aa r X i v : . [ m a t h . N T ] M a y SPLIT-CM POINTS AND CENTRAL VALUES OF HECKE L-SERIES
KIMBERLY HOPKINS
Abstract.
Split-CM points are points of the moduli space h /Sp ( Z ) corresponding to prod-ucts E × E ′ of elliptic curves with the same complex multiplication. We prove that the numberof split-CM points in a given class of h /Sp ( Z ) is related to the coefficients of a weight 3 / L ( ψ N ,
1) of a certain Hecke L -series. The Hecke character ψ N is a twist of the canonicalHecke character ψ for the elliptic Q -curve A studied by Gross, and formulas for L ( ψ,
1) as wellas generalizations were proven by Villegas and Zagier. The formulas for L ( ψ N ,
1) are easilycomputable and numerical examples are given. Introduction
Let
D < | D | prime be the discriminant of an imaginary quadratic field K with ring ofintegers O K . Suppose N is a prime which splits in O K and is divisible by an ideal N of norm N .We will define Hecke characters ψ N of K of weight one and conductor N (see Section 3). Theseare twists of the canonical Hecke characters studied by Rohrlich [Roh80a, Roh80b, Roh82] andShimura [Shi64, Shi71, Shi73b]. Denote by L ( ψ N , s ) the corresponding Hecke L -series.Our main theorem (Theorem 3.6) is a formula in the spirit of Waldspurger’s results [Wal80,Wal81]. It says approximately that(1.1) L ( ψ N ,
1) = X [ R ] X [ a ] Θ [ a ,R ] , N · h ε [ a ,R ] ( − N ) . Here the first sum is over all conjugacy classes of maximal orders R in the quaternion algebraramified only at ∞ and | D | , and the second sum is over the elements [ a ] of the ideal class groupof O K . We will see that the h ε [ a ,R ] ( − N ) are integers related to coefficients of a certain weight3 / [ a ,R ] , N are algebraic integers equal to the value of a symplectictheta function on ‘split-CM’ points (defined in Section 3) in the Siegel space h /Sp ( Z ). Weexpect the formula (1.1) to be useful for computing the central value L ( ψ N , A ( | D | ) denote a Q -curve as defined in [Gro80]. This is an elliptic curve defined overthe Hilbert class field H of K with complex multiplication by O K which is isogenous over H to its Galois conjugates. Its L -series is a product of the squares of L -series L ( ψ, s ) over the h ( D ) Hecke characters of conductor ( √ D ). A formula for the central value L ( ψ,
1) expressedas a square of linear combinations of certain theta functions was proven by Villegas in [RV91].Extensions of his result to higher weight Hecke characters were given by Villegas in [RV93] andjointly with Zagier in [RVZ93]. The Hecke character ψ N is a twist of ψ by a quadratic Dirichletcharacter of conductor ( √ D ) N . Therefore our result (1.1) gives a formula for the central valueof the corresponding twist of A ( | D | ).Our main theorem can be stated in a particularly nice form when the class number of O K is one. Then [ a ] = [ N ] = [ O K ] and so in particular Θ [ a ,R ] , N = Θ [ R ] and h ε [ a ,R ] ( − N ) = h εR ( − N ) Date : November 14, 2018.1991
Mathematics Subject Classification.Key words and phrases.
Hecke L-series, Waldspurger, quaternions, split-CM. are independent of [ a ] and N . This suggests that formula (1.1) will lead to a generating seriesfor L ( ψ N ,
1) as N varies in terms of linear combinations (with scalars in { Θ [ R ] } ) of half-integerweight modular forms.We hope to extend these results to higher weight as follows. For certain k ∈ Z ≥ it is well-known that the central value L ( ψ k N , k ) can be written as a trace over the class group of O K of a weight k Eisenstein series evaluated at Heegner points of level N and discriminant D . Itis a general philosophy (see [Zag02], for example) that such traces relate to coefficients of acorresponding modular form of half-integer weight. By the Siegel-Weil formula we can writethe central value of L ( ψ k N , s ) in terms of a sum of theta-series (1.2) L ( ψ k N , k ) · = X [ a ] X [ Q ] ω Q Θ Q ( τ a ) . Here the sum is over [ a ] in the class group of O K and over classes of positive definite quadraticforms Q : Z k −→ Z in 2 k variables and in a given genus. The point τ a ∈ h is a Heegner pointof level N and discriminant D . Analogous to the case of two variables, these quadratic formscorrespond to higher rank Hermitian forms (see [Otr71] and [HK86, HK89, HI80, HI81, HI83]).An approach to counting the number of distinct theta values in (1.2) would be to associate theHermitian forms to isomorphism classes of rank k R -modules of B , for maximal orders R of B .This paper does this for the case k = 1. Our intention here is to lay the groundwork for thegeneralization to arbitrary weight k .This paper is organized as follows. Basic notation is given in Section 2. Background anda statement of results are in Section 3. In Section 4, we analyze the endomorphisms of theprincipally polarized abelian varieties for the split-CM points, and show they form an explicitmaximal order in the quaternion algebra B . In Section 5 we identify these orders with explicitright orders in B . In Section 6 we prove the main results (Theorems 3.2, 3.3 and 3.6) andprovide numerical examples. 2. Notation
Given any imaginary quadratic field M of discriminant d <
0, we denote by O M its ring ofintegers, Cl ( O M ) its ideal class group, h ( d ) its class number, and Cl ( d ) the isomorphic classgroup of primitive positive definite binary quadratic forms of discriminant d . A nonzero integralideal of O M with no rational integral divisors besides ± primitive . Any primitiveideal a of O M can be written uniquely as the Z -module a = a Z + − b + √ d Z = [ a, − b + √ d a := N a the norm of a , and b an integer defined modulo 2 a which satisfies b ≡ d mod 4 a .Conversely any a, b ∈ Z which satisfy the conditions above determine a primitive ideal of O M .The coefficients of the corresponding primitive positive definite binary quadratic form are givenby [ a, b, c := b − D a ]. The form [ a, − b, c ] corresponds to the ideal ¯ a . We will always assume ourforms are primitive positive definite and the same for ideals. The point τ a := − b + √ d a The precise statement of this formula is simplified here for the sake of exposition. Here ω Q is the number of automorphisms of the form Q . ENTRAL VALUES OF HECKE L-SERIES 3 is in the upper half-plane h of C and is referred to in general as a CM point . A
Heegner point oflevel N and discriminant D is a CM point τ a where a is given by a form [ a, b, c ] of discriminant D such that N | a . The root of τ a is defined to be the reduced representative r ∈ ( Z / N Z ) × such that b ≡ r mod 2 N .Square brackets [ · ] around an object will denote its respective equivalence class. The unitsof a ring R are written as R × . 3. Statement of Results
We first recall some basic results for Siegel space and symplectic modular forms.Assume K is an imaginary quadratic field of prime discriminant D < −
