EEXPLICIT METHODS FOR HILBERT MODULAR FORMS
LASSINA DEMB´EL´E AND JOHN VOIGHT
Abstract.
We exhibit algorithms to compute systems of Hecke eigenvalues for spaces ofHilbert modular forms over a totally real field. We provide many explicit examples as wellas applications to modularity and Galois representations.
Contents
1. Classical (elliptic) modular forms 22. Classical Hilbert modular forms 53. Quaternionic modular forms 84. Definite method 125. Indefinite method 176. Examples 267. Adelic quaternionic modular forms 348. Definite method, arbitrary class number 439. Indefinite method, arbitrary class number 47References 52The study of modular forms remains a dominant theme in modern number theory, a conse-quence of their intrinsic appeal as well as their applications to a wide variety of mathematicalproblems. This subject has seen dramatic progress during the past half-century in an en-vironment where both abstract theory and explicit computation have developed in parallel.Experiments will remain an essential tool in the years ahead, especially as we turn fromclassical contexts to less familiar terrain.In this article, we discuss methods for explicitly computing spaces of Hilbert modularforms, refashioning algorithms over Q to the setting of totally real fields. Saving definitionsfor the sections that follow, we state our main result. Theorem.
There exists an algorithm that, given a totally real field F , a nonzero ideal N ofthe ring of integers of F , and a weight k ∈ ( Z ≥ ) [ F : Q ] , computes the space S k ( N ) of Hilbertcusp forms of weight k and level N over F as a Hecke module. This theorem is the work of the first author [15] together with Donnelly [18] combinedwith work of the second author [69] together with Greenberg [30].The outline of this article is as follows. After briefly recalling methods for classical (el-liptic) modular forms in §
1, we introduce our results for computing Hilbert modular formsin the simplest case (of parallel weight 2 over a totally real field of strict class number 1) in
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2. In brief, our methods employ the Jacquet-Langlands correspondence to relate spaces ofHilbert modular forms to spaces of quaternionic modular forms that are more amenable tocomputation; we discuss this matter in §
3, and consider two approaches (definite and indefi-nite) in §§ § §
7, then give a complete and general description of our algorithms in adeliclanguage in §§ Magma [3] and our examples are computed using this implementa-tion. Donnelly and the second author [21] are using this implementation to compute Heckedata for thousands of forms over totally real fields up to degree 6.These notes arose from lectures at the Centre de Recerca Matem`atica (CRM) in Barcelona;it is our great pleasure to thank the CRM for the invitation to speak and the hospitalityof the organizers, Luis Dieulefait and Victor Rotger. The authors would also like to thankMatthew Greenberg, Ariel Pacetti, Aurel Page, Jeroen Sijsling, and the referee for manyhelpful comments as well as Benedict Gross for his remarks which we include at the end ofSection 6. The first author is supported by a Marie-Curie Fellowship, and the second authorby an NSF Grant No. DMS-0901971.1.
Classical (elliptic) modular forms
To motivate our study of Hilbert modular forms, we begin by briefly considering algorithmsfor classical (elliptic) modular forms. For a more detailed introduction to modular forms,see the books by Darmon [11] and Diamond and Shurman [19], and for more informationon computational aspects see Cremona [9], Kilford [40], Stein [64], and the many referencescontained therein.Let H = { x + yi ∈ C : y > } denote the upper-half plane and let H ∗ = H ∪ P ( Q ) denotethe completed upper half-plane with the cusps P ( Q ). The groupGL +2 ( Q ) = { γ ∈ GL ( Q ) : det γ > } acts on H ∗ by linear fractional transformations. For N ∈ Z > , we consider the subgroup ofthose integral matrices of determinant 1 that are upper-triangular modulo N ,Γ ( N ) = (cid:26) γ = (cid:18) a bc d (cid:19) ∈ SL ( Z ) : N | c (cid:27) ⊆ GL +2 ( Z ) = SL ( Z ) ⊆ GL +2 ( Q ) . The group PΓ ( N ) = Γ ( N ) / {± } is a discrete subgroup of PSL ( R ). A modular form ofweight k ∈ Z > and level N is a holomorphic function f : H → C such that(1.1) f ( γz ) = f (cid:18) az + bcz + d (cid:19) = ( cz + d ) k f ( z )for all γ ∈ Γ ( N ) and such that f ( z ) tends to a finite limit as z tends to any cusp (i.e., f isholomorphic at the cusps).One can equivalently write this as follows. For γ ∈ GL ( R ) and z ∈ H we define j ( γ, z ) = cz + d . We then define a weight k action of GL +2 ( Q ) on the space of complex-valued functions n H by(1.2) ( f | k γ )( z ) = (det γ ) k − j ( γ, z ) k f ( γz ) . Then (1.1) is equivalent to f | k γ = f for all γ ∈ Γ ( N ).Note that the determinant factor (det γ ) k − in our definition is different from the usual(det γ ) k/ , which is an analytic normalization. Consequently, the central subgroup Q × ⊆ GL +2 ( Q ) acts by f | k γ = γ k − f . Remark . For simplicity we treat only the case of Γ ( N )-level structure in this article.If desired, one could without difficulty extend our methods to more general level structureswith characters, and so on.The C -vector space of modular forms of weight k and level N is finite-dimensional and isdenoted M k ( N ).If f ∈ M k ( N ), then f ( z + 1) = f ( z ) so f has a Fourier expansion(1.4) f ( z ) = ∞ (cid:88) n =0 a n q n = a + a q + a q + a q + . . . where a n ∈ C and q = exp(2 πiz ).We say that f is a cusp form if f ( z ) → z tends to any cusp (i.e., f vanishes at thecusps). The C -vector space of cusp forms of weight k and level N is denoted S k ( N ). Wehave M k ( N ) = S k ( N ) ⊕ E k ( N ) where E k ( N ) is spanned by the Eisenstein series of level N .Note that when k ≥
2, then (1.1) is equivalent to f ( γz ) ( d ( γz )) k − = f ( z ) ( dz ) k − so one may equivalently think of a cusp form f ∈ S k ( N ) as a holomorphic differential( k − modular curve X ( N ) = Γ ( N ) \H ∗ . (Because of our normalization, suchdifferential forms will be global sections of the line bundle corresponding to the algebraiclocal system Sym k − ( C ), corresponding to bivariate homogeneous polynomials of degree k −
2. Some authors use an analytic normalization instead.)The spaces M k ( N ) and S k ( N ) are equipped with an action of pairwise commuting diago-nalizable Hecke operators T n for each integer n ∈ Z > . The Hecke operators can be thoughtof in several different ways: they arise from correspondences on the modular curve X ( N ),as “averaging” operators over lattices of index n , or more formally from double coset de-compositions for the group Γ ( N ) inside SL ( Z ). The action of the Hecke operator T n isdetermined by the action of T p for p | n , and the latter for p (cid:45) N in weight k are given simplyby the formula ( T p f )( z ) = p k − f ( pz ) + 1 p p − (cid:88) a =0 f (cid:18) z + ap (cid:19) . (For primes p | N one omits the first term, and there are also operators called Atkin-Lehnerinvolutions .) We say therefore that S k ( N ) is a Hecke module , namely, an abelian groupequipped with an action of the
