Some explicit constructions of integral structures in quaternion algebras
aa r X i v : . [ m a t h . N T ] J a n Some explicit constructions of integralstructures in quaternion algebras
M. Ciavarella, L. TerraciniNovember 11, 2018
Abstract
Let B be an undefined quaternion algebra over Q . Followingthe explicit chacterization of some Eichler orders in B given byHashimoto, we define explicit embeddings of these orders in somelocal rings of matrices; we describe the two natural inclusions of anEichler order of leven N q in an Eichler order of level N . Moreoverwe provide a basis for a chain of Eichler orders in B and prove resultsabout their intersection. AMS Mathematics Subject Classification: 11R52
The aim of this work is to give an explicit description of the quaternionalgebras over Q and of some of their Eichler orders. Let B be a quaternionalgebra over Q of discriminant ∆ and let B q = B ⊗ Q Q q be its localizationat the prime number q . It is well known that if q is a unramified place, thenthere is an isomorphism between B q and M ( Q q ); if B is ramified at q then B q can be represented as a subalgebra of M ( Q q ), where Q q denotes thequadratic unramified extension of Q q , as described in [4]. In the generalliterature on quaternion algebras Eichler orders are defined by using theselocal isomorphisms. In [3] an explicit definition of an Eichler orders R ( N )of level N is given. The author fixes a representation of the quaternionalgebra B as a pair {− ∆ N, p } and gives a basis of the Eichler order R ( N )depending on this representation. This construction provides a very usefultool for working with Eichler orders. However, for our purposes, it hasthe limitation of not respecting the natural inclusion of an Eichler order oflevel M in an Eichler order of level N for N dividing M . Starting from the1ork of Hashimoto, we then provide an explicit description of Eichler orders R ( N ) and R ( N q ), and of the two natural inclusion maps R ( N q ) → R ( N ).More precisely, for any prime number q , we will describe an isomor-phisms ϕ q between B q and the corresponding matrix algebra and we willwrite the image of R q ( N ) under ϕ q . We characterize two copies of R ( N q )in R ( N ) by using these local isomorphisms, and we define a basis for eachof them in terms of a basis of R ( N ).As in [7] we will consider the quaternionic analogue of the congruence groupsΦ( N ); we will express them by using our characterization of Eichler ordesand we will prove some initial results for these groups.Our interest in Eichler orders and groups Φ( N ), arises from a difficultyencountered in some previous work on Galois representations and Heckealgebras arising from quaternionic groups [7], [1]: an analogue for Shimuracurves of Ihara’s lemma (which holds for modular curves) is missing. Webriefly give a sketch of this open problem; for a deep overview of the statusof art see [2].For any integer number N , Φ( N ) is defined as ( GL +2 ( R ) × ( R ( N ) ⊗ ˆ Z ) × ) ∩ B × . Let we consider the Shimura curves X ( N ) and X ( N q ) comingfrom Φ( N ) and Φ( N q ) respectively, where q is a prime number such that q ∆. There are two injective maps from Φ( N q ) in Φ( N ): the naturalinclusion and the coniugation by a certain element δ q ∈ B × . These mapsnaturally induce degeneracy maps on cohomology; their direct sum providesa map α : H ( X ( N )) → H ( X ( N q )) where cohomology has coefficients inthe ring of integers of a suitable finite extension of Q ℓ for a fixed prime ℓ .The conjecture in [2] asserts that α is injective with cokernel torsion free. Let B be an indefinite quaternion algebra over Q of dicriminant ∆ = p ...p t with t a even number. We will denote by (cid:0) ∗∗ (cid:1) the Legendre symbol and by( ∗ , ∗ ) q the Hilbert symbol at q [6]. Let N be a positive integer prime to ∆and p be a prime number such that: • p ≡ p ≡ (cid:26) | ∆1 mod 8 if 2 | N ; • (cid:16) pp i (cid:17) = − p i = 2; • (cid:16) pq (cid:17) = 1 for each odd prime factor q of N .2e observe that the last condition implies that p is a square in Z q for any q prime factor of N ; since p is not a square in Z p , then p does not divide N .Hashimoto [3] shows that then B ≃ {− ∆ N, p } (with the notations of [8]).This means that B can be expressed as B ( N, p ) = Q + Q i + Q j + Q k where i = − ∆ N , j = p , k = ij = − ji . Moreover by Theorem 2.2of [3], an Eichler order of level N of B can be expressed as the Z -lattice R ( N ) = Z e + Z e + Z e + Z e with e = 1 , e = 1 + j , e = i + k , e = a ∆ N j + kp where a ∈ Z satisfies a ∆ N + 1 ≡ p .We observe that i, j, k depend on the choice of N and p ; in the sequel,whenever will be necessary to express the dependece on N we will write i N , j N , k N instead of i, j, k and e N , e N , e N , e N instead of e , e , e , e .We consider R = R (1); then R is a maximal order in B . For any primenumber q , let we denote B q = B ⊗ Q Q q and R q ( N ) = R ( N ) ⊗ Z Z q .We start with a simple lemma which will be useful in the sequel. Lemma 2.1
Let K be a field and let B , B be two quaternion algebrasover K . If there exist a non-zero homomorphism ϕ : B → B then ϕ is anisomorphism. Proof
Since B is a central simple algebra, it does not have non-trivial bilateralideals so that ϕ is injective, Then the dimension dim K ( ϕ ( B )) = 4 and ϕ is an isomorphism. Corollary 2.1
Let K be a field and B , B be two quaternion algebrasover K . We represent B as B = K + Ki + Kj + Kk with i , j ∈ K and k = ij = − ji . Let ϕ : B → B be a K -linear map such that ϕ (1) = 1 , ϕ ( i ) = i , ϕ ( j ) = j , ϕ ( k ) = ϕ ( i ) ϕ ( j ) = − ϕ ( j ) ϕ ( i ) . Then ϕ is an isomorphism of K -algebras. We will work with K = Q or K = Q q for any place q including ∞ . Weobserve that to define in an explicit way an isomorphism of K -algebras ϕ : B Nq → B ′ it is enough to define the values ϕ ( i ) , ϕ ( j ) such that ϕ ( i ) = − ∆ N , ϕ ( j ) = p and ϕ ( i ) ϕ ( j ) = − ϕ ( j ) ϕ ( i ). If we put ϕ (1) = 1, ϕ ( k ) = ϕ ( i ) ϕ ( j ) and if we extend the map by K -linearity, then by Corollary 2.1, ϕ is a well defined isomorphism of K -algebras.3 The case of M ( Q ) If ∆ = 1 then B can be represented as B ( N,
1) = {− N, } where N isany positive integer. It is well known that there is an isomorphism ϕ N : B → M ( Q ) such that the image of the maximal order R is M ( Z ). Let weexplicitly describe such an isomorphism. We consider the Q -linear map ϕ N defined as follows: ϕ N ( i ) = (cid:18) − N (cid:19) and ϕ N ( j ) = (cid:18) − (cid:19) . Then ϕ N ( i ) = − N I ϕ N ( j ) = I where I is the identity 2 × ϕ N ( k ) = ϕ N ( i ) ϕ N ( j ) = (cid:18) − N (cid:19) (cid:18) − (cid:19) = (cid:18) − − N (cid:19) = − ϕ N ( j ) ϕ N ( i ) . It results that for any element x + yi + zj + tk ∈ B ( N,
1) with x, y, z, t ∈ Q ϕ N ( x + iy + jz + kt ) = (cid:18) x − z − y − tN ( y − t ) x + z (cid:19) and by Corollary 2.1 the map ϕ N : B ( N, p ) → M ( Q ) is an isomorphism.The image of the basis of the Eichler order R ( N ) is: ϕ N ( e ) = (cid:18) (cid:19) ϕ N ( e ) = ϕ N (cid:18) j (cid:19) = 12 (cid:20)(cid:18) (cid:19) + (cid:18) − (cid:19)(cid:21) = (cid:18) (cid:19) ϕ N ( e ) = ϕ N (cid:18) i + k (cid:19) = 12 (cid:20)(cid:18) − N (cid:19) + (cid:18) − − N (cid:19)(cid:21) = (cid:18) −
10 0 (cid:19) ϕ N ( e ) = ϕ N ( N j + k ) = (cid:18) − N N (cid:19) + (cid:18) − − N (cid:19) = (cid:18) − N − − N N (cid:19) and for any element xe + ye + ze + te ∈ R ( N ) with x, y, z, t ∈ Z ϕ N ( xe + ye + ze + te ) = (cid:18) x − N t − z − t − N t x + y + N t (cid:19) .
4e observe that if
N >
1, the reduced discriminant p | det(tr( ϕ N ( e k ) ϕ N ( e h ))) | for h, k = 1 , ..., ϕ N ( R ( N )) is N so that the image of R ( N ) via ϕ N is ϕ N ( R ( N )) = (cid:26) γ ∈ M ( Z ) | γ ≡ (cid:18) ∗ ∗ ∗ (cid:19) mod N (cid:27) . If N = 1 then R (1) is a maximal order of B = B , the reduced discriminantof ϕ ( R ) is: vuuuuut(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 (1)and its image via ϕ is M ( Z ). B ( N, and B ( M, Let B be a quaternion algebra of discriminant 1 and let B ( N,
1) = Q + Q i N + Q j N + Q k N and B ( M,
1) = Q + Q i M + Q j M + Q k M be two representationsof B where N and M are as in Section 2. We will write an isomorphismΨ MN : B ( N, → B ( M, MN as the composite ( ϕ M ) − ◦ ϕ N :Ψ MN ( i N ) = i M (cid:18) M + N M (cid:19) + k M (cid:18) M − N M (cid:19) Ψ MN ( j N ) = j M Ψ MN ( k N ) = i M (cid:18) M − N M (cid:19) + k M (cid:18) M + N M (cid:19) Proposition 3.1 If M is an integer such that M | N , then Ψ MN ( R ( N )) ⊂ R ( M ) . Proof
