Construction of Anti-Cyclotomic Euler Systems of Abelian Varieties Associated to X 1 (N)
aa r X i v : . [ m a t h . N T ] O c t Construction of Anti-Cyclotomic Euler Systems of AbelianVarieties Associated to X ( N ) Daeyeol Jeon, Byoung Du Kim, and Chang Heon Kim
Abstract.
Let K be an imaginary quadratic field, N be a positiveinteger, f ( z ) be a newform of level Γ ( N ), and A f be the abelian varietyassociated to f . For each τ ∈ K (Im τ > P τ on A f defined over an extended ring class field of K of level N . Ourconstruction generalizes Birch’s construction of the Heegner points tothe abelian varieties associated to modular forms of level Γ ( N ) andnontrivial character. Then, we show that P τ ’s satisfy the distributionand congruence relations of an Euler system, which implies that it shouldbe possible to apply the Euler system techniques to them to show arelation between the non-torsionness of P τ and the rank of A f ( K ). Contents
1. Introduction 12. Preliminaries 42.1. Modular functions b ( τ ) and c ( τ ) 42.2. Field of definitions of b ( τ ) and c ( τ ) for a CM-points τ
63. The Main Theorem of Complex Multiplication, and the action ofthe Galois groups on P τ
74. Euler systems 104.1. Hecke operators 104.2. Distribution relations 114.3. Congruence Relations 205. Appendix 21References 231.
Introduction
In this paper, we present the construction of certain points on the modularcurve X ( N ) (and by extension, on the abelian varieties over Q given byirreducible quotients of J ( N )). We will argue that our points generalizeBirch’s Heegner points ([2]) (which are defined on X ( N )) in the sense that Mathematics Subject Classification.
Primary: 11G, Secondary: 14G05, 14G35.
Key words and phrases.
Euler systems, rational points of abelian varieties associatedto modular forms. (like Birch’s construction) modular functions and class field theory play anintegral role in the construction, and show that they satisfy the conditions(the distribution and congruence relations) of an Euler system.First, we give a brief description of Birch’s construction of the Heegnerpoints. As is customary, we let H denote the upper-half plane, let Y ( N )(resp. Y ( N )) denote Γ ( N ) \ H (resp. Γ ( N ) \ H ), and X ( N ) (resp. X ( N ))denote its compactification by the addition of cusps. Let j ( τ ) ( τ ∈ H ) denotethe modular elliptic function given by the j -invariant of Λ τ = (1 , τ ) = Z · Z · τ . It is well-known that ( j ( τ ) , j ( N τ )) satisfies a certain polynomialequation P N ( X, Y ) = 0, which gives an affine model over Q of Y ( N ). If K is an imaginary quadratic field, and τ ∈ K ∩ H , then j ( τ ) generates a certainring class field extension of K by class field theory. Birch noted that for acarefully chosen τ , ( j ( τ ) , j ( N τ )) is a point on the affine model Y ( N ) / Q overthe ring class field (thus can be considered as a point on X ( N ) / Q ), whichhe called a Heegner point ([2]). A Heegner point can be also considered asa point on an elliptic curve E over Q of conductor N , as we will explainshortly.On the other hand, the authors wanted to find an explicit way to constructa point on X ( N ) also defined over a certain ring class field extension of K .Noting the role played by j ( τ ) in Birch’s construction, we looked for modularfunctions that can play a similar role.In [1], Baaziz constructed modular functions b ( τ ) , c ( τ ) of level Γ ( N ) (Sec-tion 2), which are rational functions of the Weierstrass functions ℘ ( · ; Λ τ ) , ℘ ′ ( · ; Λ τ ),and generate the function field of X ( N ). Jeon, Kim, and Lee noted ([5])that ( b ( τ ) , c ( τ )) gives an affine model (over Q ) of Y ( N ). The modularfunctions b ( τ ) , c ( τ ) seemed ideal for our purpose.Our first goal is to define a point analogous to the Heegner points: Wedefine P τ = ( b ( τ ) , c ( τ )) ∈ Y ( N )( ⊂ X ( N )) for any τ ∈ K , Im τ > P τ is defined. For an order O of K , let L O ,N be the extended ring class field of level N associated to O (see Section 2.2). We show that if O acts on Λ τ , then P τ ∈ X ( N )( L O ,N )(Corollary 2.3).Thirdly, we show that P τ satisfies the distribution and congruence rela-tions of Euler systems.Let f ( z ) = P ∞ n =1 a n ( f ) q n ( a = 1) be a newform of level Γ ( N ) withcharacter ǫ (modulo N ), A f be the abelian variety given by the quotientof J ( N ) divided by the ideal of the Hecke algebra generated by T l − a l ( f )( l ∤ N ), U l − a l ( f ) ( l | N ), h l i − ǫ ( l ) (( l, N ) = 1) where l runs over all primes,and µ f : X ( N ) → J ( N ) → A f be a modular parametrization (where themap from X ( N ) to J ( N ) is given by P ( P ) − ( ∞ )). We let P τ alsodenote µ f ( P τ ) ∈ A f by abuse of notation.Suppose K = Q ( √ D ) for some square-free negative integer D . Fix τ K = √ D if D √ D + 12 if D ≡ c prime to N , let uler Systems of X ( N ) 3 τ ′ = a + τ K c for an integer a ∈ Z . Then, as mentioned above, P τ ′ is defined over L O c ,N where O c = Z + c O K .Suppose p is a prime number prime to N · disc( K/ Q ), and let a p ( f ) bethe p -th Fourier coefficient of the q -expansion of f .If ( p, c ) = 1, p ≡ N ), and p is inert over K/ Q , thenTr L O cp,N /L O c,N P τ ′ /p = a p ( f ) P τ ′ . (1)(See Theorem 4.2.)On the other hand, if p | c , thenTr L O cp,N /L O c,N P τ ′ /p = a p ( f ) P τ ′ − ǫ ( p ) P pτ ′ . (2)(See Theorem 4.4.)Now, suppose p is a prime that is inert over K/ Q and p ≡ N ),and ( p, c ) = 1. Let λ be any prime of L O c ,N lying above p , λ ′ be any primeof L O cp ,N lying above λ , and red λ and red λ ′ be reduction maps onto thespecial fiber of the Néron model (over Z p ) of A f . Then, we have( p + 1) red λ ′ P τ ′ /p = (Frob p + p · ǫ ( p ) · Frob − p ) red λ P τ ′ (3) = a p ( f ) · red λ P τ ′ (see Theorem 4.6).Now, let’s compare them with the conditions of Kolyvagin’s Euler systemof the Heegner points ([9] Sections 1, 3). Let C τ denote Birch’s Heegnerpoint ( j ( τ ) , j ( N τ )) ∈ X ( N ). Suppose f ( z ) is a newform of level Γ ( N )and A f is the abelian variety associated to f ( z ) as in the case of X ( N )(the most prominent case being an elliptic curve over Q ). Again, we fixa modular parametrization map µ f : X ( N ) → A f defined over Q whichsatisfies µ f ( ∞ ) = 0. By abuse of notation, we let C τ denote µ f ( C τ ) ∈ A f as well. By Kolyvagin ([7], [8]), for each n ∈ Z ( n >
