LSU Digital Commons LSU Digital Commons Computing special l-values of certain modular forms with Computing special l-values of certain modular forms with complex multiplication complex multiplication

. In this expository paper, we illustrate two explicit methods which lead to special L -values of certain modular forms admitting complex multiplication (CM), motivated in part by properties of L -functions obtained from Calabi–Yau manifolds deﬁned over Q .


Introduction
In arithmetic geometry, the study of L-functions plays an important role as demonstrated by the well-known Birch and Swinnerton-Dyer (BSD) conjecture which connects the purely algebraic object, the algebraic rank of an elliptic curve E defined over Q, with a purely analytic object, the analytic rank of the Hasse-Weil L-function L(E, s) of E at the center of the critical strip. It further provides a detailed description of the leading coefficient of L(E, s) expanded at this point. In this expository paper, we will explore how to use the theory of modular forms and hypergeometric functions to compute special L-values of modular forms. The cases we compute all have a common background of Calabi-Yau manifolds admitting complex multiplication (CM).
A Calabi-Yau manifold X of dimension d is a simply connected complex algebraic variety with trivial canonical bundle and vanishing ith cohomology group H i (X, O X ) for 0 < i < d. When d = 1, X is an elliptic curve; when d = 2, X is a K3 surface. A few papers of Noriko Yui, such as [12,44], are devoted to the modularity of Calabi-Yau manifolds X defined over Q. More precisely, the action of the absolute Galois group G Q of Q on the transcendental part of the middleétale cohomology H d et (X, Q ) should correspond to an automorphic representation according to Langlands' philosophy. In particular, when the Galois representation is 2-dimensional, it should correspond to a classical modular form of weight d + 1 so that both have the same associated L-functions. Recently, in [23] the second and third authors with Noriko Yui and Wadim Zudilin studied 14 truncated hypergeometric series that are related to weight-4 cuspidal eigenforms arising from rigid Calabi-Yau threefolds defined over Q by (super)congruences. One of the 14 modular forms in [23] has CM, that is, f is invariant under the twist by a quadratic character associated to an imaginary quadratic extension K of Q. In this case there is an idèle class character χ = χ(f ) of K with algebraic type d + 1 such that L(f, s) = L χ, s − d 2 . See [22,Chapter 7,Section 4] by the first author for more detail.
As explained in [10] by Deligne, for a weight-k modular form f (τ ), the values of the attached L-function L(f, s) at the integers within its critical strip, namely L(f, n+1) for 0 ≤ n ≤ k−2, are of special interest, called the periods of f . Precisely, these values can be expressed as (cf. [20]) (1.1) Though literally L(f, n + 1) can be computed as a line integral, the computation can be unraveled as an iterated integral. We will illustrate this idea through examples. As these are periods of modular forms, the theory of modular forms plays a central role. Additionally, the theory of hypergeometric functions provides helpful perspectives for obtaining exact values. Our main results below are motivated in part by the Clausen formula for hypergeometric functions (see (2.9)), which expresses the square of a one-parameter family of integrals in terms of a oneparameter family of iterated double integrals. In terms of cusp forms with CM by Q √ −1 , the above identity can be restated as where η(4τ ) 2 η(8τ ) 2 is the weight-2 level 32 cuspidal eigenform corresponding to ψ, and η(4τ ) 6 is the weight-3 level 16 cuspidal eigenform corresponding to ψ 2 . These identities can be reformulated in terms of cusp forms with CM by Q √ −3 as Here η(6τ ) 4 is the level 36 weight-2 cuspidal Hecke eigenform corresponding to χ, η(2τ ) 3 η(6τ ) 3 is the level 12 weight-3 Hecke eigenform corresponding to χ 2 , and η(3τ ) 8 is the weight-4 Hecke eigenform of level 9 corresponding to χ 3 .
In connection with Calabi-Yau manifolds explained above, the two weight-2 forms η(4τ ) 2 η(8τ ) 2 and η(6τ ) 4 come from the elliptic curves E 1 and E 2 , respectively. The weight-3 form η(4τ ) 6 arises from any one of the elliptic K3 surfaces labeled by A in [37] by Stienstra and Beukers. One of the defining equations is y 2 + 1 − t 2 xy − t 2 y = x 3 − t 2 x 2 . The relation between such a K3 surface and the elliptic curve y 4 + x 2 = 1 is through the Shioda-Inose structure described in [35]. In [37] Stienstra and Beukers showed that the monodromy group Γ 2 of the elliptic fiberation of A is isomorphic to an index-2 subgroup of the congruence group Γ 1 (5). The group Γ 2 itself is a noncongruence group. See [2] by Atkin and the first two authors for explicit congruence relations between the unique normalized weight-3 cusp form for Γ 2 and the congruence cusp form η(4τ ) 6 .