Let N = SM with S ∈ N . Then Ψ MN ( e N ) = e M , Ψ MN ( e N ) = e M ,Ψ MN ( e N ) = e M and Ψ MN ( e N ) = (1 − S ) e M + Se M . > We fix a prime p and a positive integer N as in Section 2. We representthe quaternion algebra B of discriminant ∆ as B ( N, p ) = {− ∆ N, p } = Q + Q i + Q j + Q k . For each prime q we want to identify B q to a ring5f matrices, in such a way that the integer structure is preserved. Let wedenote by R q ( N ) the subring of M ( Z q ) containing all the matrices of theform (cid:26)(cid:18) Z q Z q N Z q Z q (cid:19)(cid:27) . We observe that if q N then R q ( N ) = M ( Z q ).We recall that every local Eichler orders of level N in M ( Q q ) is isomorphicto R q ( N ) and its reduced discriminant is equal to ∆ N .We will deal separately with the cases of unramified places and of ramifiedplaces. In this section let q be a prime number such that q ∆; since at q thequaternion algebra B ( N, p ) is not ramified, the Hilbert symbol is1 = ( − ∆ N, p ) q . (2)We shall define an isomorphism ϕ ( N,p ) q : B ( N, p ) q f → M ( Q q ) such that ϕ ( N,p ) q ( R q ( N )) = R q ( N ) . To make easier the notation we will write ϕ Nq instead of ϕ ( N,p ) q . q not dividing ∆ p such that p isnot a square in Z × q We consider the case q ∆ p such that (cid:16) pq (cid:17) = − q = 2 and q | N ). This hypotheses on q assurethat p is not a square in Z × q , thus Q q ( √ p ) is a quadratic extension of Q q and by the identity (2), the prime − ∆ N is the norm of a unit of Q q ( √ p ).We write − ∆ N = x − py with x, y ∈ Z q and we define ϕ Nq as follows: ϕ Nq ( i ) = (cid:18) x − pyy − x (cid:19) ϕ Nq ( j ) = (cid:18) p (cid:19) . Then ϕ Nq ( i ) = − ∆ N I ϕ Nq ( j ) = pIϕ Nq ( k ) = ϕ Nq ( i ) ϕ Nq ( j ) = (cid:18) − py px − x py (cid:19) = − ϕ Nq ( j ) ϕ Nq ( i ) . It results that for any h = α + βi + γj + δk ∈ B q ( N, p ) ϕ Nq ( h ) = (cid:18) α + βx − δpy − βpy + γp + δxpβy + γ − δx α − βx + δpy (cid:19) ϕ Nq : B q → M ( Q q ) is an isomorphism.The image of the basis of the local Eichler order is: ϕ Nq ( e ) = (cid:18) (cid:19) ϕ Nq ( e ) = 12 (cid:18) p (cid:19) ∈ M ( Z q ) since q = 2 ϕ Nq ( e ) = 12 (cid:18) x − py p ( x − y ) y − x − x + py (cid:19) ∈ M ( Z q ) since q = 2 ϕ Nq ( e ) = (cid:18) − y a ∆ N + x a ∆ N − xp y (cid:19) ∈ M ( Z q ) since q = p. The reduced discriminant of ϕ Nq ( R q ( N )) is ∆ N . So ϕ Nq ( R q ( N )) = M ( Z q ).For any element g = αe + βe + γe + δe ∈ R q ( N ) ϕ Nq ( g ) = (cid:18) α + β + γ ( x − py ) − δy β p + γp ( x − y ) + δ ( a ∆ N + x ) β + γ ( y − x ) + δ a ∆ N − xp α + β + γ ( py − x ) + δy (cid:19) (3) q such that p is a square in Z × q We consider the primes q ∆ such that (cid:16) pq (cid:17) = 1. We observe that thishypothesis excludes the case p = q and includes q | N and q = 2 (in fact if q = 2 then by hypothesis p ≡ § p is a square in Z × ). We define the Q q -linear map ϕ Nq as follows: ϕ Nq ( i ) = (cid:18) − ∆ N (cid:19) and ϕ Nq ( j ) = (cid:18) −√ p √ p (cid:19) where √ p is an element ω in Z × q such that ω = p . Then ϕ Nq ( i ) = − ∆ N I ϕ Nq ( j ) = pIϕ Nq ( k ) = ϕ Nq ( i ) ϕ Nq ( j ) = (cid:18) √ p ∆ N √ p (cid:19) = − ϕ Nq ( j ) ϕ Nq ( i ) . It results that for any element α + βi + γj + δk ∈ B q ϕ Nq ( α + βi + γj + δk ) = (cid:18) α − γ √ p β + δ √ p ∆ N ( − β + δ √ p ) α + γ √ p (cid:19) ϕ Nq : B q ( N, p ) → M ( Q q ) is an isomorphism.The image of a basis of the local Eichler order R q ( N ) is: ϕ Nq ( e ) = Iϕ Nq ( e ) = −√ p √ p ! ∈ R q ( N ) ϕ Nq ( e ) = √ p N ( √ p − ! ∈ R q ( N ) ϕ Nq ( e ) = 1 √ p (cid:18) − a ∆ N N a ∆ N (cid:19) ∈ R q ( N )The reduced discriminant of ϕ Nq ( R q ( N )) is ∆ N , so ϕ Nq ( R q ( N )) = R q ( N ).Then, for any element g = αe + βe + γe + δe ∈ R q ( N ) ϕ Nq ( g ) = α + (1 −√ p ) β − δa ∆ N √ p γ (1+ √ p )2 + δ √ p γ ∆ N √ p − + δ ∆ N √ p α + (1+ √ p ) β + δa ∆ N √ p ! . (4)We observe that in this case we accept that 2 | N . p If q = p then 1 = ( − ∆ N, p ) p = (cid:16) − ∆ Np (cid:17) ([6], II, § − ∆ N is a squarein Z × p . We recall that a ∈ Z was choosen in Section 2 in such a way that a ∆ N + 1 ≡ p . Let we denote by √− ∆ N the square root of − ∆ N in Z × p such that a √− ∆ N ≡ − p . Then the following identity holds:( a ∆ N − √− ∆ N ) = √− ∆ N ( a √− ∆ N − ≡ p. (5)We define the Q p -linear map ϕ Np as follows: ϕ Np ( i ) = (cid:18) −√− ∆ N √− ∆ N (cid:19) and ϕ Np ( j ) = (cid:18) p (cid:19) . Then ϕ Np ( i ) = − ∆ N I ϕ Np ( j ) = pIϕ Np ( k ) = ϕ Np ( i ) ϕ Np ( j ) = (cid:18) −√− ∆ Np √− ∆ N (cid:19) = − ϕ Np ( j ) ϕ Np ( i ) .