4. Let L be animaginary quadratic field of discriminant − N < N is a prime which splits in O K , andis divisible by an ideal N of norm N . Note h ( D ) and h ( − N ) are both odd since | D | and N areprime. Let µ : O K / N −→ Z /N Z be the natural isomorphism. Composing this with the Jacobisymbol ( · N ) : Z /N Z −→ { , ± } defines a character χ : ( O K / N ) × −→ {± } . This is an odd quadratic Dirichlet character of conductor N . Let I N denote the group of nonzerofractional ideals of K which are coprime to N , and let P N ⊂ I N be the subgroup of principalideals. The map ψ N : P N −→ K × defined by ψ N (( α )) := χ ( α ) α is a homomorphism. There are exactly h ( D ) extensions of ψ N to a Hecke character ψ N : I N −→ C × . This produces h ( D ) primitive Hecke characters of weight one and conductor N . (See[Gro84, Pac05] and [Roh80a, p.225] for more details). Fix a choice of ψ N . We can extend ψ N to a multiplicative function on all of O K by setting ψ N ( a ) := 0 if a is not coprime to N .To ψ N we associate the Hecke L -function L ( ψ N , s ) := X a ⊂ O K ψ N ( a ) N a s , Re( s ) > / . We now recall a result due to Hecke which gives the central value L ( ψ N ,
1) as a linearcombination of certain theta series evaluated at CM points. For each primitive ideal Q of O L ,the associated theta series is defined byΘ Q ( τ ) := X λ ∈ Q q N ( λ ) / N ( Q ) , q = e πiτ , τ ∈ h . It is a modular form on Γ ( N ) of weight one and character sgn( · ) (cid:0) − N | · | (cid:1) (see [Eic66, p.49], forexample).For each primitive ideal a of O K with norm prime to N , the product ideal a ¯ N is of the form[ a N, − b + √ D ] for some a , b ∈ Z . The point τ a ¯ N := − b + √ D a N ∈ h is a Heegner point of level N and discriminant D . We will write τ a or just τ for τ a ¯ N when thecontext is clear. Note that as a runs over a distinct set of representatives of Cl ( O K ), so does a ¯ N . (The fact that representatives of Cl ( O K ) can be chosen with norm prime to N is in [Cox89,Lemmas 2.3, 2.25], for example.) By a we will always mean a primitive ideal with norm primeto N as above. KIMBERLY HOPKINS
Hecke’s formula [Hec59] for the central value of L ( ψ N , s ) states(3.1) L ( ψ N ,
1) = 2 πω N √ N X [ a ] ∈ Cl ( O K ) X [ Q ] ∈ Cl ( O L ) Θ Q ( τ a ¯ N ) ψ ¯ N (¯ a )where ω N is the number of units in O L .The theta function for Q arises from a certain specialization of a symplectic theta func-tion. Let Sp ( Z ) denote the Siegel modular group of degree 2. Let Γ θ be the subgroupof (cid:0) α βγ δ (cid:1) ∈ Sp ( Z ) ( α, β, γ, δ ∈ Mat ( Z )) such that both α T γ and β T δ have even diag-onal entries. The group Γ θ inherits the action of Sp ( Z ) on the Siegel upper half plane h := (cid:8) z ∈ Mat ( C ) : T z = z, Im( z ) > (cid:9) . Define the symplectic theta function by θ ( z ) := X ~x ∈ Z exp[ πi T ~x z ~x ] , z ∈ h . The function θ satisfies the functional equation(3.2) θ ( M ◦ z ) = χ ( M )[det( γz + δ )] / θ ( z ) , M ∈ Γ θ where χ ( M ) is an eighth root of unity which depends on the chosen square root of det( γz + δ )but is otherwise independent of z . It is a symplectic modular form on Γ θ of dimension − / χ (see [Eic66, p.43] or [Mum07, p.189], for example) .Given a primitive ideal Q of O L , let Q := [ a, b, c ] represent the corresponding binary quadraticform of discriminant − N . The product of the matrix of Q with any Heegner point τ a is theSiegel point Qτ a := (cid:18) a bb c (cid:19) · τ a ∈ h . We will refer to points constructed in this way as split-CM points of level N and discriminant D . This yields the relation(3.3) Θ Q ( τ a ) = θ ( Qτ a )which can be substituted into formula (3.1) to get(3.4) L ( ψ N ,
1) = 2 πω N √ N X [ a ] ∈ Cl ( O K ) X [ Q ] ∈ Cl ( − N ) θ ( Qτ a ) ψ ¯ N (¯ a ) . If Q ∼ Q ′ in Cl ( − N ), then Qτ a ∼ Q ′ τ a in h /Sp ( Z ), and if a ∼ a ′ in Cl ( O K ), then Qτ a ∼ Qτ a ′ in h /Sp ( Z ) (see Remark 6.2 and Lemma 6.12). In addition it is shown in [Pac05, Lemma 53]that these equivalences of Siegel points sustain modulo Γ θ . The function θ/ψ ¯ N is invariant onsuch points: Lemma 3.1.
Fix an ideal a ⊂ O K and a prime ideal N ⊂ O K of norm N . Let Q be a binaryquadratic form of discriminant − N . Then the value (3.5) θ ( Qτ a ¯ N ) ψ ¯ N (¯ a ) depends only on the class [ Q ] ∈ Cl ( O L ) and the class [ a ] ∈ Cl ( O K ) .Proof. The value θ ( Qτ a ¯ N ) is independent of the class representative of [ Q ] because equivalentforms represent the same values. That (3.5) is independent of the representative of [ a ] ∈ Cl ( O K ) is a short calculation using the functional equation for θ in (3.2) and is done in [Pac05,Proposition 22]. (cid:3) The symplectic theta function is sometimes defined with extra parameters, θ ( z, u, v ) where u, v ∈ C , inwhich case the theta function above is equal to θ ( z,~ ,~ ENTRAL VALUES OF HECKE L-SERIES 5
Therefore the set of points [ Q ] τ [ a ] ¯ N as [ Q ] runs over Cl ( − N ) and [ a ] runs over Cl ( O K ) areequivalent in h / Γ θ and are identified under θ/ψ ¯ N . We refer to [ Q ] τ [ a ] ¯ N as a split-CM orbit .Thus to determine which values θ ( Qτ a ) are equal in (3.4) it is necessary to determine whichsplit-CM orbits [ Q ] τ [ a ] ¯ N are equivalent modulo Γ θ . Since h /Sp ( Z ) is a moduli space for theprincipally polarized abelian varieties of dimension two ([Mum07] or [BL04, Chp. 8]), the classesof split-CM points are determined by the isomorphism classes of the corresponding varieties.To describe these, we will recall some basic facts about quaternion algebras. Let B :=( − , D ) Q be the quaternion algebra over Q ramified at ∞ and | D | . Recall two maximal orders R , R ′ in B are equivalent if there exists x ∈ B × such that R ′ = x − Rx . Moreover, twooptimal embeddings φ : O L ֒ → R and φ ′ : O L ֒ → R ′ are equivalent if there exists x ∈ B × and r ∈ R ′× such that R ′ = x − Rx and φ ′ = ( xr ) − φ ( xr ). Let R denote the set of conjugacyclasses of maximal orders in B and let Φ R denote the set of classes of optimal embeddingsof O L into the maximal orders of B . Let R N ⊂ R denote the maximal order classes whichadmit an optimal embedding of O L . Given an optimal embedding ( φ : O L ֒ → R ) ∈ Φ R , let( ¯ φ : O L ֒ → R ) ∈ Φ R denote its quaternionic conjugate, so that φ ( √− N ) = ¯ φ ( −√− N ). Thequotient Φ R / − will denote the set Φ R modulo this conjugation. Let h R ( − N ) denote the numberof optimal embeddings of O L into R modulo conjugation by R × . This number is an invariantof the choice of representative of [ R ] in R .Our first theorem says that the classes of split-CM points in Siegel space correspond to classesof maximal orders in B . Theorem 3.2.
Fix [ a ] ∈ Cl ( O K ) , N ⊂ O K a prime ideal of norm N , and τ := τ a ¯ N . There is abijection Υ : { Qτ : [ Q ] ∈ Cl ( − N ) } /Sp ( Z ) −→ R N . This map is independent of the choice of representative a of [ a ] . Let Υ − ([ R ]) for [ R ] ∈ R N denote the pre-image class in h /Sp ( Z ) and set Υ − ([ R ]) := ∅ if[ R ] ∈ R \ R N . Our second theorem gives the number of split-CM orbits in a given class. Theorem 3.3.
Assume the hypotheses of Theorem 3.2. For any [ R ] ∈ R , (cid:8) [ Q ] τ ∈ Υ − ([ R ]) : [ Q ] ∈ Cl ( − N ) (cid:9) = h R ( − N ) / . That is, the number of split-CM orbits in the class in h /Sp ( Z ) corresponding to [ R ] underTheorem 3.2 is h R ( − N ) / . For a maximal order R of B , define S R := Z + 2 R and S R ⊂ S R to be the suborder of tracezero elements. The suborder S R is a rank 3 Z -submodule of R . Define g R to be its theta series g R ( τ ) := 12 X x ∈ S R q N ( x ) = 12 + X N> a R ( N ) q N , where a R ( N ) are defined by its q -expansion. It is well known that g R is a weight 3 / (4 | D | ). Applying [Gro87, Proposition 12 .
