually compute the term on the RHS, we recall a few facts about cubic Gauss sums. We can write π " ś pi|D π i , where π i is a generator of norm p i with π i " 1p3q. Then: Gpχ π q " ź pi|D χ π{πi pπ i qGpχ πi q.

Invariance under the Galois action
We will write below M D as a trace.
Lemma 4.17. M D P K and we can write it as a trace: where H O is the ray class field of modulus 3D, ϕ is Euler's totient function and f 1,1{6 pτ q " θ 1,1{6 pτ q{θ 0 pτ q Recall we defined T D " R D,1{6 pτ qπ´2 {3 ω k 1 and from Lemma 4.13 we have T D " R D,1{6 pτ {3qπ´2 {3 ω k0 for k 0 " k`k 1 . As T D " p´1q σ T D from Proposition 4.14, we get immediately from Lemma 4.17: Corollary 4.18. T D P Q when σpDq even and T D { ?´3 P Q when σpDq odd and we have the formula: Proof of Lemma 4.17: We can write explicitly M D " ř rPpZ{DZqˆf r pτ qχ π prqπ´2 {3 . For an ideal A " ra,´b`?´3 2 s Z with generator k a "´n a a`m a´b`?´3 2¯w ith 6|m a , n a " 1p6q, we are going to have the Galois transformation: To show this, note that from Lemma 4.7 we have f r,1{6 pτ q σ´1 A " f rna,1{6 pτ q. We compute pπ 1{3 q σ´1 A " χ π pAqπ 1{3 . Furthermore, χ π pAq "`π A˘3 "´π A¯3 "´π naa`mab¯3 and n a pn a a`m a bq 2 " a mod π, so we have´π naa`mab¯3 "´π n´1 a¯3 "´π na¯3 " χ π pn a q.
Moreover, taking the ideals Ar "´1`b˚p1´r˚q´b`?´3 2¯, where b˚" b´1 mod D and r˚" r´1 mod D, we have Nm A r " a Ar " r´1 mod 3D and n Ar " r mod 3D, and then: Define the group G 0 " trArs, r P pZ{DZqˆu. It is a subgroup of GalpH O {Kq and G 0 isomorphic to pZ{DZqˆ. We define the fixed field of G 0 in GalpH O {Kq to be H 0 " th P H O : σphq " h, @σ P G 0 u and from Galois theory this implies GalpH O {H 0 q -G 0 . Then we can rewrite the relation above as Moreover, if we take the trace further to K, we get Tr HO{K pf 1 pτ qπ´2 {3 q " # GalpHO{Kq # GalpHO{H0q Tr HO{H0 pf 1 pτ qπ´2 {3 q " ś p|D pp´1q Tr HO{H0 pf 1 pτ qπ´2 {3 q.

Integrality
In Section 3 we have showed that S D P Q. We will show below that S 1{6 P Z, thus 3c 3D S D P Z.
Recall that S 1{6 " 3c 3D S D " Tr H3D {K ΘpDωq Θpωq D 1{3 . Note that it is enough to show that D 1{3 ΘpDωq{Θpωq is an algebraic integer, as its trace would be a rational number as well as an algebraic integer, thus an integer. Moreover, it is enough to show that ΘpDωq{Θpωq is an algebraic integer.
We will use the following Lemma: Then, for z " τ , since jpτ q is an algebraic integer, we get that f pτ q is the root of a polynomial with coefficients that are algebraic integers, and thus f pτ q is an algebraic integer as well.
First we will show that 2 ΘpDωq Θpωq is an algebraic integer. We have showed that Θp´Dτ q θ 0 p´τ q 2 . Since e´2 πi{24 3 3{4 is an algebraic integer, it is enough to show that Θp´Dτ q θ0p´τ q 2 is one as well. Recall that θ 0 pzq " ηpz{3q and take f 0 pzq " ΘpDzq ηpz{3q 2 . Note that: • f 0 is a modular function for Γp36Dq; • f 0 pγzq is holomorphic on H for all γ P SL 2 pZq; • f 0 pγzq has Fourier coefficients that are algebraic integers in its Fourier expansion at 8 for all γ P SL 2 pZq.