Hecke algebra (cid:101) T = Z [ T p ] p = Z [ T , T , . . . ], a polynomial ringin countably many variables over Z indexed by the primes. Our Hecke modules will alwaysbe finite-dimensional C -vector spaces. form f ∈ S k ( N ) is an oldform (at d ) if f ( z ) = g ( dz ) for some g ∈ S k ( M ) with M | N aproper divisor and d | N/M ; we say f is a newform if f is a normalized eigenform which isorthogonal to the space of oldforms (with respect to the Petersson inner product).The space S k ( N ) consequently has a basis of eigenforms , i.e., functions that are eigen-functions for each Hecke operator T n . If f is an eigenform, normalized so that a = 1 in its q -expansion (1.4), then T n f = a n f . Moreover, the field Q ( { a n } ) = E ⊆ C is a number fieldand each Hecke eigenvalue a n is an algebraic integer in E .In this way, the system of Hecke eigenvalues ( a p ) p for a normalized eigenform f ∈ S k ( N )determine the form f : H → C . These eigenvalues also determine the L -series L ( f, s ) = ∞ (cid:88) n =1 a n n s = (cid:89) p (cid:45) N (cid:18) − a p p s + 1 p s +1 − k (cid:19) − (cid:89) p | N (cid:18) − a p p s (cid:19) − associated to f (defined for Re s > (cid:96) -adic Galois representations ρ f,(cid:96) : Gal( Q / Q ) → GL ( Z (cid:96) )associated to f with the property that for any prime p (cid:45) (cid:96)N , we haveTr( ρ f,(cid:96) (Frob p )) = a p ( f ) and det( ρ f,(cid:96) (Frob p )) = p k − . Several methods have been proposed for making the Hecke module S k ( N ) into an objectamenable to explicit computation. With a view to their generalizations to Hilbert modularforms, we mention two approaches which have seen wide application. (We neglect the methodof graphs [47] and a method which uses the Eichler-Selberg trace formula [35].) For simplicity,we restrict our discussion to the case of weight k = 2.The first method goes by the name modular symbols and has been developed by Birch,Swinnerton-Dyer, Manin, Mazur, Merel, Cremona [9], Stein [64], and many others. TheHecke operators T p act naturally on the integral homology H ( X ( N ) , Z ; cusps)—linear com-binations of paths in the completed upper half plane H ∗ whose endpoints are cusps and whoseimages in X ( N ) are linear combinations of loops—and integration defines a nondegenerateHecke-equivariant pairing which gives rise to an isomorphism (the Eichler-Shimura theorem) H ( X ( N ) , C ; cusps) ∼ = S ( N ) ⊕ S ( N )where denotes complex conjugation. The formalism of modular symbols then presentsthe space H ( X ( N ) , Z ; cusps) explicitly in terms of paths in H ∗ whose endpoints are cusps(elements of P ( Q )) and whose images in X ( N ) are a linear combination of loops. We havean explicit description of the action of the Hecke operators on the space of modular symbols,and the Manin trick (the Euclidean algorithm) yields an algorithm for writing an arbitrarymodular symbol as a Z -linear combination of a finite set of generating symbols, therebyrecovering S ( N ) as a Hecke module.The second method goes by the name Brandt matrices and goes back to Brandt, Eichler[23, 24], Pizer [48], Kohel [45], and others. In this approach, a basis for S ( N ) is obtainedby linear combinations of theta series associated to (right) ideals in a quaternion order ofdiscriminant N . These theta series are generating series which encode the number of elementsin the ideal with a given reduced norm, and the Brandt matrices which represent the actionof the Hecke operators are obtained via this combinatorial (counting) data. . Classical Hilbert modular forms
We now consider the situation where the classical modular forms from the previous sectionare replaced by forms over a totally real field. References for Hilbert modular forms includeFreitag [25], van der Geer [26] and Goren [29].Let F be a totally real field with [ F : Q ] = n and let Z F be its ring of integers. The case n = 1 gives F = Q and this was treated in the previous section, so we assume throughoutthis section that n >
1. Let v , . . . , v n : F → R be the real places of F , and write v i ( x ) = x i .For γ ∈ M ( F ) we write γ i = v i ( γ ) ∈ M ( R ).For simplicity, in these first few sections ( §§ F has strict class number1; the general case, which is more technical, is treated in § +2 ( F ) = { γ ∈ GL ( F ) : det γ i > i = 1 , . . . , n } acts naturally on H n by coordinatewise linear fractional transformations z (cid:55)→ γz = ( γ i z i ) i = (cid:18) a i z i + b i c i z i + d i (cid:19) i =1 ,...,n . For a nonzero ideal N ⊆ Z F , letΓ ( N ) = (cid:26) γ = (cid:18) a bc d (cid:19) ∈ GL +2 ( Z F ) : c ∈ N (cid:27) ⊆ GL +2 ( Z F ) ⊆ GL ( F ) . Let PΓ ( N ) = Γ ( N ) / Z × F ⊆ PGL +2 ( Z F ). Then the image of PΓ ( N ) under the embeddings γ (cid:55)→ ( γ i ) i is a discrete subgroup of PGL +2 ( R ) n .Under the assumption that F has strict class number 1, we have Z × F, + = { x ∈ Z × F : x i > i } = Z × F and hence GL +2 ( Z F ) = Z × F SL ( Z F ), and so alternatively we may identifyPΓ ( N ) ∼ = (cid:26) γ = (cid:18) a bc d (cid:19) ∈ SL ( Z F ) : c ∈ N (cid:27) / {± } in analogy with the case F = Q . Definition 2.1. A Hilbert modular form of parallel weight and level N is a holomorphicfunction f : H n → C such that (2.2) f ( γz ) = f (cid:18) a z + b c z + d , . . . , a n z n + b n c n z n + d n (cid:19) = (cid:32) n (cid:89) i =1 ( c i z i + d i ) det γ i (cid:33) f ( z ) for all γ ∈ Γ ( N ) . We denote by M ( N ) the space of Hilbert modular forms of parallel weight 2 and level N ;it is a finite-dimensional C -vector space. The reader is warned not to confuse M ( N ) withthe ring M ( R ) of 2 × R . Remark . There is no holomorphy condition at the cusps in Definition 2.1 as there wasfor classical modular forms. Indeed, under our assumption that [ F : Q ] = n >
1, this followsautomatically from Koecher’s principle [26, § ote also that if u ∈ Z × F , then γ ( u ) = (cid:18) u u (cid:19) ∈ GL +2 ( Z F ) acts trivially on H and at thesame time gives a vacuous condition in (2.2), explaining the appearance of the determinantterm which was missing in the classical case.Analogous to (1.2), we define(2.4) ( f | γ )( z ) = (cid:32) n (cid:89) i =1 det γ i j ( γ i , z ) (cid:33) f ( γz )for f : H n → C and γ ∈ GL +2 ( F ); then (3.3) is equivalent to ( f | γ )( z ) = f ( z ) for all γ ∈ Γ ( N ).The group GL +2 ( F ) also acts naturally on the cusps P ( F ) (cid:44) → P ( R ) n . We say that f ∈ M ( N ) is a cusp form if f ( z ) → z tends to a cusp, and we denote thespace of cusp forms (of parallel weight 2 and level N ) by S ( N ). We have an orthogonaldecomposition M ( N ) = S ( N ) ⊕ E ( N ) where E ( N ) is spanned by Eisenstein series of level N ; for level N = (1), we have dim E (1) = + Z F , where Cl + Z F denotes the strict classgroup of Z F .Hilbert modular forms admit Fourier expansions as follows. For a fractional ideal b of F ,let b + = { x ∈ b : x i > i = 1 , . . . , n } . Let d be the different of F , and let d − denote the inverse different. A Hilbert modular form f ∈ M ( N ) admits a Fourier expansion(2.5) f ( z ) = a + (cid:88) µ ∈ ( d − ) + a µ e πi Tr( µz ) . with a = 0 if f is a cusp form.Let f ∈ M ( N ) and let n ⊆ Z F be a nonzero ideal. Then under our hypothesis that F hasstrict class number 1, we may write n = ν d − for some ν ∈ d + ; we then define a n = a ν . Thetransformation rule (2.2) implies that a n does not depend on the choice of ν , and we call a n the Fourier coefficient of f at n .The spaces M ( N ) and S ( N ) are also equipped with an action of pairwise commutingdiagonalizable Hecke operators T n indexed by the nonzero ideals n of Z F . For example, givena prime p (cid:45) N and a totally positive generator p of p we have(2.6) ( T p f )( z ) = N ( p ) f ( pz ) + 1 N ( p ) (cid:88) a ∈ F p f (cid:18) z + ap (cid:19) , where F p = Z F / p is the residue field of p ; this definition is indeed independent of the choiceof generator p . Using the notation (2.4), we can equivalently write(2.7) ( T p f )( z ) = (cid:88) a ∈ P ( F p ) ( f | π a )( z )where π ∞ = (cid:18) p