0) we can choose anappropriate τ n ∈ K ∩ H so that C τ n is defined over the ring class field L O n of the order O n = Z + n O K , and for a prime l with ( l, N ) = 1, if l ∤ n and l is inert over K/ Q , we have the distribution relationTr L O nl /L O n C τ nl = a l ( f ) C τ n (4)([7] Proposition 1). Although, it does not appear in Kolyvagin’s work, wealso have that if l | n ,Tr L O nl /L O n C τ nl = a l ( f ) C τ n − C τ n/l (5) DAEYEOL JEON, BYOUNG DU KIM, AND CHANG HEON KIM (see the proof of [13] Proposition 6.1 although the readers should note thatRubin assumes a l ( f ) = 0).Also, where l does not divide disc( K/ Q ), v is any prime of K (1) above l ,and w is any prime of L O l above v , we have the congruence relationred w ( C τ l ) = Frob l (red v ( C τ ))(6)(see [7] Proposition 6, and [8] Proposition 1).(1) is clearly analogous to (4) with the extra condition p ≡ N ),which we believe will not make much difference in practice. (2) is alsoclearly analogous to (5). The appearance of ǫ ( p ) can be easily explainedby the fact that modular forms of level Γ ( N ) have a trivial character.(2) should be what Rubin calls the distribution relation in the p -direction([17] Remark 2.1.5), and (as Rubin points out) we believe that it can replacethe congruence relations in the Euler system techniques. Also it should benoted that (2) indicates a natural connection with Iwasawa Theory.The main goal of an Euler system is to obtain a sharp bound for the ranksof A f . For example, Kolyvagin showed that if f is a newform of level Γ ( N )and N K (1) /K C ( ∈ A f ( K )) is not torsion, then rank A f ( K ) = 1. We believethat we can apply the techniques of Euler systems to { P ( a + τ ) /c } , and obtaina similar result for A f ( K ) where f is a newform of level Γ ( N ) (and a moregeneral result in the direction of Iwasawa Theory), and we are hopeful thatsuch a result will be in our subsequent publication. Remark 1.1.
There are also Kato’s Euler systems ( [6] ) defined on J ( N ) .We note that his Euler systems are “the Euler systems over the cyclotomicfields” whereas our Euler system (as well as the Euler system of the Heeg-ner points) are “the Euler systems over anti-cyclotomic fields.” They aredifferent in the definition, construction, and application. Preliminaries
Modular functions b ( τ ) and c ( τ ) . Let Γ = SL ( Z ) be the full modular group, and for any N ≥
1, Γ( N ),Γ ( N ), and Γ ( N ) be the standard congruence groups. Let Y ( N ) / Q bethe affine curve over Q of the moduli schemes of the isomorphism classesof elliptic curves E with an N -torsion point. As well-known, Y ( N ) C is(isomorphic to) ( Y ( N ) / Q ⊗ C ) an .More explicitly, this isomorphism is given by the following: Let Λ τ =( τ,
1) be the lattice in C with basis τ and 1. Then, the above-mentionedisomorphism (of analytic curves between Y ( N ) C and ( Y ( N ) / Q ⊗ C ) an ) isgiven by τ (cid:18) C / Λ τ , N + Λ τ (cid:19) . uler Systems of X ( N ) 5 The
Tate normal form of an elliptic curve with point P = (0 ,
0) is asfollows: E = E ( b, c ) : Y + (1 − c ) XY − bY = X − bX , and this is nonsingular if and only if b = 0 . On the curve E ( b, c ) we havethe following by the chord-tangent method: P = (0 , , P = ( b, bc ) , P = ( c, b − c ) , P = b ( b − c ) c , − b ( b − c − c ) c ! , (7) 5 P = − bc ( b − c − c )( b − c ) , bc ( b − bc − c )( b − c ) ! , P = ( b − c )( b − bc − c )( b − c − c ) , c (2 b − bc − bc + c )( b − c ) ( b − c − c ) ! . In fact, the condition
N P = O in E ( b, c ) gives a defining equation for X ( N ). For example, 11 P = O implies 5 P = − P , so x P = x − P = x P , where x nP denotes the x -coordinate of the n -multiple nP of P . Eq. (7)implies that(8) − bc ( b − c − c )( b − c ) = ( b − c )( b − bc − c )( b − c − c ) . Without loss of generality, the cases b = c and b = c + c may be excluded.Then Eq. (8) becomes as follows: − b c − bc + 3 b c + 9 b c − bc − b c − b c + 3 b c − bc + c + b = 0 , which is one of the equation X (11) called the raw form of X (11). By thecoordinate changes b = (1 − x ) xy (1 + xy ) and c = (1 − x ) xy , we get thefollowing equation: f ( x, y ) := y + ( x + 1) y + x = 0 . Now we note that (cid:18) C / Λ τ , N + Λ τ (cid:19) = (cid:18) y = 4 x − g ( τ ) x − g ( τ ) , (cid:18) ℘ (cid:18) N ; Λ τ (cid:19) , ℘ ′ (cid:18) N ; Λ τ (cid:19)(cid:19)(cid:19) = (cid:16) y + (1 − c ( τ )) xy − b ( τ ) y = x − b ( τ ) x , (0 , (cid:17) , DAEYEOL JEON, BYOUNG DU KIM, AND CHANG HEON KIM where ℘ ( z ; Λ τ ) is the Weierstrass elliptic function of the period Λ τ . From[1], it follows that(9) b ( τ ) = − ( ℘ ( N ; Λ τ ) − ℘ ( N ; Λ τ )) ℘ ′ ( N ; Λ τ ) , c ( τ ) = − ℘ ′ ( N ; Λ τ ) ℘ ′ ( N ; Λ τ )are modular functions on Γ ( N ) and generate the function field of X ( N ),where the derivative ℘ ′ is with respect to z .2.2. Field of definitions of b ( τ ) and c ( τ ) for a CM-points τ . Let F N bethe extension of the function field Q ( j ( τ )) generated by the Fricke functionsindexed by r ∈ N Z / Z (see [10] Section 4), where j ( τ ) is the modularinvariant function. By the theory of modular functions, it is known that F N is the set of all functions in C ( X ( N )) whose Fourier coefficients are in Q ( ζ N ), F is simply Q ( j ( τ )), andGal( F N / F ) ∼ = GL ( Z /N Z ) / {± I } ∼ = G N · SL ( Z /N Z ) / {± I } where G N = (cid:26) (cid:20) d (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) d ∈ ( Z /N Z ) ∗ (cid:27) . The functions b ( τ ) , c ( τ ) have their Fourier coefficients in Q ( ζ N ), and theyare contained in F N . Definition 2.1. P τ = ( b ( τ ) , c ( τ )) ∈ X ( N ) . Let O be an order of conductor c in an imaginary quadratic field K . Thering class field of O , denoted by L O , is determined via the Existence Theoremof class field theory [4, Theorem 8.6] by the subgroup P K, Z ( c ) ⊂ I K ( c )generated by principal ideals α O K ∈ I K ( c ) where α ≡ a mod c O K forsome a ∈ Z . Here I K ( c ) denotes the group of all fractional ideals relativelyprime to c . This implies that Gal ( L O /K ) ∼ = I K ( c ) /P K, Z ( c ) ∼ = C ( O ) , where C ( O ) is the class group of O . Following [3] we define P K, Z ,N ( cN ) ⊂ I K ( cN )to be the subgroup generated by the principal ideals α O K ∈ I K ( cN ) where α ∈ O K satisfies α ≡ a mod cN O K for some a ∈ Z with a ≡ N .