Our guiding philosophy is in line with the special values of the Riemann zeta function: where B n denotes the nth Bernoulli number. The Bernoullli numbers satisfy Kummer congruences which are important for the development of p-adic modular forms. This was extended to the ring of Gaussian integers by Hurwitz who showed that for any positive integer k, the numbers satisfy N (k) = N (1) k · (a rational number). See Hurwitz's paper [17] or the excellent expository paper [21] by Lee, R. Murty, and Park in which they reproved Hurwitz's result using the fact that for any quadratic imaginary number τ , there is a transcendental number b Q(τ ) , depending only on the field Q(τ ) (see (2.1), the Chowla-Selberg formula), such that for any integral weight-k modular form f with algebraic coefficients, [45,Proposition 26] where q = e 2πiτ , evaluated at τ = √ −1. Along the same vein, in [40] Rodriguez-Villegas and Zagier consider the Hecke character ϕ corresponding to an elliptic curve admitting CM by the imaginary quadratic field Q √ −7 and they obtain a very nice formula relating the central values of the L-functions of ϕ 2k+1 .
A special case of Damerell's result (see [8,9,42]) says that given a CM elliptic curve defined over Q with the corresponding idèle class character ρ, for each positive integer n there is a rational number C ρ,n such that L ρ n , n/2 = C ρ,n L(ρ, 1/2) n .
It has been found that these numbers are generalizations of Bernoulli numbers with important arithmetic meanings ( [5] by Coates [20] by Kohnen and Zagier for some classical discussions on periods of modular forms, [14,15,16] by Haberland and [27] by Paşol and Popa for periods and inner products, [18] by Ono, Rolen and Sprung and [28] by Rogers, Wan, and Zucker for more recent developments. Specifically, in [28], from a different perspective the authors computed the periods L η(4τ ) 6 , 1 and L η(2τ ) 3 η(6τ ) 3 , 2 . Our two theorems determine the rational number C ρ,n explicitly for special choices of elliptic curves and positive integers n. In the end of the paper, using CM values of Eisenstein series and modular polynomials, we compute a few more C χ,n values, listed in Table 1. Our approach can be adapted to compute C ρ,n for other idèle class characters ρ associated to CM elliptic curves defined over Q; see [36, p. 483] by Silverman for the full list of such elliptic curves. Note that in each case the class number of the corresponding CM field is 1. Also we focus on the periods whose corresponding Eisenstein series are holomorphic, but other periods can be handled similarly.

The Chowla-Selberg formula
The Chowla-Selberg formula says that if E is an elliptic curve whose endomorphism ring over C is an order of an imaginary quadratic field K = Q √ −d with fundamental discriminant −d, then all periods of E are algebraic multiples of a particular transcendental number where Γ (·) stands for the Gamma function, n is the number of torsion elements in K, is the primitive quadratic Dirichlet character modulo d, that is, the quadratic character attached to K over Q, and h K is the class number of K, see [31] by Selberg and Chowla, [13] by Gross,

Gamma and beta functions
The Gamma function satisfies two important properties in addition to the functional equation Γ(x + 1)/Γ(x) = x when x is not a non-positive integer. See [1] by Andrews, Askey and Roy for details. The first one is the reflection formula: for a ∈ C The second one is the multiplication formula for Γ: for integer m ≥ 1 and a ∈ C Now recall the beta function. For a, b ∈ C with positive real parts, It is known that

Hypergeometric functions
The (generalized) hypergeometric function with parameters a i , b j and argument x is defined by where (a) k := a(a + 1) · · · (a + k − 1) is the Pochhammer symbol with the convention (a) 0 = 1, and can be written as (a) k = Γ(a + k)/Γ(a). For a i , b j such that the formal power series is welldefined, the radius of convergence for x is typically 1. In particular, a 1 a 2 · · · a n b 1 · · · b n−1 ; xy dy.
In the classic developments, the 2 F 1 functions play a vital role. They can be written using the above recipe in the following Euler integral formula (2.7) In this sense, we say the 2 F 1 value corresponds to a 1-integral, or 1-period. The Gauss summation formula [1, Theorem 2.2.2] says that for a, b, c ∈ C with Re(c−a−b)>0, A virtue of such a formula is to give the precise value of the integral (2.7) in terms of Gamma values.