9] to fundamental − N gives a R ( N ) = ω R ω N h R ( − N )where ω R is the cardinality of the set R × / < ± > .This gives immediately the following Corollary to Theorem 3.3. KIMBERLY HOPKINS
Corollary 3.4.
Assume the hypotheses of Theorem 3.3. For any [ R ] ∈ R , (cid:8) [ Q ] τ ∈ Υ − ([ R ]) : [ Q ] ∈ Cl ( − N ) (cid:9) = a R ( N ) · ω N ω R . That is, the number of split-CM orbits in the class in h /Sp ( Z ) corresponding to [ R ] underTheorem 3.2 is proportional to the N -th Fourier coefficient of the weight / modular form g R . The application of Theorems 3.2 and 3.3 to a formula for L ( ψ N ,
1) proceeds as follows. Definethe following normalization of θ given by [Pac05]:(3.6) ˆ θ ( Qτ a ¯ N ) := θ ( Qτ a ¯ N ) η ( ¯ N ) η ( O K )where η ( z ) := e ( z ) Q ∞ n =1 (1 − e πiz ) for Im( z ) > η on ideals is defined in Section 6. It is proven in [Pac05, Proposition 23] (see also [HV97])that the numbers in ˆ θ ( Qτ a ¯ N ) /ψ ¯ N (¯ a ) are algebraic integers.Define Θ [ a ,Q ] , N := ˆ θ ( Qτ a ¯ N ) ψ ¯ N (¯ a ) . This is well-defined by Lemma 3.1. The following lemma says that the theta-values whichcorrespond to a given class [ R ] ∈ R under Theorem 3.2 are all equal up to ± Lemma 3.5.
Fix [ a ] ∈ Cl ( O K ) , N ⊂ O K a prime ideal of norm N , and τ := τ a ¯ N . Let [ R ] ∈ R .Then the values (3.7) (cid:8) Θ [ a ,Q ] , N : [ Q ] τ ∈ Υ − ([ R ]) (cid:9) differ by ± . Assume Lemma 3.5 holds (see Section 5 for the proof). Given [ R ] ∈ R N and any [ Q ] τ ∈ Υ − ([ R ]), define Θ [ a ,R ] , N to be either Θ [ a ,Q ] , N or − Θ [ a ,Q ] , N so that it satisfies Re(Θ [ a ,R ] , N ) > [ a ,R ] , N := 0 if [ R ] ∈ R \ R N .We record the mysterious ± ε [ a ,R ] : (cid:8) [ Q ] τ ∈ Υ − ([ R ]) (cid:9) −→ {± } (3.8) [ Q ] τ sgn (cid:0) Re (cid:0) Θ [ a ,Q ] , N (cid:1)(cid:1) . Note Θ [ a ,Q ] , N = ± Θ [ a ,R ] , N by construction. This definition assigns, albeit somewhat arbitrarily,a fixed choice of sign for the theta-values as [ Q ] varies.We then define a corresponding twisted variant of h R ( − N ) by(3.9) h ε [ a ,R ] ( − N ) := X [ Q ] τ ∈ Υ − ([ R ]) ε [ a ,R ] ([ Q ] τ ) . The formula for L ( ψ N ,
1) can now be stated as follows.
Theorem 3.6.
Let N ⊂ O K be a prime ideal of norm N . Then (3.10) L ( ψ N ,
1) = π · η ( ¯ N ) η ( O K ) ω N √ N X [ R ] ∈ R X [ a ] ∈ Cl ( O K ) Θ [ a ,R ] , N · h ε [ a ,R ] ( − N ) . where Θ [ a ,R ] , N is an algebraic integer and h ε [ a ,R ] ( − N ) is an integer with | h ε [ a ,R ] ( − N ) | ≤ h R ( − N ) . ENTRAL VALUES OF HECKE L-SERIES 7
Remark . The signs in Lemma 3.5 and hence the function h ε [ a ,R ] ( − N ) depend on the character χ which appears in the functional equation (6.5) for θ . In particular, the values of χ depend onthe entries of the transformation matrices in Γ θ which takes one Siegel point to an equivalentone. This value is complicated to compute or even define, and is discussed in detail in [AM75,Sta82] and [Eic66, Appendix to Chp 1]. An arithmetic formula for these signs and for h ε [ a ,R ] ( − N )is yet to be determined. But since the h ε [ a ,R ] ( − N ) are a weighted count of optimal embeddings,we expect that, like the h R ( − N ), they will be related to coefficients of a half-integer weightmodular form. This will be treated in a subsequent paper.Theorem 3.6 gives us an upper bound on L ( ψ N ,
1) in terms of the computable modular formcoefficients h R ( − N ). Corollary 3.8.
Assume the hypotheses of Theorem 3.6. Then | L ( ψ N , | ≤ π · | η ( ¯ N ) η ( O K ) | ω N √ N X [ R ] ∈ R X [ a ] ∈ Cl ( O K ) | Θ [ a ,R ] , N | · h R ( − N ) . If h ( D ) = 1, then (3.10) has a particularly simple form: Corollary 3.9.
Assume the hypotheses of Theorem 3.6 and suppose h ( D ) = 1 . Then Θ [ a ,R ] , N =Θ [ R ] and h ε [ a ,R ] ( − N ) = h ε [ R ] ( − N ) are independent of a and N and L ( ψ N ,
1) = π · | η ( O K ) | ω N √ N X [ R ] ∈ R Θ [ R ] · h ε [ R ] ( − N ) . We conclude this section with a comment regarding varying N . The set [ N (cid:8) [ Q ] τ [ a ] ¯ N : [ Q ] ∈ Cl ( − N ) , [ a ] ∈ Cl ( O K ) , N ⊂ O K of norm N (cid:9) , of split-CM orbits over all prime N with D ≡ (cid:3) mod 4 N partitions into a finite number ofSiegel classes in h /Sp ( Z ). This has a natural explanation from our viewpoint. As a complextorus, X Qτ is isomorphic to a product E × E ′ of two elliptic curves E, E ′ defined over ¯ Q andwith complex multiplication by O K . (This is the reason the Qτ are called ‘split-CM’.) It is ageneral result of [NN81] that there are only finitely many principal polarizations on a givencomplex abelian variety up to isomorphism. There are also only finitely many isomorphismclasses of elliptic curves with CM by O K . Together these imply that the number of classesof Siegel points ( X Qτ , H Qτ ) for all split-CM points Qτ of discriminant D must be finite. See[Pac05, Theorem 58] as well for an alternative interpretation.4. Endomorphisms of X z preserving H z In this section we prove that the endomorphisms of the abelian varieties corresponding tosplit-CM points give maximal orders in the quaternion algebra B = ( − , D ) Q . Let V, V ′ becomplex vector spaces of dimension 2 with lattices L ⊂ V , L ′ ⊂ V ′ . The analytic and rationalrepresentations are denoted by ρ a : Hom( X, X ′ ) −→ Hom C ( V, V ′ ) and ρ r : Hom( X, X ′ ) −→ Hom Z ( L, L ′ ), respectively. Recall the periods matrices Π , Π ′ ∈ Mat × ( C ) of X, X ′ commutewith ρ a and ρ r in the following diagram(4.1) Z gρ r ( f ) (cid:15) (cid:15) Π / / C gρ a ( f ) (cid:15) (cid:15) Z g ′ Π ′ / / C g ′ KIMBERLY HOPKINS (see [BL04], for example).For any Siegel point z ∈ h , let Π z := [ z, ] ∈ Mat × ( C ) be its period matrix, L z := Π z Z be its defining lattice, and X z := C /L z be its corresponding complex torus. The Hermitianform H z : C × C → C defined by H z ( u, v ) := T u Im( z ) − ¯ v determines a principal polarizationon X z . As a point in the moduli space h /Sp ( Z ), z corresponds to the principally polarizedabelian variety ( X z , H z ). Throughout Sections 4, 5 and 6, fix a representative a of [ a ] ∈ Cl ( O K ), N ⊂ O K a prime ideal of norm N , τ := τ a ¯ N := − b + √ D a N , and a split-CM point z = Qτ of level N and discriminant D where Q := [ a, b, c ] is of discriminant − N . The endomorphisms of ( X z , H z )will be our first main object of study.We define B to be the Q -algebra of endomorphisms of X z which fix H z B := (cid:8) α ∈ End Q ( X z ) : H z ( αu, v ) = H z ( u, α ι v ) ∀ u, v ∈ C (cid:9) ;here ι is the canonical involution inherited from Mat ( K ) as defined in [Shi73a]. In terms ofmatrices, let H z := Im( z ) − denote the matrix of H z with respect to the standard basis of C .Then viewing End Q ( X z ) ⊆ Mat ( K ), the set B is B = (cid:8) M ∈ End Q ( X z ) : T ¯ M H z = H z M ι (cid:9) . The bar denotes complex conjugation restricted to K . The map ι sends a matrix M to itsadjoint, or equivalently sends M to Tr( M ) · − M .We define R z to be the Z -submodule of endomorphisms which fix H z (4.2) R z := (cid:8) M ∈ End( X z ) : T ¯ M H z = H z M ι (cid:9) . The first observation is that B is isomorphic to a rational definite quaternion algebra. Proposition 4.1. B is isomorphic to B as Q -algebras.Remark . In [Shi73a, Proposition 2 . B is a quaternion algebra over Q ina much more general setting by showing B ⊗ ¯ Q is isomorphic to Mat ( ¯ Q ). Here we give analternative proof which explicitly gives the primes ramified in B . Proof.