It then follows from the Existence Theorem that there exists an extension L O ,N called the extended ring class field of level N , with Galois group Gal ( L O ,N /K ) ∼ = I K ( cN ) /P K, Z ,N ( cN ) . uler Systems of X ( N ) 7 We note that L O , = L O and L O ,N is a Galois extension of L O . In particular,if O = O K , then L O ,N is equal to the ray class field K ( N ).A point τ ∈ K ∩ H is a root of ax + bx + c where a, b, c ∈ Z are relativelyprime with a >
0. Then the lattice L τ = [1 , τ ] is a proper ideal for the order O = [1 , aτ ] (see [4, Theorem 7.7]). As a consequence of Shimura reciprocitywe have the following theorem. Theorem 2.2. [4, Theorem 15.16]
Fix τ ∈ K ∩ H and O as above andassume that f ( τ ) is well-defined for a modular function f ∈ F N . Then f ( τ ) ∈ L O ,N . Thus we have the following immediate corollary.
Corollary 2.3.
Fix τ ∈ K ∩ H and O as above and assume that b ( τ ) , c ( τ ) are defined. Then b ( τ ) , c ( τ ) ∈ L O ,N and therefore the point P τ is definedover L O ,N .Proof. This immediately follows from Theorem 2.2 because b ( τ ) , c ( τ ) ∈ F N . (cid:3) The Main Theorem of Complex Multiplication, and theaction of the Galois groups on P τ In this section, we apply Shimura’s theory of complex multiplication to P τ to study the action of the Galois groups of extended ring class fields, andin particular, we find the field of definition of P τ by other means.As before, the lattice ( α, α ′ ) denotes Z α + Z α ′ .The following is from [18] Section 5.2 and Section 5.3.Suppose Λ is an arbitrary Z -lattice in K . For each rational prime p , let K p = K ⊗ Q Q p and Λ p = Λ ⊗ Z Z p (so that A K = Q p K p ). It is worth notingthat if p splits completely over K/ Q (so that p O K = p ¯ p ), then K p = K p × K ¯ p .For any x ∈ A ∗ K , we may speak of the p -component x p of x belongingto K ∗ p . (In other words, if p is inert, x p ∈ K ∗ p O K , if p splits completely, x p = ( x p , x ¯ p ) ∈ K ∗ p × K ∗ ¯ p , and if p is ramified, x p ∈ K ∗ p for the unique prime p of O K above p .)We observe that x p Λ p is a Z p -lattice in K p . It is well-known that thereexists a Z -lattice Λ ′ in K such that Λ ′ p = x p Λ p for every p ([18] page 116).Then, we define x Λ def = Λ ′ . The isomorphism x : K/ Λ × x → K/x
Λ is given as follows: Since Q / Z = Q p Q p / Z p canonically, we have the canonical decomposition K/ Λ = Q p K p / Λ p .There is a well-defined isomorphism given by multiplication x p : K p / Λ p × x p → DAEYEOL JEON, BYOUNG DU KIM, AND CHANG HEON KIM K p /x p Λ p for each prime p . Combining them for all p , we obtain an iso-morphism x : K/ Λ → K/x
Λ. In other words, x : K/ Λ → K/x
Λ is an iso-morphism which makes the following diagram commutative for every prime p : K p / Λ p x p −→ K p /x p Λ p ↓ ↓ K/ Λ x −→ K/x Λ . The following is by Shimura, et. al.