These properties can be checked using the properties of Θ K from the Appendix as well as the properties of ηpzq. Note that we are in the conditions of Lemma 4.20, thus f pτ q is an algebraic integer. This implies that 2 ΘpDωq Θpωq is an algebraic integer, hence 2S 1{6 is an integer. Now we will show that DS 1{6 is an integer as well by showing that D 2 is an algebraic integer. Using Lemma 4.3 for µ P t´1{2,´1{6u, ν " 1{2, a " 1, z " D´3`?´3 2 , we can rewrite: Taking the quotient by 3 2 Θpτ q " 4 ? 3|θ 0 pτ q| 2 , we get: r,1{6 pτ q θ 0 pτ qˇˇˇˇ2 .
To show the last property we can use the automorphic definitions of both θ r,µ and θ 0 . We get a Fourier expansion with coefficients in O K rζ 24 , ζ D 2 s. Thus we are in the conditions of Lemma 4.20, hence f r,µ pτ q is an algebraic integer. This implies that D Θpωq D 1{3 , which is a rational number, is indeed an integer. Since we already showed that 2S 1{6 is an integer, we get S 1{6 P Z when D is odd.
Finally, since S 1{6 " p´1q σD T 2 D and S 1{6 is an integer, from Corollary 4.18 we get: Note that this implies that ω k0 is the unique choice for a cube root of unity such that T D or is an integer.

Case of D not square free
We present below the case of D not square free. We write D " D 1 D 2 2 such that D 1 D 2 square free. In this case we use the formula (13): for D 0 " D 1 D 2 . All the details of the proof for the square-free case will follow through and we only briefly mention the steps. We apply the factorization formula and obtain the factorization from Corollary 4.6: We use this to show similarly to the proof of Proposition 4.11 that: where A s " ra s ,´b`?´3 2 s with Nmpa s q " s mod D 1 D 2 . Take π 1 a generator of D 1 and π 2 is the generator of D 2 such that´´b`?´3 2¯i s divisible by pπ 1 π 2 q 2 . Then χ D pA s q " χ π1 psqχ π2 psq.
The main difference is when we compute the complex conjugate of f r,1{6 pτ qχ π1 psqχ π2 psq, as we get R D,1{6 pτ q " Gpχπ 1 χ π 2 q π1π2 R D,1{6 pτ q and this equals R D,1{6 pτ q " p´1q σpDq ω k 1 π1 1{3 π2 2{3 π 1{3 1 π 1{3 2 R D,1{6 pτ q for a cubic root of unity ω k 1 . Then we can rewrite: 2 ω k0 , and we can show that this is the trace: where H O is the ray class field for the modulus D 0 and ω k0 is a cubic root of unity. Moreover, we can further show as in Section 4.5 that T D {3 P Z when σpDq even and T D { ?´3 P Z when σpDq odd.
Finally, we have S D ‰ 0 for D split only for D " 1 mod 9 and in this case the Tamagawa number c 3D equals 3 1`σpDq , thus S D is an integer square up to an even power of 3 and it equals:

Shimura reciprocity law
We present below some background on Shimura's reciprocity law following the exposition of Stevenhagen [25]. For more details also see Gee [6]. Let F be the field of modular functions over Q. From CM theory (see for example [25]), it is known that if τ P K X H and f P F , then we have f pτ q P K ab , where K ab is the maximal abelian extension of K. Shimura's reciprocity law gives us a way to compute the Galois conjugates f pτ q σ of f pτ q when acting with σ P GalpK ab {Kq. We recall that F " Ť N ě1 F N , where F N is the space of modular functions of level N . Moreover, F N is the function field of the modular curve XpN q " ΓpN qzH˚over Qpζ N q, where ζ N " e 2πi{N and H˚" H Y P 1 pQq. We can compute explicitly F N " Qpj, j N q, where j is the j-invariant and j N pzq " jpN zq. In particular, we have F 1 " Qpjq.
When working over Q, one has an isomorphism GalpF N {F 1 q -GL 2 pZ{N Zq{t˘1u. More precisely, if we denote by g σ the Galois action corresponding to the matrix g P GL 2 pZ{N Zq under the isomorphism above, it is enough to define the Galois action for SL 2 pZ{N Zq and for We state explicitly the two actions below: • Action of α P SL 2 pZ{N Zq on F N . We have pf pτ qq σα " f α pτ q :" f pατ q, where α is acting on the upper half plane via fractional linear transformations.