We will need the following elementary lemma.
Lemma 4.3.
Suppose Q , Q ∈ Mat ( Z ) with determinant N . Set H i := Im( Q i τ ) − and R i := (cid:8) M ∈ End ( X Q i τ ) : T ¯ M H i = H i M ι (cid:9) , i = 1 , . Let S = Z or Q and suppose there exists A ∈ GL ( S ) such that Q = (det A ) − AQ T A . Thenthe map End S ( X Q τ ) −→ End S ( X Q τ )(4.3) M AM A − and the induced map R ⊗ Z S −→ R ⊗ Z S are S -algebra isomorphisms.Proof of Lemma. Let Π i := [ Q i τ, ] be the period matrices for Q i τ , i = 1 ,
2. Suppose M ∈ End S ( X Q τ ). By (4.1), this is if and only if M Π i = Π i P for some P ∈ Mat ( S ). Set˜ A := (cid:18) (det A − ) T A A − (cid:19) ∈ GL ( S ) . Using the identity A Π ˜ A = Π gives( AM A − )Π = Π ( ˜ A − P ˜ A ) . ENTRAL VALUES OF HECKE L-SERIES 9
Clearly ˜ A − P ˜ A ∈ Mat ( S ), hence AM A − ∈ End S ( X Q τ ).Furthermore the identity H = (det A − ) T AH A implies T ( AM A − ) H = H ( AM A − ) ι bya straightforward calculation. (cid:3) Define matrices(4.4) A := 12 a (cid:18) − b a (cid:19) ∈ GL ( Q ) and Q ′ := (cid:18) N (cid:19) . By Lemma 4.3, B is isomorphic as a Q -algebra to B ′ := (cid:8) M ∈ End Q ( X Q ′ τ ) : T ¯ M H ′ = H ′ M ι (cid:9) where H ′ := Im( Q ′ τ ) − .We will compute B ′ explicitly. Let E τ := C / ( Z + Z τ ) for any τ ∈ h . Clearly X Q ′ τ ∼ = E τ × E Nτ as complex tori. The endomorphisms of X Q ′ τ are characterized as follows. Lemma 4.4.
End ( X Q ′ τ ) = (cid:18) O K Z + Z ω/NN Z + Z ¯ ω O K (cid:19) where ω := a N τ . Assuming this for a moment, we have End Q ( X Q ′ τ ) = Mat ( K ), and a quick calculationshows any M = (cid:0) α βγ δ (cid:1) ∈ Mat ( K ) satisfies T ¯ M H = HM ι if and only if δ = ¯ α and γ = − N ¯ β .Therefore B ′ = (cid:26)(cid:18) α β − N ¯ β ¯ α (cid:19) : α, β ∈ K (cid:27) ⊂ Mat ( K ) . The elements (cid:18) (cid:19) , (cid:18) √ D −√ D (cid:19) , (cid:18) − N (cid:19) , (cid:18) √ DN √ D (cid:19) form a basis of B ′ and clearly give an isomorphism to ( D, − N ) Q . We claim B ∼ = ( D, − N ) Q .This is a general fact: if p , q are primes with p ≡ q ≡ − p is a square modulo q ,then ( − p, − q ) Q is ramified at ∞ and p only, so ( − p, − q ) Q ∼ = ( − , p ) Q . Hence B ∼ = B ′ ∼ = B as Q -algebras.It remains to prove Lemma 4.4. Proof of Lemma 4.4.
For any quadratic surds τ, τ ′ ∈ K ,Hom( E τ , E τ ′ ) = { α ∈ K : α ( Z + Z τ ) ⊆ Z + Z τ ′ } . Since X Q ′ τ ∼ = E τ × E Nτ , we haveEnd( X Q ′ τ ) = (cid:18) End( E τ ) Hom( E Nτ , E τ )Hom( E τ , E Nτ ) End( E Nτ ) (cid:19) . We compute. End( E Nτ ) = O K since Z + Z a N τ = O K and [1 , N τ ] is a (proper) fractional O K -ideal. Similarly End( E τ ) = O K since Z + Z τ is a fractional O K -ideal.It is straightforward to check Z + Z a τ ⊆ Hom( E Nτ , E τ ). On the other hand, Hom( E Nτ , E τ ) ⊂ Z + Z τ by definition, and this is proper containment since otherwise Z + Z N τ would preserve Z + Z τ which is impossible since the former contains O K . Therefore Hom( E Nτ , E τ ) = Z + Z mτ for some integer m | a but a quick calculation shows m = a else it divides a , b and c whosegcd is assumed to be 1.It remains to show Hom( E τ , E Nτ ) = N Z + Z ¯ ω. First observe the ideal ( N ) in O K is contained in Hom( E τ , E Nτ ) since N ( Z + Z a N τ )( Z + Z τ ) ⊆ N ( Z + Z τ ) ⊆ Z + N Z τ. Furthermore ( N ) splits as ( N ) = N · ¯ N where N = N Z + Z ω . Therefore N · ¯ N ⊆ Hom( E τ , E Nτ ) ⊆ O K , where the last containment follows because Z + Z τ is a proper fractional O K -ideal which contains Z + Z N τ . But since O K is Noetherian, there exists a maximal order M such that N · ¯ N ⊆ Hom( E τ , E Nτ ) ⊆ M ⊆ O K . Therefore either N or ¯ N is in M . Whichever is contained in M is actually equal to M since theyare both prime and hence maximal. But Hom( E τ , E Nτ ) is not contained in N . For example,¯ ω ∈ Hom( E τ , E Nτ ) but not in N . ThusHom( E τ , E Nτ ) ⊆ ¯ N . Finally since the index [ ¯ N : ( N )] = N is prime, either Hom( E τ , E Nτ ) is equal to N or ¯ N , butwe already showed the former is impossible, hence it is the latter. (cid:3) This also completes the proof of Proposition 4.1. (cid:3)
Lemma 4.5. R z is isomorphic to an order in B as Z -algebras, and admits an optimal embed-ding of O L .Proof. The first part is immediate.The embedding is given in matrix form by QS where S := ( − ). It is straightforward tocheck that ( QS ) = − N and QS ∈ R z using definition (4.2). An embedding is optimal if itdoes not extend to any larger order in the quotient field, but this is immediate since O L is themaximal order in L . (See [Shi73a] for additional discussion of this order.) (cid:3) The next step is to prove the order R z is maximal. Theorem 4.6. R z is a maximal order.Proof. It suffices to show the local order ( R z ) p is maximal for all primes p . We do this withthe following two lemmas. Lemma 4.7. ( R z ) p is maximal for all primes p = 2 .Proof of Lemma. Define R ′ := B ′ ∩ End( Q ′ τ ) with Q ′ defined in (4.4). From Lemma 4.4 andthe definition of B ′ above it is clear that R ′ is an order given explicitly by(4.5) R ′ = (cid:26)(cid:18) α β − N ¯ β ¯ α (cid:19) : α ∈ O K , β ∈ Z + Z ω/N (cid:27) . Its discriminant is D , which can be computed using the basis(4.6) u := (cid:18) (cid:19) , u := (cid:18) ω
00 ¯ ω (cid:19) , u = (cid:18) − N (cid:19) , u = (cid:18) ω/N − ¯ ω (cid:19) . Hence R ′ is maximal. For p a , the matrix A from (4.4) is in Mat ( Z p ) and so gives anisomorphism M AM A − from ( R z ) p → R ′ p . Hence ( R z ) p is maximal for p a .There exists a form ˜ Q = (cid:16) a ˜ b ˜ b c (cid:17) properly equivalent to Q with gcd(2 a, ˜ a ) = 1 (see [Cox89,p. 25,35], for example). Applying Lemma 4.3 to the pair Q and ˜ Q gives R z ∼ = R ˜ Qτ . Hence for p | a we can apply the paragraph above to R ˜ Qτ to conclude ( R z ) p is maximal. (cid:3) Lemma 4.8. ( R z ) is maximal. ENTRAL VALUES OF HECKE L-SERIES 11
Proof of Lemma.