00 1 (cid:19) and π a = (cid:18) a p (cid:19) for a ∈ F p .If f ∈ S ( N ) is an eigenform, normalized so that a (1) = 1, then T n f = a n f , and eacheigenvalue a n is an algebraic integer which lies in the number field E = Q ( { a n } ) ⊆ C (see himura [58, Section 2]) generated by the Fourier coefficients of f . We again have notions of oldforms and newforms , analogously defined (so that a newform is in particular a normalizedeigenform).Associated to an eigenform f ∈ S ( N ) we have an L -function L ( f, s ) = (cid:88) n a n N n s and l -adic Galois representations ρ f, l : Gal( F /F ) → GL ( Z F, l )for primes l of Z F such that, for any prime p (cid:45) lN , we haveTr( ρ f, l (Frob p )) = a p ( f ) and det( ρ f, l (Frob p )) = N p . Each of these is determined by the Hecke eigenvalues a n of f , so we are again content tocompute S ( N ) as a Hecke module.We are now ready to state the first version of our main result. Theorem 2.8 (Demb´el´e [15], Greenberg-Voight [30]) . There exists an algorithm which, givena totally real field F of strict class number and a nonzero ideal N ⊆ Z F , computes thespace S ( N ) of Hilbert cusp forms of parallel weight and level N over F as a Hecke module. In other words, there exists an explicit finite procedure which takes as input the field F and the ideal N ⊆ Z F encoded in bits (in the usual way, see e.g. Cohen [7]), and outputs afinite set of sequences ( a p ( f )) p encoding the Hecke eigenvalues for each cusp form constituent f in S ( N ), where a p ( f ) ∈ E f ⊆ Q . (This algorithm will produce any finite subsequencein a finite amount of time, but in theory will produce the entire sequence if it is left to runforever.) Alternatively, this algorithm can simply output matrices for the Hecke operators T p ; one recovers the constituent forms using linear algebra. Example . Let F = Q ( √ Z F = Z [ w ] where w = (1+ √ / w − w − N = (3 w − ⊆ Z F ; we have N ( N ) = 229 is prime. We compute that dim S ( N ) = 4.There are 2 Hecke irreducible subspaces of dimensions 1 and 3, corresponding to newforms f and g (and its Galois conjugates). We have the following table of eigenvalues; we write p = ( p ) for p ∈ Z F . p (2) ( w + 2) (3) ( w + 3) ( w − N p a p ( f ) − − − − a p ( g ) t t − t + 1 − t + 2 t + 2 t − t − − t + 8 t + 1Here, the element t ∈ Q satisfies t − t − t +1 = 0 and E = Q ( t ) is an S -field of discriminant148.Recall that in the method of modular symbols, a cusp form f ∈ S ( N ) corresponds to aholomorphic differential (1-)form (2 πi ) f ( z ) dz on X ( N ) and so, by the theorem of Eichler-Shimura, arises naturally in the space H ( X ( N ) , C ). In a similar way, a Hilbert cusp form f ∈ S ( N ) gives rise to a holomorphic differential n -form (2 πi ) n f ( z , . . . , z n ) dz · · · dz n onthe Hilbert modular variety X ( N ), the desingularization of the compact space Γ ( N ) \ ( H n ) ∗ where ( H n ) ∗ = H n ∪ P ( F ). But now X ( N ) is an algebraic variety of complex dimension n nd f arises in the cohomology group H n ( X ( N ) , C ). Computing with higher dimensionalvarieties (and higher degree cohomology groups) is not an easy task! So we seek an alternativeapproach.Langlands functoriality predicts that S ( N ) as a Hecke module occurs in the cohomologyof other “modular” varieties as well. This functoriality was already evident by the factthat both modular symbols and their quaternionic variant, Brandt matrices, can be usedto compute the classical space S ( N ). In our situation, this functoriality is known as theJacquet-Langlands correspondence, which ultimately will allow us to work with varieties ofcomplex dimension 1 or 0 by considering twisted forms of GL over F arising from quaternionalgebras. In dimension 1, we will arrive at an algorithm which works in the cohomology ofa Shimura curve, analogous to a modular curve, and thereby give a kind of analogue ofmodular symbols; in dimension 0, we generalize Brandt matrices by working with thetaseries on (totally definite) quaternion orders.3. Quaternionic modular forms
In this section, we define modular forms on quaternion algebras; our main reference isHida [34]. We retain the notation of the previous section; in particular, F is a totally realfield of degree [ F : Q ] = n with ring of integers Z F .A quaternion algebra B over F is a central simple algebra of dimension 4. Equivalently, aquaternion algebra B over F is an F -algebra generated by elements i, j satisfying(3.1) i = a, j = b, and ji = − ij for some a, b ∈ F × ; we denote such an algebra B = (cid:18) a, bF (cid:19) . For more information aboutquaternion algebras, see Vign´eras [66].Let B be a quaternion algebra over F . Then B has a unique involution : B → B called conjugation such that xx ∈ F for all x ∈ B ; we define the reduced norm of x to benrd( x ) = xx . For B = (cid:18) a, bF (cid:19) as in (3.1) and x = u + vi + zj + wij ∈ B , we have x = u − ( vi + zj + wij ) and nrd( x ) = u − av − bz + abw . A Z F - lattice of B is a finitely generated Z F -submodule I of B such that F I = B . An order O of B is a Z F -lattice which is also a subring of B . A maximal order of B is an orderwhich is not properly contained in any other order. Let O (1) ⊆ B be a maximal order in B .A right fractional O -ideal is a Z F -lattice I such that its right order O R ( I ) = { x ∈ B : xI ⊆ I } is equal to O ; left ideals are defined analogously.Let K ⊃ F be a field containing F . Then B K = B ⊗ F K is a quaternion algebra over K ,and we say K splits B if B K ∼ = M ( K ).Let v be a noncomplex place of F , and let F v denote the completion of F at v . Thenthere is a unique quaternion algebra over F v which is a division ring, up to isomorphism.We say B is unramified (or split ) at v if F v splits B , otherwise we say B is ramified at v .The set S of ramified places of B is a finite set of even cardinality which characterizes B upto isomorphism, and conversely given any such set S there is a quaternion algebra over B ramified exactly at the places in S . We define the discriminant D of B to be the ideal of Z F given by the product of all finite ramified places of B . et N ⊆ Z F be an ideal which is coprime to the discriminant D . Then there is anisomorphism ι N : O (1) (cid:44) → O (1) ⊗ Z F Z F, N ∼ = M ( Z F, N )where Z F, N denotes the completion of Z F at N . Let O ( N ) = { x ∈ O (1) : ι N ( x ) is upper triangular modulo N } ;the order O ( N ) is called an Eichler order of level N . We will abbreviate O = O ( N ).We number the real places v , . . . , v n of F so that B is split at v , . . . , v r and ramified at v r +1 , . . . , v n , so that B ⊗ Q R ∼ = M ( R ) r × H n − r where H = (cid:18) − , − R (cid:19) is the division ring of Hamiltonians. If B is ramified at all real places(i.e. r = 0) then we say that B is (totally) definite , and otherwise we say B is indefinite .The arithmetic properties of the algebra B and its forms are quite different according as B is definite or indefinite, and so we consider these two cases separately. Using an adeliclanguage, one can treat them more uniformly (though to some extent this merely repackagesthe difference)—we refer to § B is indefinite, so that r >
0. The case B ∼ = M ( Q ) corresponds tothe classical case of elliptic modular forms; this was treated in §
1, so we assume B (cid:54)∼ = M ( Q ).Let ι ∞ : B (cid:44) → M ( R ) r denote the map corresponding to the split embeddings v , . . . , v r . Then the group B × + = { γ ∈ B × : det γ i = (nrd γ ) i > i = 1 , . . . , r } acts on H r by coordinatewise linear fractional transformations. Let O × + = O × ∩ B × + . Under the assumption that F has strict class number 1, which we maintain, we have O × + = Z × F O × where O × = { γ ∈ O : nrd( γ ) = 1 } . LetΓ = Γ B ( N ) = ι ∞ ( O × + ) ⊆ GL +2 ( R ) r . and let PΓ = Γ / Z × F . Then PΓ is a discrete subgroup of PGL +2 ( R ) r which can be identifiedwith PΓ ∼ = ι ∞ ( O × ) / {± } ⊆ PSL ( R ) . Definition 3.2.