8t results that for any element α + βi + γj + δk ∈ B p ( N, p ) ϕ Np ( α + βi + γj + δk ) = (cid:18) α − β √− ∆ N γ − δ √− ∆ Nγp + δp √− ∆ N α + β √− ∆ N (cid:19) and by Corollary 2.1, ϕ Np : B ( N, p ) p → M ( Q p ) is an isomorphism. Itremains to show that integer structures are preserved.The image of the basis of the local Eichler order is: ϕ Np ( e ) = Iϕ Np ( e ) = 12 (cid:18) p (cid:19) ∈ M ( Z p ) ϕ Np ( e ) = 12 (cid:18) −√− ∆ N −√− ∆ Np √− ∆ N √− ∆ N (cid:19) ∈ M ( Z p ) ϕ Np ( e ) = a ∆ N −√− ∆ Np a ∆ N + √− ∆ N ! ∈ M ( Z p ) . The reduced discriminant of ϕ Np ( R p ( N )) is ∆ N so that ϕ Np ( R p ( N )) = M ( Z p ).For any element αe + βe + γe + δe ∈ R p ( N ) the following identity holds: ϕ Np ( αe + βe + γe + δe ) = (6) (cid:18) α + β − γ √− ∆ N β − γ √− ∆ N + δp ( a ∆ N − √− ∆ N ) pβ + γp √− ∆ N + 2 δ ( a ∆ N + √− ∆ N ) 2 α + β + γ √− ∆ N (cid:19) . Since B is an indefinite quaternion algebra over Q , there exists an isomor-phism B ∞ ≃ M ( R ). We define ϕ N ∞ via i (cid:18) − ∆ N (cid:19) j (cid:18) √ p −√ p (cid:19) . Then ϕ N ∞ ( i ) = − ∆ N I ϕ N ∞ ( j ) = pIϕ N ∞ ( k ) = ϕ N ∞ ( i ) ϕ N ∞ ( j ) = (cid:18) −√ p −√ p ∆ N (cid:19) = − ϕ N ∞ ( j ) ϕ N ∞ ( i )and by Corollary 2.1 the map ϕ N ∞ : B ∞ → M ( R ) is an isomorphism.9 .3 The isomorphism at the ramified places For any prime number q such that q | ∆ we shall define, following [4], anisomorphism ϕ Nq : B q ( N, p ) f → (cid:26)(cid:18) α βqβ α (cid:19) | α, β ∈ Q q (cid:27) such that ϕ Nq ( R q ( N )) = ϕ Nq ( R q ) = (cid:26)(cid:18) α βqβ α (cid:19) | α, β ∈ Z q ) (cid:27) := O q where Q q is the quadratic unramified extension of Q q , α ¯ α is its non-trivial automorphism and Z q is its ring of integers.We have − − ∆ N, p ) q = ( − ∆ Nq , p ) q ( q, p ) q ; since ∆ is square free ( − ∆ Nq , p ) q =1 and ( q, p ) q = −
1. This means in particular that p is not a square in Q q and − ∆ Nq is a norm of a unit of Q q ( √ p ). Thus there exist x, y ∈ Z q suchthat − ∆ Nq = x − py = ( x −√ py )( x + √ py ). We can identify Q q = Q q ( √ p )and Z q = Z q ( √ p ).We define ϕ Nq as follows: ϕ Nq ( i ) = (cid:18) x − √ pyq ( x + √ py ) 0 (cid:19) ϕ Nq ( j ) = (cid:18) −√ p √ p (cid:19) . Then ϕ Nq ( i ) = ∆ N I ϕ Nq ( j ) = pIϕ Nq ( k ) = ϕ Nq ( i ) ϕ Nq ( j ) = (cid:18) √ p ( x − √ py ) −√ pq ( x + √ py ) 0 (cid:19) = − ϕ Nq ( j ) ϕ Nq ( i )and for any element α + βi + γj + δk of B q ( N, p ) with α, β, γ, δ ∈ Q q , ϕ Nq ( α + βi + γj + δk ) = (cid:18) α − γ √ p ( β + δ √ p )( x − √ py ) q ( β − δ √ p )( x + √ py ) α + γ √ p (cid:19) . By Corollary 2.1, ϕ Nq is an isomorphism.We compute the image of the local Eichler order R q ( N ): ϕ Nq ( e ) = Iϕ Nq ( e ) = 12 (cid:18) − √ p
00 1 + √ p (cid:19) ∈ O q ϕ q ( e ) = 12 (cid:18) x − √ py )(1 + √ p ) q ( x + √ py )(1 − √ p ) 0 (cid:19) ∈ O q q ( e ) = − a ∆ Np √ p − y + xp √ pq (cid:16) − y − xp √ p (cid:17) a ∆ Np √ p ! ∈ O q and the reduced discriminant of ϕ Nq ( R q ( N )) is N ∆.For any element αe + βe + γe + δe of R q ( N ) ϕ Nq ( αe + βe + γe + δe ) = α + β − √ p (cid:16) β + aN ∆ δp (cid:17) ( x − √ py ) h γ + √ p (cid:16) γ + δp (cid:17)i q ( x + √ py ) h γ − √ p (cid:16) γ + δp (cid:17)i α + β + √ p (cid:16) β + aN ∆ δp (cid:17) . Φ( N ) Let B be a quaternion algebra over Q of discriminant ∆ and let R ( N ) be anEichler order of level N of B . Then R ( N ) × × ˆ Z is a compact open subgroupof the finite adelization B × , ∞ A and it is possible to associate to it a discretesubgroup Φ( N ) of SL ( R ) byΦ( N ) = ( GL +2 ( R ) × ( R ( N ) × × ˆ Z )) ∩ B × . It is known that Φ( N ) is a co-compact congruence subgroup of SL ( R ) [8]. Lemma 5.1