Here is a version of the Clausen formula [1, p. 116 as long as both sides converge. If we write both hand sides using the integral forms via (2.6), then the Clausen formula expresses the square of a certain 1-integral as an iterated 2-integral. From the classic result of Schwarz, hypergeometric functions are tied to automorphic forms of triangle groups. See [43] by Yang for using hypergeometric functions to compute automorphic forms for genus 0 Shimura curves with three elliptic points and see [39] by the third author and Yang for some applications.

n=1
(1 − q n ) is the weight-1/2 Dedekind eta function. Many modular forms can be written as eta products or quotients. Among all modular forms, theta functions and Eisenstein series (see [29,45]) play important roles. We first recall below the classic weight-1/2 Jacobi theta functions following mainly [45] by Zagier: which satisfy the relation They can be expressed in terms of the Dedekind eta function as Then the modular λ-function is (2.11) The lambda function λ(τ ) generates the field of all meromorphic weight-0 modular forms for Γ(2), the principal level-2 congruence subgroup. The group Γ(2) has 3 cusps, 0, 1, i∞ at which the values of λ are 1, ∞, 0 respectively. Another relevant classical result is (see Borweins [3]) (2.12) Thus From [45,Proposition 7], the logarithmic derivative of the discriminant modular form ∆(τ ) = η(τ ) 24 is 1 2πi where nq n 1 − q n is the weight-2 holomorphic quasi-modular form for SL 2 (Z). (2.14) Proof . From Theorem 2.2(c) and Theorem 2.6(c) in [4] by Borwein brothers and Garvin, one has the following two expressions of Jacobi theta functions in terms of E 2 : Taking logarithmic derivative of λ gives Similarly, there are also weight-1 cubic theta functions as follows (see [4]) where ζ 3 = e 2πi/3 . They satisfy the cubic relation Parallel to (2.12) is the following identity (see [4]) A finite index subgroup Γ of SL 2 (Z) acts on the upper half plane H via fractional linear transformations. Its fundamental domain can be compactified by adding a few missing points, called the cusps, to get the compact modular curve X Γ for Γ. The meromorphic modular functions for Γ form a field and we will denote it by C(X Γ ). If X Γ has genus 0, C(X Γ ) has a generator t over C which plays a crucial role. For example, its derivative t = dt dτ is a weight-2 meromorphic modular form for Γ. When t = λ(τ ), the explicit form of t is given in Lemma 2.1. When −I / ∈ Γ, by the Galois theory in the context of modular curves, a finite extension of C(X Γ ) corresponds to a unique finite index subgroup Γ of Γ. Note that C(X Γ ) is a simple field extension of C(X Γ ). The ramifications of a generator can occur either along the cusps or along the elliptic points with specified degrees. Nevertheless, finding such a generator is equivalent to determining the group Γ . In some cases below, we start with a genus 0 group Γ with C(X Γ ) generated by t and compute an algebraic function on Γ which leads to a genus 1 subgroup Γ . In this case the invariant differential for X Γ naturally corresponds to the unique normalized weight-2 cusp form f for Γ . This process may lead to an expression of f (τ )dτ as an algebraic function R(t) times dt. Combined with formula (1.1), L(f, 1) can be computed using When X Γ has CM by an imaginary quadratic field K, by the Chowla-Selberg formula (2.1), it is expected that L(f, 1) is an algebraic multiple of b K . This approach is closely related to [28] by Rogers, Wan, and Zucker where they used elliptic integrals instead of modular forms.

Idèle class characters and modular forms
Given a number field K, denote by Σ(K) the set of places of K, and for each v It is a locally compact topological group in which the multiplicative group K × is diagonally embedded. A character ξ of I K is a continuous homomorphism from ξ v . Furthermore, owing to the topology on I K , one can show that, for almost all nonarchimedean places v of K, the character ξ v is trivial on U v , hence it is determined by its value at any uniformizer In this case we say that ξ and In this case we say that ξ and ξ v are ramified at v and the product of M fv v over ramified places v is called the conductor of ξ. When a character ξ of I K is trivial on K × , it is called an idèle class character of K. The weak approximation theorem (cf. [25, p. 117]) implies that an idèle class character ξ = v ξ v with a given conductor is determined by the local components ξ v for all but finitely many places v.