Let B be indefinite. A quaternionic modular form for B of parallel weight and level N is a holomorphic function f : H r → C such that (3.3) f ( γz ) = f (cid:18) a z + b c z + d , . . . , a r z r + b r c r z r + d r (cid:19) = (cid:32) r (cid:89) i =1 ( c i z i + d i ) det γ i (cid:33) f ( z ) for all γ ∈ Γ B ( N ) . nalogous to (2.4), we define(3.4) ( f | γ )( z ) = f ( γz ) r (cid:89) i =1 det γ i j ( γ i , z ) for f : H r → C and γ ∈ B × + ; then (3.3) is equivalent to ( f | γ )( z ) = f ( z ) for all γ ∈ Γ B ( N ).We denote by M B ( N ) the space of quaternionic modular forms for B of parallel weight 2and level N , a finite-dimensional C -vector space.A quaternionic modular form for B = M ( F ) is exactly a Hilbert modular form over F ;our presentation in these three sections has been consciously redundant so as to emphasizethis similarity. (We could recover the definition of cusp forms given in Section 1 if we alsoimpose the condition that the form vanish at the cusps.) As we will see later, this similarityis less apparent when the general and more technical theory is exposited.The Hecke operators are defined on M B ( N ) following their definition in (2.7). Let p be aprime of Z F with p (cid:45) N , and let p be a totally positive generator of p . DefineΘ( p ) = O × + \ { π ∈ O + : nrd( π ) Z F = p } = O × + \ { π ∈ O + : nrd( π ) = p } . The set Θ( p ) has cardinality N p + 1. The Hecke operator T p is then given by(3.5) ( T p f )( z ) = (cid:88) π ∈ Θ( p ) ( f | π )( z ) . The set Θ( p ) admits an explicit description as follows. As above, let F p = Z F / p be theresidue field of p , and let ι p : O (cid:44) → M ( Z F, p ) be a splitting. Then the set Θ( p ) is in bijectionwith the set of left ideals of O by π (cid:55)→ O π . This set of left ideals is in bijection [41, Lemma6.2] with the set P ( F p ): explicitly, given the splitting ι p , the left ideal corresponding to a = ( x : y ) ∈ P ( F p ) is(3.6) J a = O ι − p (cid:18) x y (cid:19) + O p . By strong approximation [66, Th´eor`eme III.4.3], each of the ideals J a is principal, so J a = O π a with nrd( π a ) = p for all a ∈ P ( F p ). Therefore, we have Θ( p ) = { π a : a ∈ P ( F p ) } .This definition reduces to the one given in (2.7) for Hilbert modular forms with the choices π ∞ = (cid:18) p
00 1 (cid:19) and π a = (cid:18) a p (cid:19) for a ∈ F p .Having treated Hilbert modular forms in the previous section, now suppose that B (cid:54)∼ =M ( F ), or equivalently that B is a division ring. Then a modular form is vacuously acusp form as there are no cusps! We then refer to quaternionic modular forms equallywell as quaternionic cusp forms and let S B ( N ) = M B ( N ). Here, a cusp form f gives aholomorphic differential r -form (2 πi ) r f ( z , . . . , z r ) dz · · · dz r on the associated quaternionicShimura variety X B ( N ) = Γ B ( N ) \H r , a complex variety of dimension r .The important case for us will be when r = 1. Then Γ B ( N ) ⊆ PGL +2 ( R ) acts on the upperhalf-plane and the quotient Γ B ( N ) \H can be given the structure of a Riemann surface, knownas a Shimura curve . In this simple case, a cusp form for B is simply a holomorphic map f : H → C such that f ( γz ) = ( c z + d ) f ( z ) for all γ ∈ Γ B ( N ), where γ = v ( γ ) = (cid:18) a b c d (cid:19) and v is the unique split real place of F . ext, suppose that B is definite, so that r = 0. Recall that O = O ( N ) ⊆ O (1) isan Eichler order of level N . A right fractional O -ideal is invertible if there exists a leftfractional O -ideal I − such that I − I = O , or equivalently if I is locally principal , i.e., foreach (finite) prime ideal p of Z F , the ideal I p is a principal right O p -ideal. If I is invertible,then necessarily I − = { x ∈ B : xI ⊆ O} .Let I, J be invertible right fractional O -ideals. We say that I and J are in the same rightideal class (or are isomorphic ) if there exists an x ∈ B × such that I = xJ , or equivalently if I and J are isomorphic as right O -modules. We write [ I ] for the equivalence class of I underthis relation and denote the set of invertible right O -ideal classes by Cl O . The set Cl O isfinite and H = O is independent of the choice of Eichler order O = O ( N ) of level N . Definition 3.7.
Let B be definite. A quaternionic modular form for B of parallel weight and level N is a map f : Cl O ( N ) → C . The space of quaternionic modular forms M B ( N ) is obviously a C -vector space of dimen-sion equal to H .A modular form for B which is orthogonal to the (1-dimensional) subspace of constantfunctions is called a cusp form for B ; the space of such forms is denoted S B ( N ).The Hecke operators are defined on M B ( N ) as follows. Let p be a prime ideal of Z F with p (cid:45) N . For a right O -ideal I with nrd( I ) coprime to p , the Hecke operator T p is given by(3.8) ( T p f )([ I ]) = (cid:88) J ⊆ I nrd( JI − )= p f ([ J ]) , the sum over all invertible right O -ideals J ⊆ I such that nrd( J ) = p nrd( I ). As in (3.5),this sum is also naturally over P ( F p ), indexing the ideals of norm index p . This definitiondoes not depend on the choice of representative I in its ideal class and extends by linearityto all of S B ( N ).Consequent to the definitions in the previous paragraphs, we may now consider the Heckemodules of quaternionic cusp forms over F for the different quaternion algebras B over F . These spaces are related to each other, and thus to spaces of Hilbert modular forms,according to their arithmetic invariants by the Jacquet-Langlands correspondence as follows.
Theorem 3.9 (Eichler-Shimizu-Jacquet-Langlands) . Let B be a quaternion algebra over F of discriminant D and let N be an ideal coprime to D . Then there is an injective map ofHecke modules S B ( N ) (cid:44) → S ( DN ) whose image consists of those Hilbert cusp forms which are new at all primes p | D .Proof. See Jacquet and Langlands [36, Chap. XVI], Gelbart and Jacquet [27, §
8] and workof Hida [32]; another useful reference is Hida [33, Proposition 2.12] who deduces Theorem3.9 from the representation theoretic results of Jacquet and Langlands. (cid:3)
Theorem 3.9 yields an isomorphism S B ( N ) ∼ = S ( N )when the quaternion algebra B has discriminant D = (1). Since a quaternion algebra mustbe ramified at an even number of places, when n = [ F : Q ] is even we can achieve this for he definite quaternion algebra B which is ramified at exactly the real places of F (and nofinite place). When n is odd, the simplest choice is to instead take B to be ramified at allbut one real place of F (and still no finite place), and hence B is indefinite (and g = 1). Remark . Note that in general a space of newforms can be realized as a Hecke mod-ule inside many different spaces of quaternionic cusp forms. Indeed, for any factorization M = DN with D squarefree and N coprime to D , we consider a quaternion algebra B ofdiscriminant D (ramified at either all or all but one real place of F ) and thereby realize S B ( N ) ∼ = S ( M ) D -new . For example, if p, q are (rational) primes, then the space S ( pq ) new of classical newforms can, after splitting off old subspaces, be computed using an indefinitequaternion algebra of discriminant 1 or pq (corresponding to a modular curve or a Shimuracurve, respectively) or a definite quaternion algebra of discriminant p or q .Our main conclusion from this section is that to compute spaces of Hilbert cusp forms itsuffices to compute instead spaces of quaternionic cusp forms. The explicit description of S B ( N ) as a Hecke module varies according as if B is definite or indefinite.4. Definite method