If we denote by R ( N ) (1) the group of reduced norm elementsof R ( N ) , then the following identity holds: Φ( N ) = R ( N ) (1) . Proof
The inclusion ⊇ is trivial since R ( N ) (1) ⊆ B × and R ( N ) (1) ⊆ GL +2 ( R ) × ( R ( N ) × × ˆ Z ).We prove the inclusion ⊆ . Let α be an element of Φ( N ); then:a) α ∈ GL +2 ( R ) × ( R ( N ) × × ˆ Z )b) α ∈ B × If we denote by n ( α ) the reduced norm of α , then by b), n ( α ) is a rationalnumber, which, by a), is a p -adic unit for every prime p , and positive. Thus n ( α ) = 1 and α ∈ R ( N ). 11 Explicit description of two conjugates to R ( N q ) in B ( N, p ) In this section we will keep the usual notation and we will represent thequaternion algebra B as B ( N, p ) = {− N ∆ , p } .Let q be a prime number such that q ∆. It is well known that by definition R ( N q ) ≃ R ( N ) ∩ ( ϕ Nq ) − ( R q ( qN )) . (7)We shall identify R ( N q ) with this subgroup of R ( N ). Let we consider theid`ele η q in B × A defined by η q = η q,ν = 1 if ν = q η q,q = ( ϕ Nq ) − (cid:18) q
00 1 (cid:19) if ν = qBy strong approximation, write η q = δ q g ∞ u , with δ q ∈ B × , g ∞ ∈ GL +2 ( R ) and u ∈ ( R ( N q ) ⊗ Z b Z ) × .We observe that η q R q ( N q ) η − q = δ q R q ( N q ) δ − q = R q ( N ) ∩ ( ϕ Nq ) − (cid:18) Z q q Z q N Z q Z q (cid:19) and δ q R ( N q ) δ − q = R ( N ) ∩ ( ϕ Nq ) − (cid:18) Z q q Z q N Z q Z q (cid:19) . (8)We will give bases for R ( N q ) and δ q R ( N q ) δ − q . We observe that the fol-lowing theorems are direct applications of the construction in [5] § ϕ Nq of a generic element of R q ( N ). Proposition 6.1
Let q be a prime number such that q ∆ p and p is not asquare in Z × q . Let − ∆ N = x − py with x, y ∈ Z q . Let c , c , c be integerssuch that c ≡ ( y − x ) mod qc ≡ p − mod qc ≡ x mod q. Then a basis of R ( N q ) in R ( N ) is: f = e , f = − c e + e , f = − c ( a ∆ N − c ) e + e , f = qe and a basis of δ q R ( N q ) δ − q in R ( N ) is: g = e , g = c e + e , g = − c ( a ∆ N + c ) e + e , g = qe . roof By the results in Section 4.1.1 and by the equality (7), we see that f , f , f , f ∈ R ( N q ) anddet − c − c ( a ∆ N − c ) q = q. By the results in Section 4.1.1 and by the equality (8) we see that g , g , g , g ∈ δ q R ( N q ) δ − q and det c − c ( a ∆ N + c ) q = q. Proposition 6.2
Let q ∆ be a prime number such that p is a square in Z × q . A basis of R ( qN ) in R ( N ) is: f = e , f = e , f = e − ce , f = qe where Z ∋ c ≡ ( p − √ p )2 − mod q. A basis of δ q R ( qN ) δ − q in R ( N ) is: g = e , g = e , g = e − c ′ e , g = qe where Z ∋ c ′ ≡ ( p + √ p )2 − mod q. Proof
By the results in section 4.1.2 and by the equality (7), we observethat f , f , f , f ∈ R ( qN ) anddet − c q = q ;by the equality (8), it is easy to verify that g , g , g , g ∈ δ q R ( qN ) δ − q anddet − c ′ q = q. roposition 6.3 Let √− ∆ N be the square root of − ∆ N in Q p such that a √− ∆ N ≡ − p . A basis of R ( N p ) in R ( N ) is: f = e , f = − c e + e , f = − a ∆ N + c ) e + pe , f = p ( Ae + Be ) and a basis of δ p R ( N p ) δ − p in R ( N ) is: g = e , g = c e + e , g = − a ∆ N − c p e + e g = pe where Z ∋ c ≡ √− ∆ N mod p , a ∈ Z is such that a ∆ N + 1 ≡ p and A, B ∈ Z are such that Ap + 2 B ( a ∆ N + c ) = 1 . Proof