Theorem 3.1 (Main Theorem of Complex Multiplication, [18] Chapter 5Theorem 5.4.) . Recall that K is an imaginary quadratic field. Let Λ ⊂ K bea lattice in K , σ be an automorphism of C invariant on K (in other words,a K -automorphism of C ), s be an element of A × K so that σ | K ab = [ s, K ] , and E be an elliptic curve so that there is an analytic isomorphism ξ : C / Λ → E .Then, there is an isomorphism ξ ′ : C /s − Λ → E σ so that the following iscommutative: K/ Λ ξ → E tors s − ↓ ↓ σK/s − Λ ξ ′ → E σtors . ( ξ ′ is uniquely determined by the above property once ξ is fixed.) Note that the precise definition of s − Λ is given above.As before, we let P τ = ( b ( τ ) , c ( τ )) for τ ∈ K ∩ H , and Λ τ = (1 , τ ).For any lattice Λ in K we have the standard invariants G n (Λ) = X ω ∈ Λ ,ω =0 ω − n , g (Λ) = 60 · G (Λ) , g (Λ) = 140 · G (Λ) . Suppose E τ is the elliptic curve given by the Weierstrass equation y = 4 x − g (Λ τ ) x − g (Λ τ )so that there is an (analytic) isomorphism ξ : C / Λ τ → E τ z ( ℘ ( z ; Λ τ ) , ℘ ′ ( z ; Λ τ )) . As in Theorem 3.1, σ is any automorphism of C invariant on K , and s ∈ A × K satisfies [ s, K ] = σ | K ab . By Theorem 3.1, there is an (analytic)isomorphism ξ ′ : C /s − Λ τ → E στ such that the diagram in Theorem 3.1commutes. As well-known, there is a lattice Λ ′ = ( ω ′ , ω ′ ) in K such that uler Systems of X ( N ) 9 g (Λ τ ) σ = g (Λ ′ ) , g (Λ τ ) σ = g (Λ ′ ) , and ξ ′′ : C / Λ ′ → E στ z ( ℘ ( z ; Λ ′ ) , ℘ ′ ( z ; Λ ′ ))is an analytic isomorphism. Then, the composite map C /s − Λ τ ξ ′ −→ E στ ξ ′′− −→ C / Λ ′ is an analytic isomorphism, which is given by C /s − Λ τ × λ −→ C / Λ ′ for some λ ∈ C ∗ (implying Λ ′ = λ · s − Λ τ ). In other words, ξ ′ ( z ) = ξ ′′ ( λ · z )for z ∈ C /s − Λ τ .Therefore, for any u ∈ K/ Λ τ , ℘ ( u ; Λ τ ) σ = ℘ ( λ · s − u ; λ · s − Λ τ ) , (10) (cid:0) ℘ ′ ( u ; Λ τ ) (cid:1) σ = ℘ ′ ( λ · s − u ; λ · s − Λ τ ) . (11)Suppose N · O K = Q ki =1 v n i i ( n i >
0) for some primes v , · · · , v k of O K .Suppose an order O c = Z + c O K acts on Λ τ for some c ∈ Z ( c >
0) with( c, N ) = 1.Suppose σ is identity on L O c . Since Gal( L O c /K ) ∼ = A ∗ K /K ∗ Q v O ∗ c,v , s ∈ A ∗ K satisfying [ s, K ] = σ | K ab should be indeed s ∈ K ∗ Q v O ∗ c,v .Write s = µ [ · · · , a v , · · · ] v where µ ∈ K ∗ and a v ∈ O ∗ c,v for each place v of K . By the Chinese remainder theorem, there is B ∈ O K so that B ≡ ( ca v i ) − (mod v n i i ) for every i = 1 , · · · , k . Let C = cB . Then, C ∈ c O K ⊂ O c , and naturally, C ≡ a − v i (mod v n i i ).Then, we have the following formula for the action of σ on P τ . Proposition 3.2.
For an L O c -automorphism σ of C , and C defined above,we have b ( τ ) σ = − (cid:18) ℘ ( C N ; Λ τ ) − ℘ ( C N ; Λ τ ) (cid:19) ℘ ′ ( C N ; Λ τ ) ,c ( τ ) σ = − ℘ ′ ( C N ; Λ τ ) ℘ ′ ( C N ; Λ τ )
20 DAEYEOL JEON, BYOUNG DU KIM, AND CHANG HEON KIM
Proof.
Since we assume a v ∈ O ∗ c,v for each place v , a − v Z l (1 , τ ) = Z l (1 , τ ).Therefore, [ · · · , a v , · · · ] − Λ τ = Λ τ , and s − Λ τ = µ − Λ τ .If l is a prime, and l ∤ N , then N ∈ Z l , thus N ≡ Z l (1 , τ ). Since a v ∈ O ∗ c,v for each v , Q v | l a − v N ∈ Λ τ ⊗ Z l , and since C ∈ O c , C N ∈ Λ τ ⊗ Z l .In other words, a − v N ≡ C N ≡ τ ⊗ Z l .If l | N and v | l for a prime l ( i.e. , v = v i for some i = 1 , · · · , k ), byconstruction C ≡ a − v i (mod v n i i ), thus C ≡ a − v i (mod N O K vi ). In otherwords, a − v i N − C N ∈ O K vi . Since O c ⊂ Λ τ and O c,v i = O K vi (because( c, N ) = 1), a − v i N − C N ∈ Λ τ ⊗ Z l . In other words, a − v i N ≡ C N moduloΛ τ ⊗ Z l .Combined we have C N = [ · · · , a v , · · · ] − N (mod Λ τ ).Then, by (10) and (11), ℘ ( 1 N ; Λ τ ) σ = ℘ ( λ · s − N ; λ · s − Λ τ ) , = ℘ ( λ · µ − C N ; λ · µ − Λ τ ) , (cid:18) ℘ ′ ( 1 N ; Λ τ ) (cid:19) σ = ℘ ′ ( λ · s − N ; λ · s − Λ τ )= ℘ ′ ( λ · µ − C N ; λ · µ − Λ τ ) . Thus, we have b ( τ ) σ = − (cid:18) ℘ ( λ · µ − C N ; λ · µ − Λ τ ) − ℘ ( λ · µ − C N ; λ · µ − Λ τ ) (cid:19) ℘ ′ ( λ · µ − C N ; λ · µ − Λ τ ) . By noting ℘ ( λµ − z ; λµ − Λ) = ( λµ − ) − ℘ ( z ; Λ) and ℘ ′ ( λµ − z, λµ − Λ) =( λµ − ) − ℘ ′ ( z ; Λ), we obtain our claim.The case for c ( τ ) σ is similar. (cid:3) Furthermore, suppose σ is identity on L O c ,N (assuming ( c, N ) = 1).Since Gal( L O c ,N /K ) ∼ = A ∗ K /K ∗ Q v ∤ N O ∗ c,v Q ki =1 v n i i , we can choose s = µ [ · · · , a v , · · · ] v so that a v i ≡ v n i i ) for i = 1 , · · · , k , thus we canchoose C which satisfies C ≡ N ). Thus by Proposition 3.2, we have b ( τ ) σ = b ( τ ) , c ( τ ) σ = c ( τ ) . Therefore, P τ is defined over L O c ,N .4. Euler systems
Hecke operators. uler Systems of X ( N ) 11 Recall that we let ( α, α ′ ) denote Z α + Z α ′ . Also, we will let Z p ( α, α ′ )denote Z p · α + Z p · α ′ . We recall that a point on X ( N ) / Q ⊗ C is given by( E, P ) where E is a generalized elliptic curve over C , and P ∈ E [ N ]. Forsimplicity, let ( E, P ) also denote the divisor (
E, P ) ∈ DivX ( N ). First, wenote that for a prime p ( p ∤ N ), T p acts on DivX ( N ) by T p : ( E, P ) X C ( E/C, P ′ )where C runs over all cyclic subgroups of E [ p ] of order p (there are p + 1such subgroups), and P ′ is the image of P under E → E/C (see [19] page 5).In particular, when (
E, P ) = ( C / Λ τ , /N ), T p ( E, P ) = p − X j =0 ( C / Λ τ + jp , /N ) + C / (1 /p, τ ) , /N ) . As in Section 3, write N = k Y i =1 v n i i for prime ideals v i of O K and integers n i >
0. We recall from Section 2.2 thatwhen O is an order of K of conductor c (i.e., O = Z + c O K ) with ( c, N ) = 1, L O ,N is the extended ring class field of level N which is characterized byGal( L O ,N /K ) ∼ = I K ( cN ) /P K, Z ,N ( cN ) ∼ = A ∗ K /K ∗ Y v ∤ N O ∗ v Y i (1 + v n i i ) . Let f ( z ) = P a n q n be a (standardized) newform of level Γ ( N ) andcharacter ǫ (mod N ), and µ f : X ( N ) → J ( N ) → A f be a modularparametrization map where the first map is given by P ( P ) − ( ∞ ) forany P ∈ X ( N ), and A f is the quotient of J ( N ) given in the standard wayassociated to f . Notation 4.1.