• Action of p 1 0 0 d q P pZ{N Zqˆon F N . Note that for f P F N we have a Fourier expansion f pzq " ř ně0 a n q n{N with coefficients a n P Qpζ N q, q " e 2πiz . If we denote u d " p 1 0 0 d q, then the action of σ u d is given by pf pτ qq σu d " f u d pτ q :" ř ně0 a σ d n q n{N , where σ d is the Galois action in GalpQpζ N q{Qq that sends ζ N Ñ ζ d N .
As the restriction maps between the fields F N are in correspondence with the natural maps between the groups GL 2 pZ{N Zq{t˘1u, we can take the projective limit to get the isomorphism: Note that the maps on F N are given by projecting GL 2 p p Zq{t˘1u Ñ GL 2 pZ{N Zq{t˘1u. To further get all the automorphisms of F we need to consider the action of GL 2 pA Q,f q. We get the exact sequence: 1 Ñ t˘1u Ñ GL 2 pA Q,f q Ñ AutpF q Ñ 1.
For this to make sense, we need to extend the action from GL 2 p p Zq to GL 2 pA Q,f q. We do this by defining the action of GL 2 pQq`on F : • Action of α P GL 2 pQq`on F . We define f α pτ q " f pατ q, where α acts by fractional linear transformations.
We extend the action of GL 2 p p Zq to GL 2 pA Q q by writing the elements g P GL 2 pA Q q in the form g " uα, where u P GL 2 p p Zq and α P GL 2 pQq`. Note that this decomposition is not uniquely determined. However, by combining the two actions of u and α, a well defined action is given by: We want to look at the action of GalpK ab {Kq inside AutpF q. From class field theory we have the exact sequence: where r¨, Ks is the Artin map.
We are going to embed AK ,f into GL 2 pA Q,f q such that the Galois action of AK ,f through the Artin map and the action of the matrices in GL 2 pA Q,f q are compatible. We do this by constructing a matrix g τ pxq for the idele x P AK ,f .
Let O be the order of K generated by τ i.e. O " Zrτ s. We define the matrix g τ pxq to be the unique matrix in GL 2 pA Q q such that xˆτ 1˙" g τ pxqˆτ 1˙.
We can compute it explicitly. To do that, consider the minimal polynomial of τ to be ppXq " X 2`B X`C. Then if we write x p P Qp in the form x p " s p τ`t p P Qp with s p , t p P Q p , we can compute g τ px p q "ˆt p´sp B´s p C s p t p˙. Using the map g τ above, we have: Theorem 5.1. (Shimura's reciprocity law) For f P F and x P AK ,f , we have: where σ x is the Galois action corresponding to the idele x via the Artin map, g τ is defined above and the action of g τ pxq is the action in GL 2 pA Q,f q.
Note that the elements of Kˆhave trivial action. This can be easily seen by embedding KˆãÑ GL 2 pQq`via k ãÑ g τ pkq. Noting that τ is fixed by the action of the torus Kˆ, we have f gτ pk´1q pτ q " f pg τ pk´1qτ q " f pτ q.
We can also rewrite the theorem for ideals in K. Let f P F N and O " Zrτ s of conductor M . Going through the Artin map, we can restate Shimura's reciprocity in this case in the form: where A is an ideal prime to M N , σ A is the Galois action corresponding to the ideal A through the Artin map, and g τ pAq :" g τ ppαq p| NmpAq q. Note that g τ pAq is unique up to multiplication by roots of unity in K. However, these have trivial action on f at the unramified places p| NmpAq.

Galois conjugates of f pωq
We denote f pzq " Θ K pDzq Θ K pzq . We are interested in finding the Galois conjugates of f pωq. First we show that f pzq is a modular function: Lemma 5.2. The function f(z) is a modular function of level 3D with integer Fourier coefficients at the cusp 8.
Proof. Since Θ K pzq is a modular form of weight 1 for Γ 1 p3q, it can be easily seen that ΘpDzq is a modular form of weight 1 for Γp3Dq. Furthermore, their ratio is modular function for Γ 0 p3Dq.