The nonarchimedean places v of K are in one-to-one correspondence with the maximal ideals P v of the ring of integers of K. Given an idèle class character ξ = v∈Σ(K) defines a character ξ on the free abelian group generated by all P v with ξ v unramified, in other words, the group of fractional ideals of K coprime to the conductor of ξ. In the literature ξ is called a Hecke Grossencharacter. Upon checking the behavior of ξ on the integral principal ideals, one finds that ξ has the same conductor as ξ. Conversely, given a Hecke Grossencharacter ξ , the above formula defines an unramified character ξ v of K × v for all but finitely many places v of K, which can be uniquely extended to an idèle class character ξ = v ξ v with the same conductor as ξ by the weak approximation theorem. So the two kinds of characters are the same. The reader is referred to Section 6, Chapter VII of the book [25] by Neukirch for more detail.
A typical example of an unramified idèle class character is ξ Here | | v is the standard valuation at v. More precisely, if v is a real place, it is the usual absolute value on R; if v is a complex place, it is the square of the usual absolute value on C; if v is a nonarchimedean place, it is given by Remark 2.2. In [22, Chapter 5, Proposition 1] it is shown that any character ξ of I K can be written as the product of | | s K for some complex number s times a unitary idèle class character ξ 1 of I K which takes values in the unit circle S 1 ⊂ C × .
To an idèle class character ξ of K, we associate an L-function defined as Note that the L-function attached to the trivial character is nothing but the Dedekind zeta function of K, which converges absolutely for Re(s) > 1. The same holds for ξ unitary. In general, by Remark 2.2, we can write ξ = | | s K ξ 1 with s ∈ C and ξ 1 a unitary idèle class character. Since L(ξ, s) = L(ξ 1 , s + s ), we conclude that the L-function attached to ξ converges absolutely to a holomorphic function on the right half-plane Re(s) > 1 − Re(s ). It suffices to understand the analytic behavior of L-functions attached to unitary idèle class characters of K. This was studied by Hecke for Grossencharacters. Hecke's result was reproved in Tate's thesis [38] using adelic language, summarized below. Theorem 2.3. Let K be a number field with different d. Let ξ be a unitary idèle class character of K with conductor f. The associated L-function L(ξ, s) defined above is holomorphic on Re(s) > 1. It can be analytically continued to a meromorphic function on the whole s-plane, bounded at infinity in each vertical strip of finite width, and holomorphic everywhere except for a simple pole at s = 1 when ξ is the trivial character. Further, there is a suitable Γ-product where W (ξ) is a constant of absolute value 1 and ξ −1 is the inverse of ξ.
Here W (ξ), called the root number of ξ, is equal to the Gauss sum of ξ divided by its absolute value. See [38] for details.
In particular, when K is an imaginary quadratic extension of Q, it has one infinite place ∞ with K ∞ = C. The Γ-factor L ∞ (ξ ∞ , s) is equal to (2π) −s Γ(s) for all unitary ξ. Such a character ξ is said to have algebraic type k if ξ ∞ maps z ∈ C × to ξ ∞ (z) = (z/|z|) n with |n| = k −1. The above theorem for ξ algebraic of type k ≥ 1 combined with the converse theorem for modular forms proved by Weil [41] implies the existence of a modular form f ξ = n≥0 a n q n of weight k such that the associated L-function L(f ξ , s) := n≥1 a n n −s satisfies the relation The form f ξ is cuspidal if ξ is nontrivial. See [22, Chapter 7, Section 4] for details. Observe that, since the L-function attached to f is Eulerian at all primes, the form f is a Hecke eigenfunction. We end this subsection by noting that for an elliptic curve E defined over Q of conductor N with CM by K, it was known to Deuring that the Hasse-Weil L-function L(E, s) attached to E obtained by counting F p -rational points on the reduction of E modulo the prime p is equal to for some nontrivial unitary idèle class character ξ of K, algebraic of type 2. The above discussion says that all positive powers of ξ also correspond to modular forms.