Suppose P τ ∈ X ( N )( L ) for some extension L of K . We let P τ also denote the image µ f ( P τ ) ∈ A f ( L ) . Distribution relations.
Recall that K = Q ( √ D ) where D is a square-free negative integer. Let τ = √ D if D √ D + 12 if D ≡ c is a positive integer prime to N , and let τ ′ = a + τc where a, c ∈ Z . We note that O c = Z + c O K acts on (1 , τ ′ ). As shown inCorollary 2.3 (and also in Section 3), P τ ′ ∈ L O c ,N .4.2.1. Proof when p is prime to the conductor. Theorem 4.2.
Suppose p is a prime number, ( c, pN ) = 1 , p ≡ N ) ,and p is unramified and inert over K/ Q . Then, Tr L O cp,N /L O c,N P τ ′ /p = a p ( f ) P τ ′ . ( P τ ′ /p can be replaced by P ( τ ′ + j ) /p for any j ∈ Z , or P pτ ′ ).Proof. Recall T p (cid:18) C / Λ τ ′ , N (cid:19) = p − X j =0 (cid:18) C / Λ τ ′ + jp , N (cid:19) + (cid:18) C / ( 1 p , τ ′ ) , N (cid:19) = p − X j =0 (cid:18) C / Λ τ ′ + jp , N (cid:19) + (cid:18) C / (1 , pτ ′ ) , pN (cid:19) = p − X j =0 (cid:18) C / Λ τ ′ + jp , N (cid:19) + (cid:18) C / (1 , pτ ′ ) , N (cid:19) (the last equality because p ≡ N )).For any prime l ( = p ) and integer j , Z l (1 , τ ′ + jp ) = Z l (1 , τ ′ + j ) = Z l (1 , τ ′ ) = Z l ( 1 p , τ ′ ) . On the other hand, Z p (1 , τ ′ + jp ) = Z p (1 , a + cj + τcp )= ( a + cj + τ ) · Z p ( 1 a + cj + τ , cp )= ( a + cj + τ ) · Z p ( a + cj + ¯ τN K/ Q ( a + cj + τ ) , p ) . Case 1. D a + cj + ¯ τN K/ Q ( a + cj + τ ) = a + cj − τ ( a + cj ) − D . uler Systems of X ( N ) 13 Since p is unramified and inert over K/ Q , ( a + cj ) − D p ) forany a + cj . Thus, Z p ( a + cj + ¯ τN K/ Q ( a + cj + τ ) , p ) = Z p ( a + cj − τ, p )= Z p ( τ, p ) . Case 2. D ≡ a + cj + ¯ τN K/ Q ( a + cj + τ ) = ( a + cj ) + ( − τ + 1)( a + cj ) + ( a + cj ) + 1 − D . For a reason similar to the above, ( a + cj ) + ( a + cj ) + 1 − D p )for any a + cj . Thus, Z p ( a + cj + ¯ τN K/ Q ( a + cj + τ ) , p ) = Z p (( a + cj ) + ( − τ + 1) , p )= Z p ( τ, p ) . In either case, Z p (1 , τ ′ + jp ) = ( a + cj + τ ) · Z p ( a + cj + ¯ τN K/ Q ( a + cj + τ ) , p )= ( a + cj + τ ) · Z p ( τ, p ) , and since ( c, p ) = 1, Z p ( τ, p ) = Z p ( a + τc , p ) = Z p ( τ ′ , p ) . For each j ∈ Z , let s j = (1 , , · · · , a + cj + τ, , · · · ) ∈ A ∗ K where a + cj + τ ∈ O ∗ K p is the p -component. We note s j ∈ Q v ∤ N O ∗ c,v Q i v n i i (because all components of s j are 1 except for the p -component, and O c,p = O K p and a + cj + τ ∈ O ∗ K p ). So far, we have shown(1 , τ ′ + jp ) = s j · ( 1 p , τ ′ ) . Lemma 4.3. (a) s j N = 1 N where N on the left is considered an elementof K/ ( p , τ ′ ) , and the one on the right an element of K/ Λ τ ′ + jp .(b) { a + cj + τ } j =0 , , ··· ,p − ∪ { } = O ∗ c,p / O ∗ cp,p ( ∼ = O ∗ K p / ( Z p + p O K p ) ∗ ) .(c) Consequently, { [ s j , K ] | L O cp,N } ∪ { id } = Gal( L O cp ,N /L O c ,N ) .Proof. (a) Simply because for every v | N , v -entry of s j is 1, and for every l ∤ N , N ∈ Z l .(b) By brute force, show the elements of the lefthand side are all distinctmodulo ( Z p + p O K,p ) ∗ . Since the righthand side has p + 1 elements, weobtain our claim.