In this section, we discuss a method for computing Hilbert modular forms using a definitequaternion algebra B . We continue with our notation and our assumption that F has strictclass number 1. We accordingly call the method in this section the definite method : it isa generalization of the method of Brandt matrices mentioned briefly in § F = Q , but the first explicit algorithm was givenby Socrates and Whitehouse [60].Let I , . . . , I H be a set of representative right ideals for Cl O , with H = O . As vectorspaces, we have simply that M B ( N ) = Map(Cl O , C ) ∼ = (cid:76) Hi =1 C I i , associating to each ideal (class) its characteristic function. Let O i = O L ( I i ) be the left orderof I i and let e i = O × i / Z × F ).The action of the Hecke operators is defined by (3.8): we define the p th- Brandt matrix T ( p ) for O to be the matrix whose ( i, j )th entry is equal to(4.1) b ( p ) i,j = { J ⊆ I j : nrd( J I − j ) = p and [ J ] = [ I i ] } ∈ Z . The Brandt matrix T ( p ) is an H × H -matrix with integral entries such that the sum of theentries in each column is equal to N p + 1. The Hecke operator T p then acts by T ( p ) on (cid:76) i C I i (on the right), identifying an ideal class with its characteristic function.The Brandt matrix is just a compact way of writing down the adjacency matrix of thegraph with vertices X = Cl O where there is a directed edge from I i to each ideal class whichrepresents an ideal of index N p in I i . Indeed, consider the graph whose vertices are right O -ideals of norm a power of p and draw a directed edge from I to J if nrd( J I − ) = p . Thenthis graph is a k -regular tree with N p + 1 edges leaving each vertex. The above adjacencymatrix is obtained by taking the quotient of this graph by identifying two ideals if they arein the same ideal class.Alternatively, we may give an expression for the Brandt matrices in terms of elementsinstead of ideals. A containment J ⊆ I j of right O -ideals with [ I i ] = [ J ] corresponds to n element x ∈ J I − i ⊂ I j I − i via J = xI i , and we have nrd( J I − ) = p if and only ifnrd( x ) Z F = p .Writing J I − i = x O i , we see that x is unique up to multiplication on the right by O × i .We have O × i = ( O i ) × Z × F and ( O i ) × ∩ Z × F = {± } , so 2 e i = O i ) × . To eliminate thecontribution of the factor Z × F , we normalize as follows: let p be a totally positive generatorfor p and similarly q i for nrd( I i ) for i = 1 , . . . , H . Then x ∈ I j I − i can be chosen so thatnrd( x )( q j /q i ) = p and is unique up to multiplication by ( O i ) × .(4.2) b ( p ) i,j = 12 e i (cid:26) x ∈ I j I − i : nrd( x ) q j q i = p (cid:27) . The advantage of the expression (4.2) is that it can be expressed simply in terms of aquadratic form. Since B is definite, the space B (cid:44) → B ⊗ Q R ∼ = H n ∼ = R n comes equippedwith the positive definite quadratic form Tr nrd : B → R . If J is a Z F -lattice, then J ∼ = Z n embeds as a Euclidean lattice J (cid:44) → R n with respect to this quadratic form. It follows thatone can compute b ( p ) i,j by computing all elements x ∈ I i I − j such that Tr( q j /q i ) nrd( x ) ≤ Tr p , a finite set.Before giving references for the technical details about how the Brandt matrices above arecomputed explicitly, we pause to give three examples. Example . Consider the quaternion algebra B = (cid:18) − , − Q (cid:19) , so that B is generated by i, j subject to i = − j = −
23 and ji = − ij . We have the maximal order O = O (1) = Z ⊕ Z i ⊕ Z k ⊕ Z ik where k = ( j + 1) /
2. We consider the prime p = 2; we have an embedding O (cid:44) → M ( Z ) where i, k (cid:55)→ (cid:18) (cid:19) , (cid:18) (cid:19) (mod 2).We begin by computing the ideal classes in O . We start with C = [ O ]. We have 3 idealsof norm 2, namely I (0:1) = 2 O + ik O , I (1:1) = 2 O + ( i + 1) k O , and I (1:0) = 2 O + k O . If one ofthese ideals is principal, then it is generated by an element of reduced norm 2. The reducednorm gives a quadratic form nrd : O → Z x + yi + zk + wik (cid:55)→ x + xz + y + yw + 6 z + 6 w We see immediately that nrd( x + yi + zk + wik ) = 2 if and only if z = w = 0 and x = y = ± I (1:1) = ( i + 1) O is principal but I (1:0) and I (0:1) are not. Note also that wefind 2 e = 4 solutions matching (4.2). We notice, however, that iI (1:0) = I (0:1) , so we havejust a second ideal class C = [ I (0:1) ].Now of the two ideals contained in I (0:1) of norm 4, we have I (4)(0:1) = 4 O + i ( k + 2) O belongsto C whereas I (4)(2:1) = 4 O + (2 i + 2 k + ik ) O gives rise to a new ideal class C = [ I (2:1) ]. Ifwe continue, each of the two ideals contained in I (4)(0:1) of norm 8 belong to C , and it followsthat H = O = 3.From this computation, we have also computed the Brandt matrix T (2) = .Indeed, the first column encodes the fact that of the three right O -ideals of reduced norm , there is one which is principal and hence belongs to C and two that belong to C . Wethink of this matrix as acting on the right on row vectors.The characteristic polynomial of T (2) factors as ( x − x + x − , ,
1) isan eigenvector with eigenvalue 3 which generates the space of constant functions and givesrise to the Eisenstein series having eigenvalues a p = p + 1 for all primes p (cid:54) = 23. The spaceof cusp forms S B (1) is correspondingly of dimension 2 and is irreducible as a Hecke module.The Hecke module S B (1) can be explicitly identified with S (23) using theta correspondence.For example, the series θ ( q ) = (cid:88) γ ∈O q nrd( γ ) = (cid:88) x,y,z,w ∈ Z q x + xz + y + yw +6 z +6 w = 1 + 4 q + 4 q + 4 q + 8 q + . . . is the q -expansion of a modular form of level 23 and weight 2 and corresponds to (thecharacteristic of) C . For more details, we refer to Pizer [48, Theorem 2.29], where the firstcomputer algorithm for computing Brandt matrices over Q is also described.Now we give an example over a quadratic field. Example . Let F = Q ( √ Z F = Z [ w ] where w = (1+ √ / w − w − F ; let N = (3 w + 7) Z F be one of the primes above it. We considerthe (Hamilton) quaternion algebra B = (cid:18) − , − F (cid:19) over F of discriminant D = (1). Wehave the maximal order O (1) = Z F ⊕ i Z F ⊕ k Z F ⊕ ik Z F , where k = (1 + w ) + wi + j , and the Eichler order O ⊆ O (1) of level N given by O = Z F ⊕ (3 w + 7) i Z F ⊕ ( − i + k ) Z F ⊕ ( w + 20 i + ik ) Z F . The class number of O is H = 3. The following ideals give a set of representatives forCl O : we take I = O , I = 2 O + (( w + 2) − (2 w + 2) i + ( − w ) ik ) O = 2 Z F ⊕ (6 w + 14) i Z F ⊕ (( w + 1) + ( − w + 5) i − k ) Z F ⊕ (1 − i + wik ) Z F and I = 2 O + (( w + 1) + (1 − w ) i + (2 − w ) k ) O .We compute the orders e i = O i / Z × F ) as e = 2, e = 5 and e = 3. For example, theelement u = (2 w ) i − k − wik ∈ O satisfies the equation u + 1 = 0, and so yields an elementof order 2 in O × / Z × F . It follows that none of these orders are isomorphic (i.e., conjugate) in B .The first few Brandt matrices are: T (2) = , T ( √
5) = , T (3) = ,T ( w + 3) = , T ( w −
4) = . We note that N ( w + 3) = N ( w −
4) = 11. or example, the first column of the matrix T (2) records the fact that of the 5 = N (2) + 1right O -ideals of norm (2), there is exactly one which is principal, two are isomorphic to I and the other two are isomorphic to I .The space S B ( N ) of cusp forms is an irreducible 2-dimensional Hecke module, representedby a constituent form f with corresponding eigenvector (2 , w, − w − T f = Z [ a p ] restricted to f is equal to T f = Z [ w ], by coincidence. Wehave the following table of eigenvalues for f . p (2) ( w + 2) (3) ( w + 3) ( w − N p a p ( f ) 2 w − − w + 1 − w − w − − w For further discussion of the geometric objects which arise from this computation, see thediscussion in Section 6.Finally, one interesting example.