Eisenstein series
We now recall some useful facts of Eisenstein series of general level. Define the level N holomorphic Eisenstein series as follows. For details see [7,11]. For a fixed pair of integers (a 1 , a 2 ), for k ≥ 3, we let The series is a weight k holomorphic modular form on Γ(N ) = {γ ∈ SL 2 (Z) : γ ≡ I 2 mod N } . When N ≥ 3, Γ(N ) does not contain −I 2 . For k = 1 or 2, the series does not converge absolutely so we adopt the following approach using the more general non-holomorphic Eisenstein series We now recall some basic properties of this function. For details see [7, Proposition 5.2.2] by Cohen and Strömberg. Firstly the series G * k,(a 1 ,a 2 ;N ) (s, τ ) converges absolutely and uniformly on any compact subset of the upper half complex plane when Re(2s+k) > 2 thus it is continuous at s = 0 when k ≥ 3. Also for a fixed τ , there exists a meromorphic continuation of G * k,(a 1 ,a 2 ;N ) (s, τ ) to the whole s-plane which is parallel to the Fourier series stated in Proposition 2.4 below. In our later application, we are interested in the following series Thus when k ≥ 3, the series G * k,(a 1 ,a 2 ;N ) (τ ) and G k,(a 1 ,a 2 ;N ) (τ ) coincide. For integers a ≥ 0, N ≥ 1 and k ≥ 1, we can define the series where W k (z; s) is defined inductively as follows: being a K-Bessel function [7, Definition 3.2.8] and for k ≥ 1, z ∈ C with Im(z) = 0, Proposition 2.5 ([7, Lemma 3.5.6(b)]). When k > 0, s = 0, we have In addition, we have to deal with when s → 0.
Theorem 2.6. We have the Fourier expansions and for integer k ≥ 3, Proof . For any fixed integer k ≥ 1 and Re(s) > 0, we have and similarly, Hence, when k = 2, we have As s → 0 + , To get the first equality, we use the facts that Γ 1 2 = √ π and ζ(x, s) has a simple pole at s = 1 with residue 1. Notice that when n < 0, for n ∈ {2πn(N m + a)τ, 2πn(N m + N − a)τ }, the term W 2 (n , 0) is 0 since the imaginary part of n is negative.
If k = 1, we have As s → 0 + , For the first equality, we use ζ( goes to 0 for both A = a and A = N − a, the claim follows.
The cusps of Γ 0 (8) and their behaviors in Γ 0 (32) are summarized below: This means the covering X 0 (32) of X 0 (8) ramifies completely at the cusps 1 2 and 0; splits completely at the cusp i∞; and splits at the cusp 1 4 into two cusps 1 4 and 3 4 , each with ramification degree 2. Thus is a modular function for Γ 0 (32). Consequently, a defining equation for the genus 1 modular curve X 0 (32) is The unique up to scalar holomorphic differential 1-form on X 0 (32) is given by dx y 3 . By Lemma 2.1, one has the following expression of dx y 3 as a function of τ : where f 32 (τ ) := η(4τ ) 2 η(8τ ) 2 is the unique weight-2 level 32 normalized cuspidal newform.
Proof . By (1.1), Since the elliptic curve y 4 = 1 − x 2 has CM by the imaginary quadratic field K = Q(i), there is a character ψ of the idèle class group of K, algebraic of type 2 at the complex place ∞, such that L(f 32 , s) = L ψ, s − 1 2 . See Section 2.5 or [22, p. 145] by the first author for more details. The type 2 condition (see Section 2.5) means that ψ ∞ (z) = z/|z| for z ∈ C × . The field K has only one place P dividing 2 and norm N (P) = 2. As the norm of the different d of K over Q is equal to 4, the absolute value of the discriminant of K over Q, and 32 is equal to the norm of the product of d and the conductor of ψ, we conclude that the conductor of ψ is P 3 . Write M P for the maximal ideal of the ring of integers of the completion K P of K at P. It is principal, generated by π P = 1 + i. Then ψ restricted to the group of units U P = 1 + M P at P has kernel 1 + M 3 P . Note that the quotient group (1 + M P )/(1 + M 3 P ) is isomorphic to Z/4Z, generated by the image of i. Write ψ = v ψ v , where v runs through all places of K. As ψ is an idèle class character unramified outside P and i is a unit everywhere, it follows from P has order 4. This in turn implies that ψ 2 has conductor P 2 and at the complex place it is algebraic of type 3. In particular, ψ 2 L ψ v , s − 1 2 explicitly, we distinguish two cases according to the residual characteristic p at v.