(c) Because Gal( L O cp ,N /L O c ,N ) ∼ = O ∗ c,p / O ∗ cp,p . (cid:3) Now, choose σ j ∈ Aut( C ) so that σ j | K ab = [ s j , K ] for j = 0 , , · · · , p − P τ ′ = ( b ( τ ′ ) , c ( τ ′ ))where b ( τ ′ ) = − ( ℘ ( 1 N ; Λ τ ′ ) − ℘ ( 2 N ; Λ τ ′ )) ℘ ′ ( 1 N ; Λ τ ′ ) ,c ( τ ′ ) = − ℘ ′ ( 2 N ; Λ τ ′ ) ℘ ′ ( 1 N ; Λ τ ′ ) . As noted earlier, T p P τ ′ = p − X j =0 P τ ′ + jp + P pτ ′ as divisors on X ( N ).Suppose E pτ ′ : y = 4 x − g x − g with g = g (( 1 p , τ ′ )) , g = g (( 1 p , τ ′ )),and ϕ be the following analytic isomorphism: C / ( 1 p , τ ′ ) → E pτ ′ z (cid:18) ℘ ( z ; ( 1 p , τ ′ )) , ℘ ′ ( z ; ( 1 p , τ ′ )) (cid:19) . uler Systems of X ( N ) 15 By Theorem 3.1, there is an analytic isomorphism ψ : C /s j · ( 1 p , τ ′ ) → E σ − j pτ ′ so that the following diagram is commutative: K/ ( 1 p , τ ′ ) ϕ −→ ( E pτ ′ ) tor × s j ↓ ↓ σ − j K/s j · ( 1 p , τ ′ ) ψ −→ ( E σ − j pτ ′ ) tor Similar to the argument in Section 3, there is some x ∈ C ∗ so that ψ isthe composite map ψ : C /s j · ( 1 p , τ ′ ) × x −→ C /x · s j · ( 1 p , τ ′ ) ( ℘,℘ ′ ) −→ E σ − j pτ ′ . Recall s j · ( 1 p , τ ′ ) = Λ τ ′ + jp (see the discussion before Lemma 4.3), and s j · N = 1 N (Lemma 4.3 (a)). By substituting these into b ( · ) and c ( · ), wehave b ( τ ′ + jp ) = − ( ℘ ( s j · N ; s j · ( 1 p , τ ′ )) − ℘ ( s j · N ; s j · ( 1 p , τ ′ ))) ℘ ′ ( s j · N ; s j · ( 1 p , τ ′ )) which is equal to − ( ℘ ( x · s j · N ; x · s j · ( 1 p , τ ′ )) − ℘ ( x · s j · N ; x · s j · ( 1 p , τ ′ ))) ℘ ′ ( x · s j · N ; x · s j · ( 1 p , τ ′ )) because ℘ ( xz ; x Λ) = x − ℘ ( z ; Λ), ℘ ′ ( xz ; x Λ) = x − ℘ ′ ( z ; Λ). By the abovecommutative diagram, this is equal to − ( ℘ ( 1 N ; ( 1 p , τ ′ )) − ℘ ( 2 N ; ( 1 p , τ ′ ))) ℘ ′ ( 1 N ; ( 1 p , τ ′ )) σ − j = − ( ℘ ( pN ; Λ pτ ′ ) − ℘ ( 2 pN ; Λ pτ ′ )) ℘ ′ ( pN ; Λ pτ ′ ) σ − j = b ( pτ ′ ) σ − j . (The last equality is because p ≡ N ).)Similarly, c ( τ ′ + jp ) = c ( pτ ′ ) σ − j .Thus, P σ − j pτ ′ = P τ ′ + jp , which shows T p ( P τ ′ ) = p − X j =0 ( P pτ ′ ) σ − j + ( P pτ ′ ) = Tr L O cp,N /L O c,N ( P pτ ′ )(12)(as divisors) by Lemma 4.3 and by the assumption that σ j | L O cp,N = [ s j , L O cp ,N /K ].Since T p acts as multiplication by a p ( f ) on the abelian variety A f , and sinceGal( L O cp ,N /L O c ,N ) acts transitively on (cid:26) P τ ′ + jp (cid:27) ∪ { P pτ ′ } , we obtain Theo-rem 4.2. (cid:3) Proof when p divides the conductor. Theorem 4.4.
Suppose p is a prime number such that p | c and p ∤ N disc( K/ Q ) .Then, Tr L O cp,N /L O c,N P τ ′ /p = a p ( f ) P τ ′ − ǫ ( p ) P pτ ′ . ( P τ ′ /p can be replaced by P ( τ ′ + j ) /p for any j ∈ Z ) where ǫ is the (Nebentypus)character of f .Proof. Similar to Theorem 4.2, we have the following equality of divisors: T p (cid:18) C / Λ τ ′ , N (cid:19) = p − X j =0 (cid:18) C / Λ τ ′ + jp , N (cid:19) + (cid:18) C / (1 , pτ ′ ) , pN (cid:19) = p − X j =0 (cid:18) C / Λ τ ′ + jp , N (cid:19) + h p i (cid:18) C / (1 , pτ ′ ) , N (cid:19) where h·i is the diamond operator (for the action of h·i , see [19] Section 2).Similar to Theorem 4.2, for any prime l ( = p ) and integer j , Z l (1 , τ ′ + jp ) = Z l (1 , τ ′ ) = Z l (1 , τ ′ p ) . When l = p , we note Z p (1 , τ ′ + jp ) = Z p (1 , a + cj + τcp ) . Now, we have uler Systems of X ( N ) 17 τ ′ + jpτ ′ p = ( a + cj ) + τa + τ = [( a + cj ) + τ ][ a + ¯ τ ] N K/ Q ( a + τ ) . We consider τ ′ + jpτ ′ p to be an element of K ⊗ Q p (and in an appropriatecontext, the p -component of the adele A ∗ K ). In other words, if p is inert, it isan element of K p , and if p splits completely so that p O K = p ¯ p , an elementof K p × K ¯ p . Similarly, Z p (1 , τ ′ + jp ) is considered a lattice inside K ⊗ Q p . Case 1. D τ ′ + jpτ ′ p = [( a + cj ) + τ ][ a + ¯ τ ] N K/ Q ( a + τ )= ( a − D + acj ) − cjτa − D = 1 + aca − D j − ca − D jτ = 1 + aca − D j − ca − D j ( cτ ′ − a )= 1 + 2 aca − D j − c a − D jτ ′ Noting a − D p ), it follows that τ ′ + jpτ ′ p Z p (1 , τ ′ p ) = Z p (1 + 2 aca − D j − c a − D jτ ′ , τ ′ + jp )= Z p (1 + 2 aca − D j − c a − D jτ ′ + ( c pa − D j ) · ( τ ′ + jp ) , τ ′ + jp )= Z p (1 + 2 aca − D j + c j a − D , τ ′ + jp )= Z p (1 , τ ′ + jp )(the last line because p | c and p ∤ a − D ).Noting τ ′ + jτ ′ = 1 + aca − D j − ca − D jτ as shown above, what we haveshown is equivalent to (1 + aca − D j − ca − D jτ ) Z p (1 , τ ′ p ) = Z p (1 , τ ′ + jp ) . Case 2. D ≡ N K/ Q ( a + τ ) = a + a + 1 − D p ) . [( a + cj ) + τ ][ a + ¯ τ ] = a ( a + cj ) + 1 − D aτ + ( a + cj )¯ τ = a + acj + 1 − D aτ + ( a + cj )(1 − τ )= ( a + acj + 1 − D a + cj ) − cjτ So, similar to Case 1, τ ′ + jpτ ′ p = ( a + acj + − D + a + cj ) − cjτa + a + − D = 1 + acj + cja + a + − D − cja + a + − D τ = 1 + acj + cja + a + − D − cja + a + − D ( cτ ′ − a )= 1 + 2 acj + cja + a + − D − c ja + a + − D τ ′ , thus τ ′ + jτ ′ · Z p (1 , τ ′ p ) = Z p (1 + 2 acj + cja + a + − D − c ja + a + − D τ ′ , τ ′ + jp )= Z p (1 + 2 acj + cj + c j a + a + − D , τ ′ + jp )= Z p (1 , τ ′ + jp )(the last line because 2 acj + cj + c j a + a + 1 − D ≡ p )).Noting τ ′ + jτ ′ = 1+ acj + cja + a + − D − cja + a + − D τ as shown above, whatwe have shown is equivalent to uler Systems of X ( N ) 19 acj + cja + a + − D − cja + a + − D τ ! · Z p (1 , τ ′ p ) = Z p (1 , τ ′ + jp )for each j ∈ Z .Let s j = (1 , , · · · , τ ′ + jτ ′ , , · · · ) ∈ A ∗ K for j = 0 , , · · · , p − τ ′ + jτ ′ is the p -component. We have shown that in both Case 1 and Case 2, s j Λ τ ′ p = Λ τ ′ + jp . We note that in both Case 1 and Case 2, τ ′ + jτ ′ ∈ c ( O K ⊗ Z p ) ⊂ ( O c ⊗ Z p ) ∗ , which implies s j ∈ Q v ∤ N O ∗ c,v Q i v n i i . Lemma 4.5. (a) s j N = 1 N where N on the left is in K/ Λ τ ′ p , and the oneon the right in K/ Λ τ ′ + jp .(b) { τ ′ + jτ ′ } j =0 , , ··· ,p − ∼ = (( O c ) ⊗ Z p ) ∗ / (( O cp ) ⊗ Z p ) ∗ .(c) Consequently, { [ s j , K ] | L O cp,N } j =0 , , ··· ,p − = Gal( L O cp ,N /L O c ,N ) .Proof. Similar to Lemma 4.3 except for (b). As for (b), first suppose p n k c .We show that for any A, B ∈ Z p (( B, p ) = 1), 1 + Ap n j + Bp n jτ for j =0 , , · · · , p − (cid:16) O p n +1 ⊗ Z p (cid:17) ∗ , and thus { Ap n j + Bp n jτ } j =0 , , ··· ,p − = ( O p n ⊗ Z p ) ∗ / ( O p n +1 ⊗ Z p ) ∗ by noting that both sideshave the same number of elements. (cid:3) Now, choose σ j ∈ Aut( C ) so that σ j | K ab = [ s j , K ] for j = 0 , , · · · , p − b ( τ ′ + jp ) = b ( τ ′ p ) σ − j c ( τ ′ + jp ) = c ( τ ′ p ) σ − j Thus, P σ − jτ ′ p = P τ ′ + jp , which shows T p ( P τ ′ ) = p − X j =0 ( P τ ′ p ) σ − j + h p i ( P pτ ′ ) = Tr L O cp,N /L O c,N ( P τ ′ p ) + h p i ( P pτ ′ ) . Since T p acts as multiplication by a p ( f ) on the abelian variety A f , h p i as mul-tiplication by ǫ ( p ), and Gal( L O cp ,N /L O c ,N ) acts transitively on { P τ ′ + jp } j =0 , ··· ,p − ,we obtain Theorem 4.4. (cid:3) Congruence Relations.
Where L is a local field of residue characteristic p which is prime to N , ν is the maximal ideal of O L , F is O L /ν , and A is the Néron model of A f over Z p (which exists because ( p, N ) = 1), we note that there is the standardreduction map red ν : A f ( L ) → A / F p ( F ). Also let J / F p denote the fiber ofthe Néron model of J ( N ) / Q over Z p (which exists because ( p, N ) = 1).Recall that τ = √ D if D √ D +1) / D ≡ τ ′ = a + τc with ( c, N ) = 1. Theorem 4.6.
Suppose p is a prime that is inert over K/ Q , p ≡ N ) ,and ( p, c ) = 1 . Let λ be any prime of L O c ,N lying above p , and λ ′ be anyprime of L O cp ,N lying above λ . Then, we have ( p + 1) red λ ′ P τ ′ /p = (Frob p + p · ǫ ( p ) · Frob − p ) red λ P τ ′ = a p ( f ) red λ P τ ′ . Remark 4.7.