To check this let g "`a b c d˘P Γp3Dq, and then it is easy to see that f pgzq " From CM-theory, if f P F 3D and τ a generator of O K , we have f pτ q P H O the ray class field of modulus 3D. Recall H 3D is the ring class field for the the order O 3D " Z`3DO K , and we actually have: We check that f pωq is invariant under the action of KˆU p3Dq. Using Shimura's reciprocity law, we want to show: f pωq " f gωpsq pωq, for all s P KˆU p3Dq. We noted before that the action of Kˆis trivial. Thus it is enough to show the result for all elements l " pA p`Bp ωq p P U p3Dq. By the definition of U p3Dq, this implies that A p`Bp ω P pZ p rωsqˆfor all p and A 3 " 1 mod 3, B 3 " 1 mod 3, B p " 0 mod D for all p|D.
Since the action for p ∤ 3D is trivial, l has the same Galois action as l D " pA p`Bp ωq p|3D P U p3Dq. Moreover, this has the same action as l 0 " pA`Bωq p|3D , where A`Bω P O K and A " A p mod 3DZ p and B " B p mod 3DZ p for all p|3D.
Note further that we can pick A, B such that pA`Bωq generates a primitive ideal A in O K . Moreover, from above we have 3D|B and A " 1 mod 3. Recall that we can rewrite any primitive ideal in the form A " ra,´b`?´3 2 s Z , where a " Nm A and b 2 "´3 mod 4a. Then the generator of the ideal A is A`Bω " ta`s´b`?´3 2 for t, s P Z and 3D|s. Now note that f pωq " f pτ q, where τ "´b`?´3 2 . Thus from Shimura's reciprocity law, we have: pf pτ qq σ l´1 " f gτ pl p|3D q pτ q " f gτ pl0q pτ q.
Here g τ pl 0 q "`t a´sb´sca s ta˘p|3D , where ca " b 2`3 4 . Then we can rewrite the action of g τ pl 0 q explicitly as: Since 3D|s, the matrix`t a´sb´sc s t˘P Γ 0 p3Dq and f pzq is invariant under its action. Finally, since pa, 3Dq " 1 and f has rational Fourier coefficients at 8, the action of p 1 0 0 a q p|3D is trivial. Thus f pωq is invariant under the Galois action coming from U p3Dq and this finishes the proof. Now we would like to compute the Galois conjugates of f pωq under the action of GalpH 3D {Kq. We will first show the following general result: Lemma 5.4. Let F P F N be a modular function of level N with rational Fourier coefficients in its Fourier expansion at 8. Let τ "´b`?´3 2 be a CM point and let A " " a,´b`?´3 2 ı Z be a primitive ideal prime to N . Then we have the Galois action: Proof. From Shimura's reciprocity law, we have F pτ q σ´1 A " F gτ pAq pτ q. Note that the minimal polynomial of τ is p τ pXq " X 2`b X`b is an element of SL 2 pZ p q for p ∤ N , it has trivial action. Then we have F gτ pAq pτ q " F p 1 0 0 a q p|a pτ q. We rewrite the matrix p 1 0 0 a q p|a "`1 0 0 1{a˘p ∤a p 1 0 0 a q Q , where`1 0 0 1{a˘p ∤a P GL 2 p p Zq and p 1 0 0 a q Q P GL 2 pQq`. Note that the action of`1 0 0 1{a˘p ∤a is the same as the action of`1 0 0 1{a˘p |N . However, since F has rational Fourier coefficients in its Fourier expansion, this action is trivial. Thus we are left with F gτ pAq pτ q " F p 1 0 0 a q Q pτ q " F pτ {aq, which finishes the proof.
We apply the lemma above to our case: Proposition 5.5. Take the primitive ideals A " " a,´b`?´3 2 ı Z to be the representatives of the classes of the ring class group ClpO 3D q such that all norms a " Nm A are relatively prime to each other and b 2 "´3 mod 4a for all the norms a. and it acts as: f´t a´sb 1´s c 1 s t¯p|6D r,µ pτ 1 q " f r,µ p`t a´sb 1´s c 1 s t˘τ 1 q.