Case (I) p ≡ 3 mod 4. Then p is inert in K, N v = p 2 , and we may choose −p as a uniformizer π v at v. By definition ψ ∞ (−p) = −1. It follows from 1 = ψ( Case (II) p ≡ 1 mod 4. Then p = a 2 + b 2 is a sum of two squares. It splits in K, N v = p, and we may assume as a uniformizer π v = a + bi. The choice is unique by requiring a ≡ 1 mod 4 and b even. Noting that π 3 P = 2(−1 + i), we compute a + bi = b 2 π 3 P + a + b ≡ 1 + b mod M 3 P so that ψ P (a + bi) = ψ P (1 + b) = (−1) b/2 . Similar computation as above, using ψ ∞ (a + bi) = (a + bi)/ Let v be the other place of K with residual characteristic p. Then Denote by S 1 the set of integral ideals of Z[i] coprime to 2. Then each I in S 1 is a principal ideal generated by a unique element a + bi with a ≡ 1 mod 4 and b even. Conversely any a + bi of this form generates an integral ideal in S 1 . The discussion above shows that defines a Hecke Grossencharacter on S 1 (denoted | | −1/2 K ψ in notation of Section 2.5) and which in turn gives a q-expansion of f 32 as Note that the above formula gives a precise expression of the CM modular form η(4τ ) 2 η(8τ ) 2 . This can be obtained from the multiplier of η(τ ), see [32] by Serre for more details.
Combining the above two lemmas, we have as asserted in Theorem 1.1.

A proof of the first identity in Theorem 1.2 using hypergeometric functions and properties of CM modular forms
For the degree-3 Fermat curve F 3 : X 3 + Y 3 = 1, another modular parametrization is given by It defines a projection from F 3 to the modular curve X Γ(2) ramified only above the three cusps of Γ(2). This makes F 3 a modular curve for the Fermat group Φ 3 , an index-9 subgroup of Γ(2), with the field of C-rational functions being the degree-9 abelian extension C 3 λ(τ ), 3 1 − λ(τ ) of C(λ). The differential 1-form ω := dx/y 2 on F 3 gives rise to a weight 2 cusp form f for the Fermat group Φ 3 . Substituting (4.1) into this differential yields different expressions for ω: Proof . One can prove it by using (2.14) and writing (λ(1 − λ)) 1/3 as eta quotients. Precisely, Remark 4.2. As explained at the end of Section 2.4, the Fermat curve F 3 is in fact isomorphic to X 0 (27) over Q. The above lemma shows that the invariant differential of F 3 is expressed as the Q-rational form η(τ ) 4 times an irrational multiple 2 1/3 /3. We know that η(6τ ) 4 is a cusp form for Γ 0 (36). To obtain a Q-rational differential, we may use the algebraic model E 2 : x 3 +y 3 = 1/4 for the Fermat curve so that it is isogenous to X 0 (36) over Q.
We now compute the period L η(6τ ) 4 , 1 using Lemma 4.1 and the fact λ(i∞) = 0, λ(0) = 1. Proof . Using Lemma 4.1, we compute the period L η(6τ ) 4 , 1 as follows: The last equality follows from B(1/3, The elliptic curve X 0 (36) has CM by Q √ −3 . Consequently there is a character χ of the idèle class group of Q( √ −3), algebraic of type 2, such that L(χ, s − 1/2) is equal to the Hasse-Weil L-function of X 3 + Y 3 = 1/4. The weight-2 normalized Hecke eigenform of level 36, denoted f 36 , is the cusp form such that L(f 36 , s) = L(χ, s − 1/2). Similarly, higher powers of χ also give rise to modular forms of higher weights. We shall identify some of these forms explicitly. We begin with the description of χ.
Let ζ n = e 2πi/n be a primitive nth root of unity. The imaginary quadratic field F = Q(ζ 6 ) has class number 1 and discriminant −3 so that 3 is the only prime ramified in F . Further a prime p is inert in F if and only if p ≡ 2 mod 3. The ring of integers of F is Z[ζ 6 ]. Let S 2 be the set of nonzero integral ideals of Z[ζ 6 ] coprime to 6. Proof . The uniqueness of such kind of generator is straightforward to check. We prove the existence. Observe that it suffices to show that I has a generator of the form m + n √ −3 with m, n ∈ Z. If so, then N (I) = m 2 + 3n 2 , being coprime to 3, is congruent to 1 modulo 3, implying m ≡ ±1 mod 3. Hence we have (a, b) = (m, n) or (−m, −n) according as m ≡ 1 or −1 mod 3. Further, since N (I) is odd, exactly one of a, b is even.
Remark 4.5. The above argument shows that any nonzero ideal I of Z[ζ 6 ] with norm coprime to 3 is generated by an element x = a + b √ −3 with a, b ∈ Z and a ≡ 1 mod 3. If, in addition, N (I) is even, then x, xζ 3 and xζ 2 3 are the only generators of I of this form.