As we will see in the proof, P τ ′ /p can be replaced by P ( τ ′ + j ) /p for any j ∈ Z , or P pτ ′ .We state Theorem 4.6 with Frob p + p · ǫ ( p ) · Frob − p to make it appearsimilar to Kolyvagin’s congruence relation ((6) in Section 1), but in practice,the statement with a p ( f ) red λ P τ ′ might be more suitable.Proof. As we see in the proof of Theorem 4.2 (see (12)), T p ( P τ ′ ) = Tr L O cp,N /L O c,N ( P pτ ′ ) . As noted in Theorem 4.2, P pτ ′ can be replaced by P ( τ ′ + j ) /p for any j ∈ Z because they are transitive under the action of Gal( L O cp ,N /L O c ,N ). To beconsistent with the notation in the theorem, we replace P pτ ′ with P τ ′ /p .Our condition implies that ( p ) splits completely over L O c ,N /K , and λ ′ is totally ramified over L O cp ,N /L O c ,N (therefore O L O cp,N /λ ′ ∼ = O L O c,N /λ ∼ = O K / ( p )). Thus, every σ ∈ Gal( L O cp ,N /L O c ,N ) is the identity modulo λ ′ .Hence, red λ T p ( P τ ′ ) = ( p + 1) red λ ′ P τ ′ /p . On the other hand, by the Eichler-Shimura relation, T p = Frob p + h p i Ver p uler Systems of X ( N ) 21 as action on J / F p . Here Frob p is the Frobenius and Ver p is the Verschiebungof the group scheme J / F p ([19] Section 4, (4.1)). We recall that h p i acts asmultiplication by ǫ ( p ) as an action on A / F p .By recalling that Ver p is characterized by Frob p Ver p = Ver p Frob p = p ,we obtain ( p + 1) red λ ′ P τ ′ /p = (Frob p + p · ǫ ( p ) · Frob − p ) red λ P τ ′ . (We may consider Frob p as an automorphism of finite fields.) Thus, weobtain the first part of our equation.Then, note the standard result Frob p + p · ǫ ( p ) · Frob − p = a p ( f ) on A / F p .(Or, we can simply argue that T p acts as multiplication by a p as an actionon A / F p .) Thus, we also obtain the second part of the equation. (cid:3) In his original work, Kolyvagin does not make any use of the seconddistribution relation (5) (see Section 1), which in fact does not seem toappear in his work. This relation is what Rubin calls the distribution relationin the p -direction. Instead, he uses congruence relations.Rubin points out ([16], and [17] Remark 2.1.5, and also Section 4.8) thatthe congruence relations are often unnecessary, or can be derived from otherdistribution relations if “the Euler system satisfies distribution relations inthe p -direction” (such as (2) in Section 1, which is Theorem 4.4).On the other hand, suppose one insists on using a congruence relation.Theorem 4.6 has a more restrictive condition that p is inert over K/ Q and p ≡ N ) (whereas Kolyvagin’s congruence condition applies to allprimes prime to the discriminant of K/ Q ). However, a careful reading of[14] Section 4 (which is another work that uses the congruence conditions)indicates that one does not need a congruence relation for every prime, andin fact, it is probably enough that it holds for inert primes. Adding thecondition p ≡ { P a + τc } satisfies enough relations to be anEuler system. 5. Appendix
Our description of the action of the Galois groups on P τ in Section 3 isnot necessarily efficient for practical computation. In this section, we applyKoo, Shin, and Yoon’s work ([10] Section 4, [11] Section 2) on the action ofthe Galois groups of class field extensions on the singular values of modularfunctions to obtain formulas which may be more readily useful in practice.Following [10], when K is an imaginary quadratic field of discriminant d K , let θ = (cid:26) √ d K / d K ≡ − √ d K ) / d K ≡ . We let F N be the extension of the function field Q ( j ( τ )) generated by theFricke functions indexed by r ∈ N Z / Z (see [10] Section 4). By the theoryof modular functions, it is known that F N is the set of all functions in C ( X ( N )) whose Fourier coefficients are in Q ( ζ N ), F is simply Q ( j ( τ )), andGal( F N / F ) ∼ = GL ( Z /N Z ) / {± I } ∼ = G N · SL ( Z /N Z ) / {± I } where G N = (cid:26) (cid:20) d (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) d ∈ ( Z /N Z ) ∗ (cid:27) . We note b ( τ ) , c ( τ ) ∈ F N ([5]).The action of Gal( F N / F ) on F N can be made explicit through GL ( Z /N Z ) / {± I } ∼ = G N · SL ( Z /N Z ) / {± I } as follows ([11] Section 2): (cid:20) d (cid:21) ∈ G N acts on F N by X n>> −∞ c n q nN τ X n>> −∞ c σ d n q nN τ where σ d ∈ Gal( Q ( ζ N ) / Q ) is given by ζ σ d N = ζ dN .And, γ ∈ SL ( Z /N Z ) / {± I } acts on h ∈ F N by h γ ( τ ) = h (˜ γτ )where ˜ γ ∈ SL ( Z ) is a lifting of γ .Following the notation of [10], we let H and K ( N ) be the Hilbert classfield, and the ray class field modulo N O K of K respectively. (Note that K ( N ) is L O K ,N in our earlier notation.) The action of Gal( K ( N ) /H ) is givenexplicitly through the singular values of F N as follows ([10] Section 2): Letmin( θ, Q ) = x + B θ x + C θ ∈ Z [ x ], and let W N,θ = (cid:26)(cid:20) t − B θ s − C θ ss t (cid:21) ∈ GL ( Z /N Z ) : t, s ∈ Z /N Z (cid:27) . By [10] Proposition 4.1, we can identify K ( N ) with the field K ( h ( θ ) | h ∈ F N is defined and finite at θ ) . Koo, Shin, and Yoon identify Gal( K ( N ) /H ) with the image of W N,θ by thefollowing surjection:
Proposition 5.1 ([10] Proposition 4.2) . W N,θ → Gal( K ( N ) /H )) α ( h ( θ ) h α ( θ ))If d K ≤ −
7, then its kernel is {± I } . By applying this to b ( τ ) and c ( τ ),we can compute the action of Gal( K ( N ) /H ) on P τ . (The computationaladvantage is that we only need to know W N,θ .) uler Systems of X ( N ) 23 The action of Gal(
H/K ) (and by extension, the action of Gal( K ( N ) /K ))is given as follows ([10] Proposition 4.3). Let C ( d K ) the form class groupof discriminant d K , which is the set of primitive positive definite quadraticforms aX + bXY + cY ∈ Z [ X, Y ] under a proper equivalence relation. (See[10] p.351 for an explicit classification of the elements of C ( d K ).) We notethat C ( d K ) ∼ = Gal( H/K ) ([4] Theorem 7.7).For a reduced quadratic form Q = aX + bXY + cY of disc Q = d K , weset θ Q = ( − b + p d K ) / a, and set β Q = ( β p ) p ∈ Q p GL ( Z p ) (where p runs over all primes) for β p givenin [10] p.351-352. Proposition 5.2 ([10] Proposition 4.3) . Assume d K ≤ − and N ≥ .(Note that by this assumption, W N,θ / {± I } ∼ = Gal( K ( N ) /H ) .) Then, wehave a bijective map W N,θ / {± I } × C ( d K ) → Gal( K ( N ) /K )( α, Q ) (cid:16) h ( θ ) h αβ Q ( θ Q ) (cid:17) As [10] notes, there is β ∈ GL +2 ( Q ) ∩ M ( Z ) with β ≡ β p (mod N Z p ) forall primes p dividing N , and the action of β Q is understood to be that of β .The bijection in Proposition 5.2 is not a group isomorphism (but the mapin Proposition 5.1 is a group homomorphism), but from the computationalperspective, it should not matter.This bijection gives a computational description of the action of Gal( K ( N ) /K )on our functions b ( τ ) , c ( τ ) when τ = θ . [10] does not explicitly say muchwhen τ is not θ , but its authors told us that their result can be generalizedto any τ ∈ K . In such a case, the action will be that of Gal( L O n ,N /K ) where O n is the order of K acting on Λ τ , which is compatible with our work. References [1] H. Baaziz,
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