We first observe that if we fix √− ∆ N the square roots in Q p suchthat a √− ∆ N ≡ − p , then p | ( a ∆ N − c ) and p ( a ∆ N + c ). Thenthe existence of A, B ∈ Z such that Ap + 2 B ( a ∆ N + c ) = 1 is ensured.By the results in Section 4.1.3 and by the equality (7), we observe that f , f , f , f ∈ R ( N p ) anddet − c − a ∆ N + c ) pA p pB = p ;by the equality (8), we see that g , g , , g ∈ δ p R ( N p ) δ − p anddet c − a ∆ N − c p p = p. Ψ MN Let we fix ∆ as in Section 2; by the classification theorem, up to isomorphismthere exists only one quaternion algebra B over Q with discriminant ∆.Let B ( M, p ) = {− ∆ M, p } and B ( N, p ) = {− ∆ N, p } be two representationsof B , with p, N, M as in Section 2. Then there is an isomorphism Ψ MN : B ( N, p ) → B ( M, p ). This implies that there exists an element h ∈ B ( M, p )14uch that h = − N ∆; we observe that h is of the form h = i M β + j M γ + k M δ where ( β, γ, δ ) ∈ Q is a solution of the equation M ∆ β − pγ − p ∆ M δ = N ∆ . Lemma 7.1
Let f be the quadratic form on Q defined as f = M β − pM δ ;then f represents N . Proof
By the Hasse-Minkowski theorem (see for example [6]), f represents N in Q if and only if f represents N in Q ℓ at any place ℓ , that is ( N, p ) ℓ =( M, p ) ℓ for any prime number ℓ .We write: N = ℓ a u , p = ℓ b v , ǫ ( ℓ ) ≡ ℓ − mod 2.If ℓ = 2 then ( N, p ) ℓ = ( − abǫ ( ℓ ) (cid:16) uℓ (cid:17) b (cid:16) vℓ (cid:17) a • If ℓ pN then ( N, p ) ℓ = 1. • If ℓ = p then ǫ ( p ) = 0 , v = 1 , b = 1 so ( N, p ) p = (cid:16) up (cid:17) b (cid:16) vp (cid:17) a = (cid:16) up (cid:17) .By the hypothesis on the prime factors q of N , by the law of recipocityand since p ≡ (cid:18) up (cid:19) = Y q | u (cid:18) qp (cid:19) = Y q | u (cid:18) pq (cid:19) ( − ( q − p − / = 1 . • If ℓ | N and ℓ = p then b = 0 , v = p so( N, p ) ℓ = (cid:16) pℓ (cid:17) a = 1by the hypothesis on the prime factors of N .If ℓ = 2 then b = 0 and v = p ; we know that( N, p ) = ( − ǫ ( u ) ǫ ( v )+ aω ( p )+ bω ( u ) where ǫ ( v ) = 0 , ω ( p ) ≡ p − mod 2. So( N, p ) = ( − aω ( p ) . • if a = 0 then ( N, p ) = 1; • if a = 0 then 2 | N and p ≡ ω ( p ) = 0 and ( N, p ) = 1 . N and M satisfy the same hypotheses, then ( N, p ) ℓ = ( M, p ) ℓ = 1 forany prime number ℓ .We define the Q -linear map Ψ MN : B ( N, p ) → B ( M, p ) as:Ψ MN ( i N ) = h, Ψ MN ( j N ) = j M where h = βi M + δk M with ( β, γ ) ∈ Q solution of M β − pM δ = N (byLemma 7.1 such an element exists). Then Ψ MN ( i N ) = − N ∆, Ψ MN ( j N ) = p and Ψ MN ( i N )Ψ MN ( j N ) = hj M = k M β + i M pδ = − Ψ MN ( j N )Ψ MN ( i N ) . By the Corollary 2.1, the map Ψ MN is an isomorphism.We observe that if N = M S then β − pδ = S (9)so S is the norm of an element β + √ pδ of the ring of integer O = (cid:26)
12 ( a + √ pb ) : a, b ∈ Z with the same parity (cid:27) of Q ( √ p ).We denote by a M , a N the integer numbers as in Section 2, such that a M ∆ M +1 ≡ p and a N ∆ N + 1 ≡ p . Lemma 7.2 If N = M S ∈ N then we can choose β ∈ Z (cid:2) (cid:3) satisfying theidentity (9) such that a M ≡ a N β mod p . Proof
By definition of a N , a M , since p ∆ M , we find that a N S − a M ≡ p , that is by (9) a N β − a M ≡ p . In particular a N β − a M ≡ p or a N β + a M ≡ p . If we are in the second situation, thenwe can take − β instead of β , so we have that a M ≡ a N β mod p .In the sequel when M | N we choose a M as in the above lemma. Proposition 7.1
Let B ( N, p ) and B ( M, p ) be two representations of thequaternion algebra B defined over Q with discrminant ∆ . Let we considerthe isomorphism Ψ MN : B ( N, p ) → B ( M, p ) defined above. If M | N then Ψ MN ( R ( N )) ⊂ R ( M ) . Proof