Note gcd(2 a, b ) = 1 because N is prime and b is odd. Define U := (cid:0) − cx − by (cid:1) and V := (cid:0) y − bx a (cid:1) where x, y ∈ Z such that 2 ay + bx = 1. Then U QV = Q ′ where Q ′ wasdefined in (4.4). Define ˆ H := T U − HU − , ˆ B := n M ∈ End Q ( X Q ′ τ ) : T ¯ M ˆ H = ˆ HM ι o , andˆ R := ˆ B ∩ End( X Q ′ τ ). The period matrix Π ′ := [ Q ′ τ, ] satisfies Π ′ = U Π z ˜ V where ˜ V := (cid:0) V U − (cid:1) ∈ Mat ( Z ). Hence the map M U M U − from R z → ˆ R is an isomorphism over Z .Therefore ( R z ) p ∼ = ˆ R p for all primes p . We will show ˆ R is maximal.By Lemma 4.3 and the isomorphism B ∼ = B ′ , a basis for B is given by the set { A − u i A } with A defined in (4.4) and u i in (4.6). Hence by above the set { v i := U A − u i AU − } gives abasis for ˆ B over Q . Replace v i with 2 av i for i = 2 , v by 2 aN v . Then explicitly, v = (cid:18) (cid:19) v = (cid:18) aω − N x ( b + 2 ω ) 2 a ¯ ω (cid:19) v = (cid:18) aN x a − N ( N x + 1) − aN x (cid:19) v = (cid:18) aN xω a ω − N ( N x ω + ¯ ω ) − aN xω (cid:19) . By Lemma 4.4 we see v i ∈ ˆ R , i = 1 , . . . ,
4. To prove ˆ R is maximal we will use the elements { v i } to construct a basis of ˆ R whose discriminant is a unit modulo ( Z ) .Associate any matrix M := ( m ij + n ij ω ) ∈ Mat ( Q ( ω )) with m ij , n ij ∈ Q to the vector ~v M := T ( m , n , m , n , m , n , m , n ) ∈ Q . Denote the vector ~v v i by ~v i for simplicity. Let M bas ∈ Mat × ( Z ) be the matrix whose i -thcolumn is ~v i for i = 1 , . . . ,
4. Given M ∈ Mat ( K ), M ∈ ˆ B if and only if(4.7) ~v M = M bas · ~α M for some ~α M ∈ Q . Moreover M := ( m ij + n ij ω ) is in End( Q ′ τ ) if and only if(4.8) m , n , m , N n , m − b n N , n , m , n ∈ Z by (4.5). Let M end ∈ Mat × ( Q ) be the matrix which describes the conditions in (4.8) so that M ∈ End( Q ′ τ ) if and only if M end · ~v M ∈ Z . Therefore the elements M of ˆ R correspondprecisely under (4.7) to ~α M ∈ Q such that(4.9) M end · M bas · ~α M ∈ Z . To show the discriminant of ˆ R is 1 mod ( Z ) amounts to finding solutions ~β ∈ Z such that M end · M bas · ~β ≡ ~α := ~β/ ~α are given by the vectors ~α := T (0 , , , / , ~α := T (0 , , , / , and ~α := T (2 , , , / . Therefore ~v i := M bas · ~α i gives an element in ˆ R for i = 5 , ,
7. Consider the set S := { ~v , ~v , ~v , ~v } . Observe the relations ~v = ~v / , ~v = ( ~v + ~v ) / , and ~v = ( ~v + ~v ) / . These imply ~v generates ~v , while ~v and ~v generate ~v , and finally ~v and ~v generate ~v .Accordingly, replace ~v and ~v in S with ~v and ~v so that S = { ~v , ~v , ~v , ~v } . Now S is a set oflinearly independent vectors over Z and contained in ˆ R , hence a basis. A computation (usingPARI/GP [PAR08]) of the discriminant of ˆ R with respect to this basis shows it is D · N · a .This is a unit modulo ( Z ) since we may assume a is odd. Hence ˆ R is maximal. (cid:3) This concludes the proof that R z is a maximal order. (cid:3) The next step is to prove R z is the right order of an explicit ideal in B . We first recall aresult of Pacetti which constructs Siegel points from certain ideals of B .5. Split-CM points and right orders in B In this section we identify R z with an explicit right order in B . Let M be a maximal orderof B such that there exists u ∈ M with u = D . (Such an order must exist by Eichler’s massformula). Two left M -ideals I and I ′ are in the same class if there exists b ∈ B × such that I = I ′ b . The number n of left M -ideal classes is finite and independent of the choice of maximalorder M . Let I be the set of n left M -ideal classes, and recall R is the set of conjugacy classesof maximal orders in B . (Equivalently, R is the set of conjugacy classes of right orders withrespect to M , taken without repetition.) The cardinality t of R is less than or equal to n andis called the type number .Recall B ∼ = ( D, − N ) Q and let 1 , u, v, uv be a basis for B where u = D , v = − N , and uv = − vu . Define the Z -module(5.1) I z := (cid:28)(cid:18) b − u a N (cid:19) av, (cid:18) b − u a N (cid:19)(cid:18) N + bv (cid:19) , b − v , − a (cid:29) Z . It is proven in [Pac05, p. 369-372] that I z is a left ideal for a maximal order M a , [ N ] which isindependent of the class representative of [ N ] and of the form Q , and contains the element u .Let R z denote the right order of I z . It is maximal because M a , [ N ] is maximal.We will show that the right order R z has a natural identification with the maximal order R z .To do this, we recall a result of [Pac05] which associates ideals of B to Siegel points. Namely,let ( I R , R ) be a pair consisting of a left M -ideal I R with maximal right order R . Define the4-dimensional real vector space V := B ⊗ Q R , so that V /I R is a real torus. The linear map J : V → Vx u p | D | · x induces a complex structure on V . Hence the data ( V /I R , J ) determines a 2-dimensional com-plex torus. Define a map E R : V × V → R by E R ( x, y ) := Tr( u − x ¯ y ) / N ( I R ) , where N ( I R ) is the norm of the ideal I R and the ‘bar’ denotes conjugation in B . It is straight-forward to check that E R is alternating, satisfies E R ( J x, J y ) = E R ( x, y ) for all x, y ∈ V , isintegral on I R , and that the form H R : V × V → C defined by(5.2) H R ( x, y ) := E R ( J x, y ) + i E R ( x, y ) , x, y ∈ V is positive definite (see [Pac05] for details). Thus E R is a Riemann form and so there exists asymplectic basis { x , x , y , y } of I R with respect to E R . The matrix E R of E R with respect tothis basis has determinant det( E R ) = N ( I R ) − N ( u ) − disc( I R ) , where we have used the fact that disc( I R ) = (det( u i u j )) ij for any basis { u , . . . , u } of I R . Butthe fact that R is maximal implies disc( I R ) = D N ( I R ) [Piz80], [Pac05, Proposition 32], hencedet( E R ) = 1. This implies E R is of type 1, its matrix is E R = (cid:0) − (cid:1) , and H R is a principalpositive definite Hermitian form. ENTRAL VALUES OF HECKE L-SERIES 13