We can further compute this transformation and we will do this explicitly in Lemma 6.5. As we have 3D 2 |s and ta´sb " 1 mod 6 we are in the conditions of this lemma. Applying the transformation for θ r,µ and θ 0 and moreover noting that 9|sc 1 we get precisely: f r,µ p`t a´sb 1´s c 1 s t˘τ 0``t a´sb 1´s c 1 s t˘τ 1˘" f pta´sb 1 qr,µ pτ 1 q.
Since pta´sb 1 qt " 1 mod D 2 , we can rewrite this as f t 1´1 r,µ pτ 1 q for t " t mod D and t 1 " 1 mod 6. Note that t is prime to D. Thus we have showed so far that the Galois conjugates of f r,µ pτ q are the terms f s,µ pτ q, where gcdps, Dq " gcdpr, Dq. Moreover, we have nontrivial Galois action only for k 0 " ta`s´b`?´3 2 with t ı 1 mod D. Furthermore, it implies that f pτ q P H O , the ray class field of modulus 3D.
Finally, we would like to express the Galois action using ideals. For A " ra,´b`?´3 2 s Z a primitive ideal prime to 6D with a generator pk A q " pn a a`m a´b`?´3 2 q with n a " 1 mod 6 and 3|m a , we have the correspondence map between ideles and ideals given by x " pk A q p∤6D Ø A " pk A q. Picking the representatives k A as above, we have: where n a " n 1 a mod 3D and n 1 a odd. After changing r Ñ n 1´1 a r, we get the result of the Galois action from the proposition.
6 Appendix: Properties of theta functions 6.1 Properties of Θ K and η We have a functional equation for the theta function (see [13]): Furthermore, we can compute the transformation of Θ K pz˘1{3q in the lemma below: Proof. For k " 1, we can split the sum Θ`z`1 3˘" ř m,nPZ e 2πipm 2`n2´m nqpz`1 3 q in two parts, depending on whether or not the ideal pm`nωq is prime to p ?´3 q. The part of the sum for which p ?´3 q|pm`nωq gives us ř m,nPZ e 2πipm 2`n2´m nqp3z`1q " Θp3z`1q " Θp3zq. Going back to our initial computation, we get Θ pz`1{3q " Θp3zq`ωΘpzq´ωΘp3zq " p1´ωqΘp3zq`ωΘpzq, and this finishes the proof of the first formula. We can show the case k " 2 by applying the equality for k " 1 and z :" z´1{3.
Then from Lemma 3.8, we have the RHS equal to π. Furthermore, we can pick b " b 1 mod 8D 2 , b " 0 mod 3 and b 1 " 1 mod 3. Denote τ 1 "´b 1`?´3 2 . We can pick without loss of generality b, b 1 such that pb 2`3 q{D 2 and pb 12`3 q{D 2 are prime to D. Then we can find ideals A, A 1 prime to D such that Apπq 2 p ?´3 q " pτ q and A 1 pπq 2 " pτ 1 q. Let a " Nm A, a 1 " Nm A 1 and then we have: Similarly we compute ηpτ q η pτ {D 2 q " Note that we also have η´τ 1 D 2η pτ 1 q " ηp τ D 2 q ηpτ q , and thus we have η`τ D 2η pτ q " p´1q D´1 6 π.
Proof. Let φ " φ µ,r . Recall that θpg z{2 , φq " e´π i Frac2 µ p´1q r py{2q 1{2 θ r,µ pzq.. Note that mp ? 2q´1`a b c d˘" a b{2 2c d¯m p ? 2q´1. As θpg, φq is invariant under the action of SL 2 pQq, we have We will compute separately the LHS and the RHS using the definition of the Weil representation. We compute first the RHS. In order to do this, we rewrite the matrix´d´b {2 2c a¯"
2q´1qr`a b c d˘r pg z qφ 8 pxq " py{2q 1{4 sgnpdq c 1 cz`d e πip az`b cz`d qx 2 , and thus we have: From (29) and (30) we get the result of the lemma.
It follows immediately by applying the lemma above for θ r,µ and θ 0 that: Lemma 6.6. f r,1{2 is a modular function for Γp18D 2 q and f r,1{6 is a modular function for Γp6D 2 q.
Remark 6.7. Also from Lemma 6.5 it is easy to see that f r,µ pz`9D 2 q " f r,µ pzq.
We take the separate sums depending on n mod 3.
Taking Z " 3z we get the result of the lemma.