Let N = M S where S ∈ N . We recall that p ≡ MN ( e Nℓ ) ∈ R ( M ) for ℓ = 1 , , , MN : Ψ MN ( e N ) = 1 = e M MN ( e N ) = 1 + j M e M Ψ MN ( e N ) = A e M + B e M + C e M + D e M where A = δ (1 − p ) a M ∆ M ∈ Z , B = δ ( p − a M ∆ M ∈ Z , C = δp + β ∈ Z and D = δp − p ∈ Z .Ψ MN ( e N ) = A e M + B e M + C e M + D e M where B = − A = p [∆ M ( a N S − a M β + pδa M ], C = 2 δ ∈ Z and D = β − pδ ∈ Z . We observe that B ∈ Z (and A ∈ Z ), infact by Lemma 7.2: a N S − a M β ≡ a N S − a N β mod p ≡ a N S − a N ( S + pδ ) mod p ≡ p By using the local isomorphisms given in Section 4, we will prove some newresults for the Eichler orders. Let B be a quaternion algebra over Q of fixeddiscriminant ∆.Let B ( N, p ) = {− ∆ N, p } = Q + Q i N + Q j N + Q k N be a representation of B ; we will write R ( N ) ⊂ B ( N, p ) to denote the Eichler order of level N ofHashimoto [3]: R ( N ) = Z e N + Z e N + Z e N + Z e N with e N = 1 , e N = 1 + j N , e N = i N + k N , e N = a ∆ N j N + k N p where a ∈ Z satisfies a ∆ N + 1 ≡ p . By abuse of notation, in thissection we will write R ( M ) instead of Ψ NM ( R ( M )). In this way, if N | M theinclusion R ( M ) ⊂ R ( N ) in B ( N, p ) is true.
Lemma 8.1
Let B be a quaternion algebra over Q of discriminant ∆ ; let N be a positive integer prime to ∆ and q be a prime number not dividing ∆ . Then the Z -rank of T n ∈ N R ( N q n ) is equal to the Z -rank of T n ∈ N R ( q n ) . Proof
Let B (1 , p ) be a representation of B where p is as in Section 2. It isobvious that \ n R ( N q n ) = \ n R ( q n ) ∩ R ( N ) ⊂ R (1) . (10)17ince the rank is invariant by isomorphism and R ( N ) has maximal rankover Z , then rk \ n ∈ N R ( N q n ) ! = rk \ n ∈ N R ( q n ) ! . Let B (1 , p ) be a representation of B anf let q ∆ be a prime number; weconsider the chain of Eichler orders ... ⊂ R ( q n ) ⊂ ... ⊂ R ( q ) ⊂ R ( q ) ⊂ R (1)in B (1 , p ). We will characterize the intersection A q = T n ∈ N R ( q n ) as Z -lattice. Since R ( q ) ≃ R (1) ∩ ( ϕ q ) − (cid:18) Z q Z q q Z q Z q (cid:19) where ϕ q : B (1 , p ) q → M ( Q q ) is a local isomorphism, then A q ≃ R (1) ∩ "\ n ( ϕ q ) − (cid:18) Z q Z q q n Z q Z q (cid:19) = R (1) ∩ ( ϕ q ) − (cid:18) Z q Z q Z q (cid:19) . (11) Proposition 8.1
Let B be a quaternion algebra over Q of discriminant ∆ ;let we fix a prime number q not dividing ∆ . The intersection A q = \ n ∈ N R ( q n ) has rank 2 over Z . Proof
Let we fix a prime number p is as in Section 2. We will distinguish thefollowing cases:1. q ∆ p such that (cid:16) pq (cid:17) = − q ∆ such that (cid:16) pq (cid:17) = 1;3. q = p .1. Let q ∆ be a prime number such that (cid:16) pq (cid:17) = −
1. Let N be a positiveinteger prime to ∆ such that (cid:0) ps (cid:1) = 1 for all s | N and (cid:16) − ∆ Nq (cid:17) = 1. This lastcondition on N implies that there exists x ( N, q ) ∈ Z q such that − ∆ N = x ( N, q ) . Let we represent B as B ( N, p ) = {− ∆ N, p } .18f q is such that − ∆ is a square in Z q , then we can take N = 1 and A q ⊂ R (1) in B (1 , p ); if h ∈ A q then by (11) there exist α, β, γ, δ ∈ Z suchthat h = αe + βe + γe + δe where { e , e , e , e } is the Hashimoto basisof R (1) in B (1 , p ). Moreover, by the identity (3) x ( N, q ) (cid:20) − γ − δp (cid:21) + β δa ∆ p = 0 . Then A q ⊂ R (1) can be expressed as the Z -lattice A q = Z e + Z e where e = − a ∆ e − e + pe .If q is such that − ∆ is not a square in Z q , then we take N such that (cid:16) Nq (cid:17) = −
1. If h ∈ T n R ( N q n ) ≃ R ( N ) ∩ ( ϕ Nq ) − (cid:18) Z q Z q Z q (cid:19) in B ( N, p )then there exist α, β, γ, δ ∈ Z such that h = αe N + βe N + γe N + δe N where { e N , e N , e N , e N } is the Hashimoto basis of R ( N ) in B ( N, p ). Moreover, bythe identity (3) x ( N, q ) (cid:20) − γ − δp (cid:21) + β δa ∆ Np = 0 . Then T n R ( N q n ) ⊂ R ( N ) can be expressed as the Z -lattice T n R ( N q n ) = Z e N + Z e N where e N = − a ∆ N e − e + pe . By Lemma 8.1, rk ( T n R ( q n )) =rk ( T n R ( N q n )) = 2.2. Let q ∆ be a prime number such that (cid:16) pq (cid:17) = 1; we represent thequaternion algebra B as B (1 , p ) = {− ∆ , p } . If h ∈ A q , then by (11) and bythe identity (4), h = αe + βe + γe + δe where α, β, γ, δ ∈ Z satisfy theequation √ p ( − γ ) + ( γp + 2 δ ) = 0 . (12)Then γ = δ = 0 and A q ⊂ R (1) can be expressed as the Z -lattice A q = Z e + Z e .3. Let q = p ; we represent the quaternion algebra B as B (1 , p ) = {− ∆ , p } .If h ∈ A p , then by (11) and by the identity (6), h = αe + βe + γe + δe where α, β, γ, δ ∈ Z satisfy the equation √− ∆( γp + 2 δ ) + ( βp + 2 a ∆ δ ) = 0 . (13)Then A q ⊂ R (1) can be expressed as the Z -lattice A p = Z e + Z e where e = − a ∆ e − e + pe . Proposition 8.2
Let B a quaternion algebra over Q of discriminant ∆ and let B (1 , p ) be a representation of B . Let q, s ∆ p be two prime numbersuch that (cid:16) pq (cid:17) = 1 and (cid:0) ps (cid:1) = − . Then A q ∩ A p = A s ∩ A p = A q ∩ A s = Z . roof Let { e , e , e , e } be the Hashimoto basis of R (1) in B (1 , p ).If h ∈ A q ∩ A p then h = αe + βe + γe + δe where α, β, γ, δ ∈ Z satisfythe equations (12) and (13). This imply that β = γ = δ = 0 . If h ∈ A s ∩ A p then h = αe + βe + γe + δe where α, β, γ, δ ∈ Z satisfythe equation (13) and by the identity (3) x (cid:20) − γ − δp (cid:21) + y h γ i + β δ a ∆ p = 0 . (14)where x, y ∈ Z q are such that − ∆ = x − py . This imply that β = γ = δ = 0 . If h ∈ A q ∩ A s then h = αe + βe + γe + δe where α, β, γ, δ ∈ Z satisfythe equations (12) and (14). This imply that β = γ = δ = 0 . Corollary 8.1
Let B a quaternion algebra over Q of discriminant ∆ ; then A := \ N R ( N ) = Z where N runs over the set of positive integer numbers primes to ∆ . As corollary, by Lemma 5.1, the following result holds:
Corollary 8.2
Let Φ( N ) be the group defined in Section 5, then: \ N Φ( N ) = {± } where N runs over the set of positive integer numbers primes to ∆ . Using a mathematical problem-solving environment as Maple, wich workwith p -adic numbers, it is possible to produce some examples.Let we consider the quaternion algebra B over Q with discriminant ∆ = 35;following Hashimoto we can represent it as B (3 ,
13) = {− , } . A basisover Z of the Eichler order R (3) of B (3 ,
13) is e = 1 , e = 1 + j , e = i + k , e = 525 j + k .
20f we consider q = 11, then q ∆ and p = 13 is not a square in Z × ; thus byProposition 6.1, a basis of the Eichler order R (33) in B (3 ,
13) is: f = 1 , f = −
52 + i − j + k f = − − j + 113 k, f = 112 + 112 j. A basis of δ R (33) δ − in B (3 ,
13) is: g = 1 , g = 52 + i j + k g = − − j + 113 k g = 112 + 112 j. Moreover let B (17 ,
13) = {− , } = Q + Q i (17) + Q j (17) + Q k (17) bethe quaternion algebra over Q with discriminant 35 and N = 17; then( i (17) ) = −
595 and ( j (17) ) = 13. It is possible to write the isomorphismΨ : B (3 , → B (17 ,
13) described in Section 7:Ψ ( j ) = j (17) , Ψ ( i ) = 817 i (17) + 117 k (17) . References [1] Ciavarella Miriam: Congruences between modular forms and relatedmodules, arXiv:0710.4677v1 [math. NT] 25 Oct. 2007.[2] Ciavarella Miriam and Terracini Lea: Analogue of Ihara` s lemma forShimura Curves, Submitted paper, 2007.[3] Hashimoto Ki-ichiro: Explicit form of Quaternion modular embed-dings, Osaka J. Math., 32 n.3, 533-546, 1995.[4] Pizer Arnold: On the arithmetic of quaternions algebras II, J. Math.Soc. Japan, 28, 676-688, 1976.[5] Samuel Pierre: Th´eorie Alg´ebrique des Nombres, Hermann Paris, 1971.[6] Serre Jean-Pierre: Cours d’arithm´etique, Presses Universitaires deFrance, 1970.[7] Terracini Lea: A Taylor-Wiles system for quaternionic Hecke algebras,Compositio Mathematica, 137, 23-47, 2003.218] Vign´eras, Marie-France, Arithm´etique des alg`ebres de quaternions,Lecture Notes Math., 800, Springer, 1980.