The conclusion is that the data ( I R , J, E R ) determines a Siegel point in h /Sp ( Z ). Theaction of a γ ∈ Sp ( Z ) on ( I R , J, E R ) is given as a Z -linear isomorphism I R → γ ( I R ), whichsends J → γ − ◦ J ◦ γ , and E R → E R ◦ γ .Left M -ideals with the same right order class determine equivalent Siegel points under thisconstruction [Pac05, p. 364]. In other words, there is a well-defined map R −→ h /Sp ( Z ) . This can be seen as follows. Let I and I ′ be two left M -ideals with the same right order class [ R ].Assume first that they are equivalent, that is, I = I ′ b for some b ∈ B × . Then multiplicationon the right by b determines a Z -linear isomorphism γ : I −→ I ′ x x · b. Furthermore E ( γ ( x ) , γ ( y )) = Tr( u − x · b ( y · b )) N ( I ) = E ( x, y ) · N ( b ) N ( I ) = E ′ ( x, y ) , and since J is a multiplication on the left, and b on the right, clearly γ − ◦ J ◦ γ = J . Therefore( I, J, E ) ∼ ( I ′ , J, E ′ ) for I ∼ I ′ . Now suppose I and I ′ are not equivalent. Then uI has thesame left order and right order class as I but is not equivalent to I (see Lemmas 6.7 and 6.9below). Since there are at most two classes of left M -ideals with the same right order class, itmust be that uI ∼ I ′ ∼ uIu − . It is straightforward to check that the map from I to uIu − viaconjugation by u gives ( I, J, E ) ∼ ( uIu − , J, E ) and so by the above case, ( I, J, E ) ∼ ( I ′ , J, E ′ ).The ideal I z in (5.1) corresponds to the Siegel point z under this construction. This is left asan exercise in [Pac05] but can be seen as follows. Let { x , x , y , y } denote the basis, taken inorder, of I z given in (5.1). A straightforward calculation done by Pacetti shows { x , x , y , y } is symplectic with respect to E , and of principal type. Then { y , y } is a basis for the complexvector space ( V, J ), and the period matrix for the complex torus (
V /I z , J ) is the coefficientmatrix of the basis of { x , x , y , y } in terms of { y , y } . It suffices to show this period matrixis Π z := [ z, ]. Thus one needs to verify x = 2 a ˜ τ y + b ˜ τ y x = b ˜ τ y + 2 c ˜ τ y , where ˜ τ := − b + √ | D | J a N is given by the complex multiplication J . This is a simple calculationusing the relations D = b − a c N and − N = b − ac .Note this construction determines an isomorphism σ : I z −→ L z by x (cid:18) ab (cid:19) τ, x (cid:18) b c (cid:19) τ, y (cid:18) (cid:19) , y (cid:18) (cid:19) , which maps J i . In particular H R z ( x, y ) = H z (cid:12)(cid:12) L z × L z ( σ ( x ) , σ ( y )) for all x, y ∈ I z .The elements of R z and R z can now be related as follows. Any b ∈ R z preserves I z (onthe right) as well as the complex structure J and hence defines an endomorphism f b of X z .Likewise, any M ∈ R z defines an endomorphism f M of the torus X z by definition. We claimthese rings give the same endomorphisms of X z : Proposition 5.1.
As endomorphisms, R z is identified with R z . Proof.
Suppose f b ∈ End( X z ) for some b ∈ R z . To show f b comes from R z , it suffices to show ρ r ( f b ) preserves H z (cid:12)(cid:12) L z × L z . Equivalently by the map σ it suffices to show H R z ( x · b, y ) = H R z ( x, y · b ι ) . But this is immediate since Tr( u − ( xb )¯ y ) = Tr( u − x (¯¯ b ¯ y ) = Tr( u − x ( y ¯ b )) and ¯ b = b ι in B .Therefore as endomorphisms R z is contained in R z . Conversely any f M ∈ End( X z ) for M ∈ R z defines a linear map from I z to itself which commutes with the complex structure J , hencecorresponds to an element in R z . (cid:3) Corollary 5.2. R z is isomorphic to the maximal right order R z in the quaternion algebra B ,and this map sends QS v .Proof. The first part follows immediately from the proposition. Regarding the embedding, therational representation in Mat ( Z ) of the endomorphism QS ∈ R z is b +12 c − a − b − b a − c b +12 . Its action on the basis x , x , y , y of I z shows immediately that it is the linear transformationgiven by multiplication on the right by v . (cid:3) Formula for the central value L ( ψ N , Proof of Theorems 3.2 and 3.3.
Fix [ a ] ∈ Cl ( O K ), N ⊂ O K a prime ideal of norm N , τ := τ a ¯ N .Throughout the rest of this section, fix z := Qτ and z ′ := Q ′ τ where Q, Q ′ are binary quadraticforms of discriminant − N . DefineΥ : { Qτ : [ Q ] ∈ Cl ( − N ) } /Sp ( Z ) −→ R N (6.1) [ Q ] τ [ R Qτ ]Given an R Qτ , let φ Q : O L ֒ → R Qτ be the optimal embedding defined in Lemma 4.5 andCorollary 5.2. Define a second mapΥ : Cl ( − N ) −→ Φ R / − (6.2) [ Q ] [ φ Q : O L ֒ → R Qτ ] . We will start by showing that the maps Υ and Υ are well-defined. First note Υ is injective:if R Qτ ∼ R Q ′ τ in B , then we saw in the last section that Pacetti’s map R −→ h /Sp ( Z )sends R Qτ Qτ . After proving the maps are well-defined, we will prove Υ is a bijection andindependent of the choice of representative a of [ a ]. This will simultaneously prove Theorems3.3 and 3.2. Lemma 6.1. If z ∼ z ′ in h / Γ θ , then R z ∼ R z ′ in R .Remark . Note that if Q ∼ Q ′ with Q = T AQ ′ A for some A ∈ SL ( Z ), then Qτ ∼ Q ′ τ asSiegel points via the matrix (cid:0) T A A − (cid:1) ∈ Γ θ . Proof.
Recall z ∼ z ′ in h /Sp ( Z ) if and only if the abelian varieties ( X z , H z ) and ( X z ′ , H z ′ )are isomorphic. Write X, X ′ , H, H ′ for X z , X z ′ , H z , H z ′ , respectively. Suppose f : X −→ X ′ is ENTRAL VALUES OF HECKE L-SERIES 15 an isomorphism of (
X, H ) with ( X ′ , H ′ ), so that H ′ ( f ( x ) , f ( y )) = H ( x, y ) for all x, y ∈ C . Weclaim the isomorphism End( X ) −→ End( X ′ )(6.3) α f ◦ α ◦ f − induces an isomorphism of R z and R z ′ . This follows immediately from the calculation H ′ ( f ◦ α ◦ f − ( x ) , y ) = H ( α ( f − ( x )) , f − ( y ))= H ( f − ( x ) , α ι ( f − ( y ))) (since α ∈ R z )= H ′ ( x, f ( α ι ( f − ( y ))))= H ′ ( x, ( f ◦ α ◦ f − ) ι ) . The last equality follows because, as a matrix, ρ a ( f ) ι = ρ a ( f ) − det( ρ a ( f )) and so the deter-minants in ( f ◦ α ◦ f − ) ι cancel out. Therefore R z ′ = f ◦ R z ◦ f − and so by Proposition 5.1, R z ∼ R z ′ in B . (cid:3) Lemma 6.3. If Q ∼ Q ′ in Cl ( − N ) , then the corresponding optimal embeddings v +12 ֒ → R z and v +12 ֒ → R z ′ are equivalent.Proof. Suppose Q ∼ Q ′ with Q ′ = AQ T A for some A ∈ SL ( Z ). Then by Lemma 4.3, themap R z → R z ′ by M AM A − is a Z -algebra isomorphism, and extends to a Q -algebraisomorphism from B → B ′ . In particular it sends QS A ( QS ) A − = Q ′ S . By Corollary5.2, this induces a Z -algebra isomorphism of R z → R z ′ which sends v to v , and extends toa Q -algebra automorphism of B . Hence by the Skolem-Noether theorem, the map R z → R z ′ must be conjugation by some unit of B . (cid:3) We now turn to proving Υ is a bijection. The following six lemmas will be needed to proveΥ is injective. Let Q denote the ideal in L which corresponds to Q . Lemma 6.4. I z ∼ = ¯ Q ⊕ ¯ Q as right O L -modules.Proof of Lemma. Define v := x , v := x , v := y , v := − y where x i , y j is the basis of I z defined in Section 4. The { v i } also form a basis for I z . The map f : I z −→ ¯ Q ⊕ ¯ Q defined by v ( a, v ( b − √− N , v (0 , a ) v (0 , b − √− N Z -linearly is an isomorphism of Z -modules. To show it is an O L -module isomor-phism, it suffices to show f (cid:18) v i (cid:18) b + v (cid:19)(cid:19) = f ( v i ) (cid:18) b + √− N (cid:19) for all i = 1 , , , . For this, use the identities: v (cid:18) b + v (cid:19) = bv − av v (cid:18) b + v (cid:19) = cv v (cid:18) b + v (cid:19) = cv v (cid:18) b + v (cid:19) = − av + bv . (cid:3) Lemma 6.5.