Example . Let F = Q ( √
15) and let N = (5 , √ S ( N ) such that no single Hecke eigenvaluegenerates the entire field E of Hecke eigenvalues. E gal = Q ( √ , i, √ , √ u ) E = Q ( √ , i, √ u ) (cid:105)(cid:105)(cid:105)(cid:105)(cid:105)(cid:105)(cid:105)(cid:105)(cid:105) (cid:86)(cid:86)(cid:86)(cid:86)(cid:86)(cid:86)(cid:86)(cid:86) Q ( i, √ (cid:111)(cid:111)(cid:111)(cid:111)(cid:111) (cid:85)(cid:85)(cid:85)(cid:85)(cid:85)(cid:85)(cid:85)(cid:85)(cid:85)(cid:85)(cid:85) Q ( √ , √ u ) Q ( √ , √− u ) (cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104) Q ( √− (cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80) Q ( i ) Q ( √ (cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104) Q Here, u = (5 + √ / E gal is the Galois closure of E . Each Hecke eigenvalue a p forthis form f generates a proper subfields of E . (There are also examples of this phenomenonover Q , and they are related to the phenomenon of inner twists; this was analyzed over Q by Koo, Stein, and Wiese [44].)With these examples in hand, we now give an overview of how these computations areperformed; for more detail, see work of the first author [15]. It is clear we need severalalgorithms to compute Brandt matrices. First, we need a basic suite of algorithms forworking with quaternion orders and ideals; these are discussed in work of Kirschmer and thesecond author [41, Section 1], and build on basic tools for number rings by Cohen [7]. Aspart of this suite, we need a method to compute a maximal order, which is covered by workof the second author [70]. Next, we need to compute a set of representatives for Cl O andto test if two right O -ideals are isomorphic: this is covered by Kirschmer and the secondauthor [41, Sections 6–7], including a runtime analysis. To compute a set of representatives,we use direct enumeration in the tree as in Example 4.3 and a mass formula due to Eichleras an additional termination criterion. To test for isomorphism, we use lattice methods tofind short vectors with respect to the quadratic form Tr nrd. n this method, to compute with level N one must compute a set of representatives Cl O =Cl O ( N ) anew. The first author has given an improvement on this basic algorithm, allowingus to work always with ideal classes Cl O (1) belonging to the maximal order at the smallprice of a more complicated description of the Hecke module. The proof of correctness forthis method is best explained in the adelic language, so we refer to Section 8 for more detail.Let I , . . . , I h be representatives for Cl O (1) and suppose that N is relatively prime tonrd( I i ) for each i —this is made possible by strong approximation. As before, let p be a primeof Z F with p (cid:45) D . Let p be a totally positive generator for p , and let q i be a totally positivegenerator for nrd( I i ). By our notation, we have O L ( I i ) = O (1) i . Let Γ i = O (1) × i / Z × F . Foreach i, j , consider the setΘ( p ) i,j = Γ j (cid:15) (cid:26) x ∈ I j I − i : nrd( x ) q i q j = p (cid:27) where Γ j acts by the identification O (1) × i / Z × F = O (1) × / {± } . Via a splitting isomorphism ι N : O (1) (cid:44) → O (1) ⊗ Z F, N ∼ = M ( Z F, N ) , the group O (1) × acts on P ( Z F / N ) and since O (1) ⊗ Z F, N ∼ = O (1) i ⊗ Z F, N for each i (sincenrd( I i ) is prime to N ), the group Γ i = O (1) × i / Z × F similarly acts on P ( Z F / N ).We then define a Hecke module structure on (cid:76) hi =1 C [Γ i \ P ( Z F / N )] via the map C [Γ j \ P ( Z F / N )] → C [Γ i \ P ( Z F / N )]Γ j x (cid:55)→ (cid:88) γ ∈ Θ( p ) i,j Γ i ( γx )on each component. It is a nontrivial but nevertheless routine calculation that this Heckemodule is isomorphic to the Hecke module M B ( N ) defined by the Brandt matrices at thebeginning of this section. Example . We keep the notations of Example 4.4. The quaternion algebra B has classnumber 1, thus the maximal order O (1) is unique up to conjugation. We have Γ = O (1) × / Z × F has cardinality 60. We consider the splitting map¯ ι N : O (1) → O (1) ⊗ Z F ( Z F / N ) ∼ = M ( Z F / N )given by ¯ ι N ( i ) = (cid:18)
11 037 50 (cid:19) , ¯ ι N ( k ) = (cid:18)
47 5818 33 (cid:19) , ¯ ι N ( ik ) = (cid:18)
29 2816 14 (cid:19) . (One directly verifies that ¯ ι N ( ik ) = ¯ ι N ( i )¯ ι N ( k ).) We let Γ act on P ( Z F / N ) on the left via¯ ι N . By the above discussion, we have M B ( N ) ∼ = C [Γ \ P ( Z F / N )] . The action of Γ on P ( Z F / N ) has three orbits which are represented by x = (1 : 0), x = (1 : 1) and x = (23 : 1) whose stabilizers have cardinality e = 2, e = 5 and e = 3.Thus M B ( N ) is a free module generated by the orbits Γ x , Γ x and Γ x . Writing down theHecke action in that basis, we obtain the same Hecke operators as in Example 4.4. Remark . The approach presented above has some advantages over the usual definitionof Brandt matrices above. First of all, it is better suited for working with more generallevel structures, such as those that do not come from Eichler orders. For example, adding a haracter in this context is quite transparent. Secondly, when working over the same numberfield, a substantial amount of the required data can be precomputed and reused as the levelvaries, and consequently one gains significantly in the efficiency of the computation.5. Indefinite method
In this section, we discuss a method for computing Hilbert modular forms using a indefinitequaternion algebra B with r = 1. We accordingly call our method the indefinite method .The method is due to Greenberg and the second author [30]. We seek to generalize themethod of modular symbols by working with (co)homology. We continue to suppose that F has strict class number 1, and we assume that B (cid:54)∼ = M ( Q ) for uniformity of presentation.Recall that in this case we have defined a group Γ = Γ B ( N ) ⊆ GL +2 ( R ) such thatPΓ = Γ / Z × F ⊆ PGL +2 ( R ) is discrete; the quotient X = X B ( N ) = Γ \H is a Shimura curveand quaternionic cusp forms on B correspond to holomorphic differential 1-forms on X . Inte-gration gives a Hecke-equivariant isomorphism which is the analogue of the Eichler-Shimuratheorem, namely S B ( N ) ⊕ S B ( N ) ∼ −→ H ( X B ( N ) , C ) . We recover S B ( N ) by taking the +-eigenspace for complex conjugation on both sides. Puttingthis together with the Jacquet-Langlands correspondence, we have S ( DN ) D − new ∼ = S B ( N ) ∼ = H ( X B ( N ) , C ) + . We have the identifications H ( X, C ) = H ( X B ( N ) , C ) ∼ = H (Γ B ( N ) , C ) = Hom(Γ B ( N ) , C ) = Hom(Γ , C ) . To complete this description, we must relate the action of the Hecke operators. Let p (cid:45) DN be prime and let p be a totally positive generator of p . As in (3.5), let(5.1) Θ( p ) = O × + \ { π ∈ O + : nrd( π ) = p } and choose representatives π a for these orbits labeled by a ∈ P ( F p ). Then any γ ∈ Γ byright multiplication permutes the elements of Θ( p ), and hence there is a unique permutation γ ∗ of P ( F p ) such that for all a ∈ P ( F p ) we have π a γ = δ a π γ ∗ a with δ a ∈ Γ. For f ∈ H (Γ , C ) = Hom(Γ , C ), we then define(5.2) ( T p f )( γ ) = (cid:88) π a ∈ Θ( p ) f ( δ a ) . In a similar way, we compute the action of complex conjugation T ∞ via the relation( T ∞ f )( γ ) = f ( δ )where µγ = δµ and µ ∈ O × \ O × .We begin with two examples, to illustrate the objects and methods involved. Example . Let F = Q ( √ Z F = Z [ w ] is the ring of integers of F where w =(1 + √ / w − w − F has strict class number 1 and u = w + 2 isa fundamental unit of F . et B = (cid:18) − , uF (cid:19) , so B is generated over F by i, j subject to i = − j = u , and ji = − ij . The algebra B is ramified at the prime ideal 2 Z F and the nonidentity real placeof B (taking √ (cid:55)→ −√
29) and no other place. The identity real place gives an embedding ι ∞ : B (cid:44) → B ⊗ F R ∼ = M ( R ) i, j (cid:55)→ (cid:18) − (cid:19) , (cid:18) √ u −√ u (cid:19) = (cid:18) . ... − . ... (cid:19) Let O = O (1) = Z F ⊕ Z F i ⊕ Z F j ⊕ Z F k where k = (1 + i )( w + 1 + j ) /