Note that if I = a + b √ −3 and J = c + d √ −3 are two ideals in S 2 with generators of the desired form, then a + b √ −3 c + d √ −3 is the desired generator of IJ. Denote by v 2 (resp. v 3 ) the only place of F above 2 (resp. 3); it has norm N v 2 = 4 (resp. N v 3 = 3). For j ∈ {2, 3} denote by M j the maximal ideal at v j , and U j the group of units at v j . Let χ be the idèle class character of F so that L χ, s − 1 2 is the Hasse-Weil L-function of the elliptic curve X 0 (36) over Q. Since the conductor 36 of the elliptic curve X 0 (36) is the product of the absolute value of the discriminant −3 and the norm of the conductor of χ, we conclude that χ has conductor M 2 M 3 . Denote by χ v the restriction of χ at the place v of F . By definition, at the complex place ∞ of F , χ ∞ (z) = z/|z|. Since χ v 3 is nontrivial on U 3 /(1 + M 3 ), which is generated by −1, we have χ v 3 (−1) = −1. As ζ 3 is a unit of F of order 3, we have χ v 3 (ζ 3 ) = 1. The group U 2 /(1 + M 2 ) is cyclic of order 3, generated by ζ 3 . Therefore we have . Parallel to what we did for ψ, we determine the value of χ v (π v ) at a uniformizer π v at the place v of F not equal to v 2 , v 3 , ∞ according to its residual characteristic p.
Case (I). p ≡ 2 mod 3 and p = 2. Then p is odd and N v = p 2 . Choose π v = −p so that Case (II). p ≡ 1 mod 3. Then p splits in F so that N v = p. By the proposition above, we may choose π v = a + b √ −3 with a, b ∈ Z, a ≡ 1 mod 3 and exactly one of a, b is even.
Then L χ, s − 1 2 = I∈S 2χ (I)N (I) −s , whereχ, equal to | | −1/2 F χ in the notation of Section 2.5, is the Grossencharacter on S 2 whose value at I = a + b √ −3 with a ≡ 1 mod 3 is given bỹ Since L(f 36 , s) = L χ, s − 1 2 , we obtain an expression for the q-expansion of f 36 : (4.2) Next we identify the modular forms corresponding to χ k , predicted by the converse theorem, for the first couple values of k. When k = 2, there is a cusp form h 3 of weight 3 so that L(h 3 , s) = L χ 2 , s − 1 . When k = 3, there is a cusp form h 4 of weight 4 so that L(h 4 , s) = L χ 3 , s − 3 2 . Note that χ 2 has conductor M 2 so that h 3 has level 12 while χ 3 has conductor M 3 so that h 4 has level 9. For each case the corresponding spaces of cusp forms are 1-dimensional from which one can conclude h 3 = η(2τ ) 3 η(6τ ) 3 and h 4 = η(3τ ) 8 right away. Here we give a bit more details. To write the corresponding L-functions, we choose π v 3 = √ −3 and π v 2 = −2 and compute This in turn gives a q-expansion for h 3 as follows: Similarly, χ 3 v 2 (π v 2 ) = −1 and which in turn yields a q-expansion of h 4 as The reader is referred to [32] by Serre for discussions of q-expansions of powers of η(τ ) when they satisfy complex multiplication.
Using the identification f 36 (τ ) = η(6τ ) 4 , we restate Lemma 4.3 as We proceed to compute the period L(h 3 , 2). First, using notation in Section 2.4 and setting s(τ ) = c(τ )/a(τ ), we rewrite h 3 (τ /2) as Using (2.17), one has This expression leads to the period L(h 3 , 2) by applying (1.1): In the above computation, the 4th equality results from the change of variable replacing τ by −1/3τ , and the 5th equality uses the transformation identity for η function Note that the 4th and the 5th steps combined relates L(h 3 , 2) to L(h 3 , 1), which could be computed using the functional equation for L χ 2 , s or L(h 3 , s) as we did in the previous section.
Here we took advantage of the fact that h 3 (τ /2) is an η-product to compute directly the relation between h 3 (τ ) and h 3 (−1/(12τ )), which is the inverse Mellin transform of the functional equation relating L(h 3 , s) and L(h 3 , 3 − s).
To finish, note the relation between special values The first equality above results from the Clausen formula (2.9), and the Gauss summation formula (2.8) is used to prove the second equality. Therefore, As observed before, L(f 36 , 1) = L(χ, 1/2) and L(h 3 , 2) = L χ 2 , 1 , this proves the first identity in Theorem 1.2 for both idèle class characters and cusp forms.