Suppose S := I z x where x ∈ B × commutes with v +12 . Then S ∼ = ¯ Q ⊕ ¯ Q , as right O L -modules.Proof of Lemma. By Lemma 6.4 and the hypotheses on x , the composition from S → ¯ Q ⊕ ¯ Q given by g ( v i x ) := f ( v i ) is an isomorphism of O L -modules. (cid:3) Lemma 6.6.
Suppose ¯ Q ⊕ ¯ Q ∼ = ¯ Q ′ ⊕ ¯ Q ′ as right O L -modules, and h ( − N ) is odd. Then Q ∼ Q ′ in Cl ( − N ) .Proof of Lemma. By a classical theorem of Steinitz [Mil71, Theorem 1.6], ¯ Q ⊕ ¯ Q ∼ = ¯ Q ′ ⊕ ¯ Q ′ asright O L -modules if and only if [ ¯ Q ′ ] = [ ¯ Q ] as classes in the ideal class group of O L . This is ifand only if [ ¯ Q ′ / ¯ Q ] = [id] where id is the identity class. But since the class number h ( − N ) isodd, this implies [ Q ] = [ Q ′ ] in Cl ( − N ). (cid:3) The next three lemmas we need are general results for quaternion algebras. Assume forLemmas 6.7, 6.8, and 6.9 below that B is a quaternion algebra ramified precisely at ∞ and aprime p . In addition, assume M and R are maximal orders and there exists u ∈ M such that u = − p . Lemma 6.7. uM u − = M. Proof.
This is clear locally at primes q = p because u − = − u/p . This is also clear locally at p because there is a unique maximal order in the division algebra B p (see [MR03, Theorem 6.4.1,p.208] or [Vig80] for example). (cid:3) Lemma 6.8.
Suppose
I, I ′ are left M -ideals with right order R . In addition assume R admitsan embedding of a ring of integers O of some imaginary quadratic field. Set J := I ( I ′ ) − . Then J I ′ ∼ = I ′ as right O -modules.Proof. First note J is a bilateral M -ideal. Since u ∈ M , uM = M u by Lemma 6.7 and so is aprincipal M -ideal of norm p . Hence it is the unique integral bilateral M -ideal of norm p , andso every bilateral M -ideal is equal to uM · m for some m ∈ Q [Eic73, Proposition 1, p. 92]. Inparticular, this implies the bilateral M -ideals are principal. Therefore J = tM = M t for some t ∈ B × , and the map, f : I ′ −→ J I ′ w → tw is a Z -module isomorphism. Since the multiplication by t is on the left, f is an isomorphism ofright O modules. (cid:3) Lemma 6.9.
Suppose I is a left M -ideal with right order R . Then uI is also a left M -idealwith right order R . Furthermore, any left M -ideal with right order R is equivalent to I or uI (or both). ENTRAL VALUES OF HECKE L-SERIES 17
Proof.
The right order of uI is clearly R . The left order is uM u − = M by Lemma 6.7.Suppose J is any left M -ideal with right order R . The ideal I − J is R -bilateral, hence I − J = P i m, i = 0 , , m ∈ Q where P is the unique bilateral R -ideal of norm p [Eic73, Proposition 1, p. 92].If I − J is principal, then I ∼ J . Otherwise i = 1. Then since the ideal I − uI is R -bilateralof norm p , by uniqueness I − uI = P and so I − J = I − uI · m. Multiplying through by I we see J ∼ uI as left M -ideals. (cid:3) Now the injectivity of Υ can be proven. Proposition 6.10.
Suppose ( R z , v ± ) ∼ ( R z ′ , v ± ) . Then Q ∼ Q ′ in Cl ( − N ) .Proof. The assumption ( R z , v ± ) ∼ ( R z ′ , v ± ) implies there exists x ∈ B × such that x − R z x = R z ′ and r ∈ R × z ′ such that ( xr ) − (cid:18) v + 12 (cid:19) xr = v + 12 . The proof is broken up into two cases.
Case 1.
Assume I z ∼ I z ′ . Then I z x ∼ I z ′ and they both have right order R z ′ . Set J := I z xI − z ′ .Then J I z ′ ∼ = I z ′ as right O L -modules by Lemma 6.8. Combining with Lemma 6.4 applied to I z ′ implies J I z ′ ∼ = ¯ Q ′ ⊕ ¯ Q ′ as right O L -modules.On the other hand, J I z ′ = I z x . Since r is a unit, I z x = I z xr , so replacing x by xr if necessarywe may assume r = 1 and x − (cid:0) v +12 (cid:1) x = v +12 . Lemma 6.5 applied to I z x gives J I z ′ ∼ = ¯ Q ⊕ ¯ Q as right O L -modules. Hence Q ∼ Q ′ by Lemma 6.6. Case 2.
Assume I z I z ′ . For each maximal order R , there can be at most two left M -idealclasses with right orders in the class [ R ]. Therefore since I z ′ has right order R z ′ ∈ [ R z ], but I z I z ′ , by Lemma 6.9 it must be that uI z ∼ I z ′ ;note uI z is a left M -ideal by Lemma 6.7. Then uI z x ∼ I z ′ and they have the same right order.Let J := uI z xI − z ′ and use the same argument from Case 1, noting that Lemmas 6.4 and 6.5hold with I z replaced by uI z since the multiplication by u is on the left. This concludes theproof that Υ is injective. (cid:3) It remains to show that Υ is a surjection. This follows from the fact: Lemma 6.11. h ( − N ) = R / − . Proof of Lemma.