2. Then O is a maximal order of B with discriminant D = 2 Z F .Let Γ = ι ∞ ( O × ) ⊆ SL ( R ) be as above and let X = X (1) C = Γ \H be the associated Shimuracurve.Although they are not an intrinsic part of our algorithm, we mention that the area of X (normalized so that an ideal triangle has area 1 /
2) is given by A = 4(2 π ) n d / F ζ F (2)Φ( D ) = 4(2 π ) √ ζ F (2)(4 −
1) = 32where Φ( D ) = (cid:81) p | D ( N p + 1), and the genus of X is given by the Riemann-Hurwitz formulaas A = 2 g − (cid:88) q e q (cid:18) − q (cid:19) where e q is the number of elliptic cycles of order q ∈ Z ≥ in Γ. An explicit formula for e q given in terms of class numbers and Legendre symbols yields e = 3 and e q = 0 for q > g − g = 1. For more details on these formulas and further introduction, seework of the second author [68] and the references given there.Next, we compute a fundamental domain for Γ, yielding a presentation for Γ; we considerthis as a black box for now. The domain, displayed in the unit disc, is as follows. We obtain the presentationΓ ∼ = (cid:104) γ, γ (cid:48) , δ , δ , δ : δ = δ = δ = [ γ, γ (cid:48) ] δ δ δ = 1 (cid:105) where γ = − w − i − j + kγ (cid:48) = 2 + 2 i + ( w − j − ( w − kδ = (2 w + 2) + wi + j + 4 kδ = iδ = ( w + 1) + (2 w + 3) i − j − k. The above method gives the isomorphisms of Hecke modules S (2 Z F ) new ∼ = S B (1) = { f : H → C : f ( gz ) d ( gz ) = f ( z ) dz for all g ∈ Γ } ∼ = H (Γ , C ) + . We compute that H (Γ , C ) + ∼ = Hom(Γ , C ) + = C f where f is the characteristic function of γ , i.e., f ( γ ) = 1, f ( γ (cid:48) ) = 0 and f ( δ i ) = 0 for i = 1 , , T p for p odd according the definition (5.2). Let p =( w + 1) Z F . Then N p = p = 5. We compute the action of T p on H (Γ , C ) + given by T p f = a p ( f ) f . The Hecke operators act as a sum over p + 1 left ideals of reduced norm p ,indexed by P ( F ). Let ι p : O (cid:44) → M ( Z F, p ) ∼ = M ( Z ) i, j, ij (cid:55)→ (cid:18) − (cid:19) , (cid:18) − (cid:19) , (cid:18) − − (cid:19) (mod 5) . Specifying the images modulo p gives them uniquely, as they lift to M ( Z ) using Hensel’slemma. (Note that j = u ≡ w + 1).) et J ∞ , J , . . . , J be defined by J a = J ( x : y ) = O ι − p (cid:18) x y (cid:19) + p O as in (3.6). Then J a = O π a are principal left O -ideals by strong approximation. For example, J = O ( i − ij ) + ( w + 2) O = O π where π = ( − w + 3) + wi + j + ij .We compute the Hecke operators as in (5.2). For f : Γ → C and γ ∈ Γ, we computeelements δ a ∈ Γ indexed by a ∈ P ( F p ) and γ ∗ a permutation of P ( F p ) such that π a γ = δ a π γ ∗ a for all a ∈ P ( F p ); then ( T p f )( γ ) = (cid:88) a ∈ P ( F p ) f ( δ a ) . The contribution to the sum for f simply counts the number of occurrences of γ in theproduct π a γπ γ ∗ a = δ a ∈ Γ. Carrying out this computation for various primes, we obtain thefollowing table. N p a p ( f ) 1 − − − − − N p + 1 − a ( p ) 5 10 5 15 30 20 55 50 60 80 70Here we list only the norm of the prime as the eigenvalue does not depend on the choiceof prime p of the given norm. This suggests that f corresponds to a base change of a formfrom Q .So we look through tables of elliptic curves over Q whose conductor is divisible only by 2and 29. We find the curve E labelled , where 1682 = 2 · , given by E : y + xy = x + x − x − . Let E F denote the base change of E to F . We compute that the twist E (cid:48) F of E F by − u √ E (cid:48) F : y + ( w + 1) xy = x + ( − w + 1) x + ( − w − x + (23 w + 52)has conductor 2 Z F . Since the extension F/ Q is abelian, by base change theorems we knowthat there exists a Hilbert cusp form associated to E (cid:48) F over F which is new of level 2 Z F ,which therefore must be equal to f . This verifies that the Jacobian J (1) of X (1) is isogenousto E (cid:48) F . We verify that E (cid:48) F ( F p ) = N p + 1 − a ( p ) and (as suggested by the table) that E hasa 5-torsion point, ( − , − w −
5) (and consequently so too does J (1)). Example . To illustrate the Jacquet-Langlands correspondence (Theorem 3.9) in action,we return to Example 4.4. Recall F = Q ( √
5) and w = (1 + √ /
2. We find the quaternionalgebra B = (cid:18) w, − (3 w + 7) F (cid:19) which is ramified at N = (3 w + 7) Z F , a prime of norm 61,and one infinite place. The order O = O (1) with Z F -basis 1 , i, k, ik , where k = (( w + 1) + wi + j ) /
2, is maximal. As above, we compute that A ( X ) = 10 and g ( X ) = 2 = dim S B (1) =dim S ( N ) new . Now we have the following fundamental domain: We obtain the presentationΓ(1) ∼ = (cid:104) γ , γ (cid:48) , γ , γ (cid:48) : [ γ , γ (cid:48) ][ γ , γ (cid:48) ] = 1 (cid:105) . (In particular, Γ(1) is a torsion-free group.) We have dim H ( X, C ) = 4, and on the basisof characteristic functions given by γ , γ (cid:48) , γ , γ (cid:48) , the action of complex conjugation is givenby the matrix − − − − : this is computed in a way We thus obtain a basis for H = H ( X, C ) + ∼ = S B (1). Computing Hecke operators as in the previous example, we findas in Example 4.4 that S B (1) is irreducible as a Hecke module, and find that H | T = (cid:18) − − (cid:19) , H | T √ = (cid:18) − −
33 4 (cid:19) , H | T = (cid:18) − − − (cid:19) ,H | T w +3 = (cid:18) − − (cid:19) , H | T w − = (cid:18) − −
11 1 (cid:19) . Happily, the characteristic polynomials of these operators agree with those computed usingthe definite method.We now give an overview of how these computations are performed: for more details, seethe reference by Greenberg and the second author [30]. To compute effectively the systemsof Hecke eigenvalues in the cohomology of a Shimura curve, we need several algorithms.First, we need to compute an explicit finite presentation of Γ with a solution to the wordproblem in Γ, i.e., given δ ∈ Γ, write δ as an explicit word in the generators for Γ. Secondly,we need to compute a generator (with totally positive reduced norm) of a left ideal I ⊆ O .The first of these problems is solved by computing a Dirichlet domain; the second is solvedusing lattice methods. We discuss each of these in turn. et p ∈ H have trivial stabilizer Γ p = { g ∈ Γ : gp = p } = { } . The Dirichlet domain centered at p is the set D ( p ) = { z ∈ H : d ( z, p ) ≤ d ( gz, p ) for all g ∈ Γ } where d denotes the hyperbolic distance. In other words, we pick in every Γ-orbit the closestpoints z to p . The set D ( p ) is a closed, connected, hyperbolically convex fundamental domainfor Γ whose boundary consists of finitely many geodesic segments, called sides , and comesequipped with a side pairing , a partition of the set of sides into pairs s, s ∗ with s ∗ = g ( s ) forsome g ∈ Γ. (We must respect a convention when a side s is fixed by an element of order 2,considering s to be the union of two sides meeting at the fixed point of g .)The second author has proven [67] that there exists an algorithm which computes a Dirich-let domain D for Γ, a side pairing for D , and a finite presentation for Γ with a minimal setof generators together with a solution to the word problem for Γ. This algorithm computes D inside the unit disc D , and we consider now Γ acting on D by a conformal transforma-tion which maps p →
0. We find then that D can be computed as an exterior domain ,namely, the intersection of the exteriors of isometric circles I ( g ) for elements g ∈ Γ, where I ( g ) = { z ∈ C : | cz + d | = 1 } where g = (cid:18) a bc d (cid:19) ∈ SU (1 , p .The solution to the word problem comes from a reduction algorithm that finds elementswhich contribute to the fundamental domain and removes redundant elements. Given afinite subset G ⊂ Γ \ { } we say that γ is G -reduced if for all g ∈ G , we have d ( γ , ≤ d ( gγ , G -reduced element, which wedenote γ (cid:55)→ red G ( γ ): if d ( γ , > d ( gγ ,
0) for some g ∈ G , set γ := gγ and repeat. Whenthe exterior domain of G is a fundamental domain, we have red G ( γ ) = 1 if and only if γ ∈ Γ.This reduction is analogous to the generalized division algorithm in a polynomial ring overa field.