5 Computing L-values using Eisenstein series and a proof of the second identity in Theorem 1.2 Now we switch our approach to use CM values of Eisenstein series discussed in Section 2.6. This will allow us to compute more values of L(χ k , k/2) systematically. Denote by Σ the set of nonzero ideals of Z[ζ 6 ] coprime to M 3 . Then Σ = ∪ i≥0 (−2) i S 2 = S 2 ∪ (−2)Σ.
Recall from the discussion in Section 4 that except for the local factors at v 2 and v 3 , L χ k , s− k/2 agrees with I∈S 2χ k (I)N (I) −s . Furthermore, the conductor of χ k is Observe that, for k ≡ 0 mod 3, χ k is unramified at v 2 , and we can extend the definition ofχ k on S 2 to Σ by the same formula, that is, for I = a + b √ −3 ∈ Σ with a, b ∈ Z and a ≡ 1 mod 3, let This is well-defined also for I with even norm because in this case, by Remark 4.2, the three generators of the above form differ by a power of ζ 3 , and the difference disappears after raising to the kth power. In this case, noting Σ = ∪ i≥0 (−2) i S 2 and using the multiplicativity ofχ k on Σ, we have To continue, note that the Fourier expansion of f 36 given in (4.2) can be extended to The extra terms come from m and n having opposite parity. In this case x = 3m + 1 − n √ −3 generates an ideal in Σ with even norm (3m + 1) 2 + 3n 2 , and by Remark 4.5, xζ 3 and xζ 2 3 are the two other elements of this form and with the same norm. Since these three elements sum to zero, no additional contributions result. Now consider the following generalization to k ≥ 1: which is a partial sum of whose associated Dirichlet L-series is Comparing these two q-series, we note that, just like the argument for f 36 above, each extra term in G k (q) from m, n with opposite parity will have x = 3m+1−n √ −3 generate an ideal in Σ with even norm (3m + 1) 2 + 3n 2 , and xζ 3 and xζ 2 3 are precisely the other two elements of the same form and with the same norm. Together they contribute x k + (xζ 3 ) k + xζ 2 3 k to the coefficient of q (3m+1) 2 +3n 2 in G k (q), which is zero if k is not a multiple of 3, and 3 3m + 1 − n √ −3 k otherwise. In conclusion, we have shown • When k is not a multiple of 3, • When k ≡ 0 mod 3, using the extendedχ k on Σ = S 2 ∪ (−2)Σ, we can write if k ≡ 0 mod 6. (5.5) Thus for k = 2, the constant Remark 5.1. As noted in Section 2.5, corresponding to each χ k with k ≥ 1 there is a modular form h k+1 = n≥0 a k+1 (n)q n of weight k + 1 such that L(h k+1 , s) = L χ k , s − k 2 . The function G k (q) defined above is obtained from h k+1 by removing the Fourier coefficients a k+1 (n) with n not coprime to 6. So it is also a modular form of weight k + 1. When k ≡ ±1 mod 6, it is equal to h k+1 , and for other k its level is higher than that of h k+1 .
The relations between L χ k , k/2 and L(G k (q), k) described in (5.5) are useful for determining the exact values of L χ k , k/2 since L(G k (q), k) can be computed using CM values of explicit Eisenstein series.
The Recall the Eisenstein series defined in Section 2.6, which has analytic continuation to s = 0 for a fixed τ . By taking τ to be 3, we find, for Re(s) > 2, This gives another expression of the analytic continuation of L(f 36 , s) and, by letting s → 0 + , we obtain the relation which gives another way to compute the value L(f 36 , 1). When k ≥ 2, we have, for Re(s) > k, This leads to a uniform expression for the L-value of G k at k for k ≥ 1: is defined in Section 2.6. Notice that when k ≥ 3, G * k,(1;3) (τ ) = G k,(1;3) (τ ) is a holomorphic Eisenstein series, and the double series converges absolutely so that there is no need to take limit to get the value L(G k (q), k).
Hence, the values of φ(τ 0 ) and φ 1 (τ 0 ) can be determined by the following η-values and the value of .
The first one can be proved immediately since for n = p ≡ 1 mod 3, a 3 (p) = a 2 (p) 2 − 2p and a 2 (p) ≡ −1 mod 3. Likewise, the other congruences can be obtained case by case from the expression for primes p ≡ 1 mod 3. Deeper relations are subject to future investigation.