For [ R ] ∈ R , let h R ( − N ) denote the number of optimal embeddings of O L into R , modulo conjugation by R × . Then R / − = 12 X [ R ] ∈ R h R ( − N ) by definition,= h ( − N ) by Eichler’s mass formula [Gro84, (1 . . (cid:3) The last task is to prove the maps Υ and Υ are independent of the choice of representative a of [ a ]. In fact we will prove a slightly stronger result regarding the right orders: Lemma 6.12. If a ∼ a ′ in Cl ( O K ) then R Qτ a ¯ N = R Qτ a ′ ¯ N .Proof. The hypothesis a ∼ a ′ implies a ¯ N ∼ a ′ ¯ N . Suppose ¯ N corresponds to a form [ N, b, c ].Then we can choose bases so that the products a ¯ N , a ′ ¯ N both correspond to forms with middlecoefficient congruent to b mod 2 N (see [RV91, Lemma 2.3], for example). The CM-points τ a ¯ N , τ a ′ ¯ N are Heegner points of level N and discriminant D by construction, and by the commentabove they have the same ‘root’ b mod 2 N of √ D mod 4 N . Hence there exists M := (cid:0) α βγ δ (cid:1) ∈ Γ ( N ) such that M ( τ a ¯ N ) = τ a ′ ¯ N . Set ˜ M := (cid:18) ˜ α ˜ β ˜ γ ˜ δ (cid:19) where ˜ α := α · , ˜ β := β · Q, ˜ γ := γ · Q − , ˜ δ := δ · . It is shown in [AM75, p.233], for example, that ˜ M ∈ Γ θ ⊆ Sp ( Z ). Therefore the relation˜ M ( Qτ a ¯ N ) = Qτ a ′ ¯ N implies Qτ a ¯ N ∼ Qτ a ′ ¯ N in h / Γ θ . Let τ := τ a ¯ N and τ ′ := τ a ′ ¯ N . An isomorphism f M : X Qτ ′ −→ X Qτ is given by T (˜ γQτ + ˜ δ )[ Q ′ τ, ] = [ Qτ, ] T (cid:18) ˜ α ˜ β ˜ γ ˜ δ (cid:19) . The analytic representation of this isomorphism, which we will also denote by f M , is f M = T (˜ γQτ + ˜ δ ) = ( γτ + δ ) · , where recall γ, δ ∈ Z . Therefore the mapEnd( X Qτ ′ ) −→ End( X Qτ ) A f M Af − M = A is the identity map, hence End( X Qτ ′ ) = End( X Qτ ). Moreover the equivalence T ¯ AH Qτ = H Qτ A ι ⇔ T ¯ AQ ι = Q ι A ι implies the relation on the left hand side is independent of τ . Hence R Qτ = R Qτ ′ . (cid:3) It follows immediately since R Qτ = R Qτ ′ that the maps Υ and Υ are independent of thechoice of representative a of [ a ].This completes the proofs of Theorems 3.2 and 3.3. (cid:3) ENTRAL VALUES OF HECKE L-SERIES 19
Recall the definitions of: the normalized theta values Θ [ a ,R ] , N in (3.6), the sign function ε [ a ,R ] on the embeddings in (3.8), and the twisted number of optimal embeddings h ε [ a ,R ] ( − N ) in (3.9).The η function in (3.6) is defined on an ideal a = [ a, − b + √ D ] of O K by(6.4) η ( a ) := e ( a ( b + 3)) · η (cid:0) − b + √ D a (cid:1) where e n ( x ) := exp(2 πix/n ) for n ∈ Z , x ∈ C , and η ( z ) := e ( z ) Q ∞ n =1 (1 − e πiz ) for Im( z ) > [ a ,R ] , N is an algebraic integer (see [Pac05, Proposition 23, p. 355] and [HV97]).We now prove Lemma 3.5. Proof of Lemma 3.5.
Theorem 31 of [Pac05] says that if Qτ a ¯ N ∼ Q ′ τ a ¯ N in h / Γ θ , thenΘ [ a ,Q ] , N = ± Θ [ a ,Q ′ ] , N . The lemma therefore follows immediately by this fact and Theorem 3.2. (cid:3)
We now prove Theorem 3.6.
Proof of Theorem 3.6.
The remaining step in deriving formula (3.10) for L ( ψ N ,
1) is to deter-mine how θ behaves on equivalent split-CM points. The following is a special case of [Pac05,Theorem 31] but we give a slightly simplified proof. Lemma 6.13.
Let Q and Q ′ be binary quadratic forms of discriminant − N . If Qτ ∼ Q ′ τ in h / Γ θ , then θ ( Qτ ) = ± θ ( Q ′ τ ) .Proof of Lemma. Suppose Qτ ∼ Q ′ τ in h / Γ θ . Then there exists M := (cid:0) α βγ δ (cid:1) ∈ Γ θ such that M ( Qτ ) = Q ′ τ . Recall the functional equation for θ is(6.5) θ ( M ◦ z ) = χ ( M )[det( γz + δ )] / θ ( z ) , M ∈ Γ θ where χ ( M ) is a certain 8th root of unity.Then θ ( Q ′ τ ) θ ( Qτ ) = χ ( M )[det( γQτ + δ )] / . Applying Smith Normal Form, there exists
U, V ∈ SL ( Z ) such that U QV = ( N ), and U ′ , V ′ ∈ SL ( Z ) such that U ′ Q ′ V ′ = ( N ). These give isomorphisms f U : X Qτ → E τ × E Nτ and f U ′ : X Q ′ τ → E τ × E Nτ respectively. From the relation M ( Qτ ) = Q ′ τ , we also get anisomorphism f M : X Q ′ τ → X Qτ given by T ( γQτ + δ )[ Q ′ τ, ] = [ Qτ, ] T (cid:18) α βγ δ (cid:19) . Thus the composition f U ◦ f M ◦ f − U ′ : E τ × E Nτ −→ E τ × E Nτ is an automorphism, and the determinant of its analytic representation is a unit and an algebraicinteger. This last fact follows from linear algebra or can be deduced directly using Lemma 4.4.Since U and U ′ are both in SL ( Z ), we get det( γQτ + δ ) ∈ O × K . Since D < − γQτ + δ ) = ±
1. Therefore [det( γQτ + δ )] / = ±√± θ ( Q ′ τ ) θ ( Qτ ) = ±√± · χ ( M ). But by Theorem 17 of [Pac05], the ratio of theta valueson the left is an algebraic integer in the Hilbert class field of K . Hence ±√± · χ ( M ) is an 8throot of unity and an algebraic integer in the Hilbert class field of K , which does not contain i .Therefore ±√± · χ ( M ) = ± . (cid:3) The theorem follows immediately from Lemma 6.13 and Theorems 3.2 and 3.3. (cid:3) Examples
This section provides tables for two class number one examples. All calculations were donein gp/PARI [PAR08]. Given D of class number one, for each admissable N we compute a form[ N, b , c ] corresponding to N . We set a N = N since Cl ( O K ) is trivial, and τ a N := τ N := − b + √ D N to be a Heegner point of level N and discriminant D . We choose [1 , − b + √ D ] for a basis of O K so that following definition (6.4), η ( N ) η ( O K ) := e ( N ( b + 3) ) · η (cid:0) − b + √ D N (cid:1) · η (cid:0) − b + √ D (cid:1) . From left to right, the columns of the table are N , the absolute values of the integers Θ [ R ] foreach [ R ] ∈ R , the number, denoted [ R ] , of classes [ Q ] ∈ Cl ( − N ) with value ± Θ [ R ] (thisequals h R ( − N ) by Theorem 3.3), and the values h ε [ a ,R ] ( − N ).For D = −
7, the type number is 1 and so [ R ] = h R ( − N ) = h ( − N ) gives the N -thcoefficient of the weight 3 / D form + ω R P N> H D ( N ) q N defined by the modifiedHurwitz invariants H D ( N ) (see [Gro84, p. 120] for their definition). N Θ [ R ] [ R ] h ε [ a ,R ] ( − N ) N Θ [ R ] [ R ] h ε [ a ,R ] ( − N )11 1 1 -1 107 1 3 -323 1 3 -1 127 1 5 143 1 1 1 151 1 7 -167 1 1 -1 163 1 1 171 1 7 -3 179 1 5 -379 1 5 -1 191 1 13 -5 Table 1. D = − N ≤ t = 1. ENTRAL VALUES OF HECKE L-SERIES 21 N Θ [ R ] [ R ] h ε [ a ,R ] ( − N ) N Θ [ R ] [ R ] h ε [ a ,R ] ( − N )23 0 2 2 103 0 3 32 1 1 2 2 231 0 2 2 163 0 1 12 1 -1 2 0 047 0 3 3 179 0 2 22 2 2 2 3 159 0 2 2 191 0 8 82 1 -1 2 5 167 0 0 0 199 0 5 52 1 -1 2 4 471 0 4 4 223 0 4 42 3 -3 2 3 3 Table 2. D = − N ≤ t = 2. Acknowledgments
I am deeply grateful to Fernando Rodriguez Villegas for his continuing guidance and supportand for sharing his ideas that have enriched this work. I would also like to thank John Voightand Ariel Pacetti for helpful discussions on this subject. Thanks to Jeffrey Stopple for his carefulreading of the manuscript. This research was partially funded by the Donald D. HarringtonEndowment Fellowship and a Wendell Gordon Endowed Fellowship at the University of Texasat Austin.
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