Example . Let F be the (totally real) cubic subfield of Q ( ζ ) with discriminant d F = 169.We have F = Q ( b ) where b + 4 b + b − F has strict class number 1.)The quaternion algebra B = (cid:18) − , bF (cid:19) has discriminant D = (1) and is ramified at 2 ofthe 3 real places of F . We take O to be an Eichler order of level p = ( b + 2), a prime idealof norm 5; explicitly, we have O = Z F ⊕ ( b + 2) i Z F ⊕ b + ( b + 4) i + j Z F ⊕ b + ( b + 4) α + αβ Z F . We compute a fundamental domain for the group Γ = Γ (1)0 ( p ) = ι ∞ ( O ∗ ) / {± } . We take p = 9 / i ∈ H .We first enumerate elements of O by their absolute reduced norm. Of the first 260 ele-ments, we find 29 elements of reduced norm 1. Let G be the set of elements which contribute to the boundary. For each g (cid:54)∈ G , wecompute red G ( g ). Each in fact reduces to 1, so we are left with 8 elements. We next enumerate elements in O moving the center in the direction of the infinite vertex v . We find an enveloper , an element g such that v lies in the interior of I ( g ), and reduceto obtain the following. The domain now has finite area. We now attempt to pair each vertex to construct a sidepairing. v v I ( g ) I ( g ) = I ( g − ) For example, the first vertex v pairs with v , pairing the isometric circle I ( g ) of g with I ( g ). We continue, but find that v does not pair with another vertex. v v v v v v v v v v v v I ( g ) = I ( g − ) Indeed, g ( v ) does not lie in the exterior domain. We compute the reduction red G ( g ; v ),analogously defined. We then obtain a domain with the right area, so we are done! v v v v v v v v v v v v v v v (In order to get an honest side pairing, we agree to the convention that a side I ( g ) whichis fixed by an element g , necessarily of order 2, is in fact the union of two sides which meetat the unique fixed point of g ; the corresponding vertex then appears along such an side.)As to the second problem, that of principalizing ideals, we refer to work of Kirschmer andthe second author [41, § N varies, by changing the coefficient moduleone can work always with the group Γ(1) associated to the maximal order O (1) rather thanrecomputing a fundamental domain for Γ = Γ ( N ) each time. This is a simple applicationof Shapiro’s lemma: the isomorphism H (Γ , C ) ∼ = H (Γ(1) , Coind
Γ(1)Γ C ) , s an isomorphism of Hecke modules: we give Coind Γ(1)Γ C = Hom(Γ(1) / Γ , C ) the naturalstructure as a Γ(1)-module, and for a cochain f ∈ H (Γ , Coind
Γ(1)Γ C ), we define(5.6) ( T p f )( γ ) = (cid:88) π a ∈ Θ( p ) f ( δ a )as in (5.2). We require still that Θ( p ) ⊂ O + ; after enumerating a set of representatives forΓ \ Γ(1), we can simply multiply elements π a ∈ O (1) by the representative for its coset.To conclude, we compare the above method to the method of modular symbols introducedwhen F = Q . The Shimura curves X = X B ( N ) do not have cusps, and so the method ofmodular symbols does not generalize directly. However, the side pairing of a Dirichlet domainfor Γ gives an explicit characterization of the gluing relations which describe X as a Riemannsurface, hence one obtains a complete description for the homology group H ( X, Z ). Pathsare now written { v, γv } for v a vertex on a side paired by γ ∈ G . The analogue of the Manintrick in the context of Shimura curves is played by the solution to the word problem in Γ,and so in some sense this can be seen as a partial extension of the Euclidean algorithm tototally real fields. Our point of view is to work dually with cohomology, but computationallythese are equivalent [30, § Examples
In this section, we provide some detailed examples to illustrate applications of the algo-rithms introduced in the previous sections.Our first motivation comes from the association between Hilbert modular forms andabelian varieties. The Eichler-Shimura construction (see Knapp [38, Chap. XII] or Shimura[59, Chapter 7]) attaches to a classical newform f ( z ) = (cid:80) n a n q n ∈ S ( N ) new an abelian vari-ety A f with several properties. First, A f has dimension equal to [ E f : Q ] where E f = Q ( { a n } )is the field of Fourier coefficients of f and End Q ( A f ) ⊗ Q ∼ = E f . Second, A f has good reduc-tion for all primes p (cid:45) N . Last, we have an equality of L -functions L ( A f , s ) = (cid:89) σ ∈ Hom( E f , C ) L ( f σ , s )where f σ is obtained by letting σ act on the Fourier coefficients of f , so that L ( f σ , s ) = ∞ (cid:88) n =1 a σn n s . The abelian variety A f arises as the quotient of J ( N ) by the ideal of End J ( N ) generatedby T n − a n .The simplest example of this construction is the case of a newform f with rational Fouriercoefficients Q ( { a n } ) = Q . Then A f is an elliptic curve of conductor N obtained analyticallyvia the map X ( N ) → A f τ (cid:55)→ z τ = 2 πi (cid:90) τi ∞ f ( z ) dz = ∞ (cid:88) n =1 a n n e πiτ . he equality of L -functions is equivalent to the statement A f ( F p ) = p +1 − a p for all primes p , and this follows in the work of Eichler and Shimura by a comparison of correspondencesin characteristic p with the Frobenius morphism.The Eichler-Shimura construction extends to the setting of totally real fields in the formof the following conjecture (see e.g. Darmon [11, Section 7.4]). Conjecture 6.1 (Eichler-Shimura) . Let f ∈ S ( N ) new be a Hilbert newform of parallel weight and level N over F . Let E f = Q ( { a n } ) be the field of Fourier coefficients of f . Then thereexists an abelian variety A f of dimension g = [ E f : Q ] defined over F of conductor N g suchthat End F ( A f ) ⊗ Q ∼ = E f and L ( A f , s ) = (cid:89) σ ∈ Hom( E f , C ) L ( f σ , s ) . In particular, if f has rational Fourier coefficients, Conjecture 6.1 predicts that one canassociate to f an elliptic curve A f defined over F with conductor N such that L ( A f , s ) = L ( f, s ).Conjecture 6.1 is known to be true in many cases. It is known if f appears in the coho-mology of a Shimura curve (as in Sections 3 and 5 above; see also Zhang [74]): this holdswhenever n = [ F : Q ] is odd or if N is exactly divisible by a prime p . If f appears in thecohomology of a Shimura curve X and f has rational Fourier coefficients, then when theconjecture holds there is a morphism X → A f and we say that A f is uniformized by X .However, Conjecture 6.1 is not known in complete generality: by the Jacquet-Langlandscorrespondence, one has left exactly those forms f over a field F of even degree n and squarefull level (i.e. if p | N then p | N ), for example N = (1). One expects still that theconjecture is true in this case: Blasius [1], for example, shows that the conjecture is trueunder the hypothesis of the Hodge conjecture.The converse to the Eichler-Shimura construction (Conjecture 6.1) over Q is known as theShimura-Taniyama conjecture. An abelian variety A over a number field F is of GL -type if End F A ⊗ Q is a number field of degree dim A . The Shimura-Taniyama conjecture statesthat given an abelian variety A of GL -type over Q , there exists an integer N ≥ J ( N ) → A . This conjecture is a theorem, a consequence of the proofof Serre’s conjecture by Khare and Wintenberger [39]. In the setting of totally real fields,the analogous conjecture is as follows. Conjecture 6.2 (Shimura-Taniyama) . Let A be an abelian variety of GL -type over a totallyreal field F . Then there exists a Hilbert newform f of parallel weight such that E f ∼ =End F ( A ) ⊗ Q and L ( A, s ) = (cid:89) σ ∈ Hom( E f , C ) L ( f σ , s ) . If both of these conjectures are true, then by Faltings’ isogeny theorem, the abelian variety A in Conjecture 6.2 would be isogenous to the abelian variety A f constructed in Conjecture6.1. An abelian variety of GL -type over a totally real field F which satisfies the conclusionof Shimura-Taniyama conjecture (Conjecture 6.2) is called modular .There has been tremendous progress in adapting the techniques initiated by Wiles [71]—which led to the proof of the Shimura-Taniyama conjecture in the case F = Q —to totally realfields. For example, there are modularity results which imply that wide classes of abelian arieties of GL -type are modular: see for example work of Skinner and Wiles [61, 62],Kisin [42, 43], Snowden [63] and Geraghty [28], and the results below. However, a completeproof of Conjceture 6.2 remains elusive.With these conjectures in mind, we turn to some computational examples. We begin with F = Q ( √
5) and let w = (1 + √ /
2. For some further discussion on this case, see work ofthe first author [14].
Example . We consider the spaces S ( N ) with N ( N ) ≤ F which is ramified at no finite place. We find that dim S ( N ) = 0 for all ideals N of Z F with N ( N ) ≤
30 and dim S ( N ) = 1 for N with norm 31.Now let N be either of the primes above 31. By the Jacquet-Langlands correspondence(Theorem 3.9) and the accompanying discussion, the space S ( N ) can also be computed usingthe indefinite quaternion algebra B ramified at N and one real place. We find agreeably thatthe Shimura curve X B (1) has genus one and thus its Jacobian is an elliptic curve. By asearch, we find the elliptic curve A : y + xy + wy = x − (1 + w ) x of conductor N = (5 + 2 w ) of norm 31.We now show that A is modular. In brief, we verify that the Galois representation ρ A, :Gal( F /F ) → GL ( F ) ⊆ GL ( F ) is irreducible, solvable, and ordinary since A ( F ) = 8(so 3 (cid:45) a ( A ) = 9 + 1 − A is modular by Skinner and Wiles [61]. Since A hasconductor p and the space S ( p ) has dimension 1, it follows by Falting’s isogeny theoremthat A is isogeneous to the Jacobian of the Shimura curve X B (1).In particular, the elliptic curves over F = Q ( √
5) with smallest conductor have primeconductor dividing 31 and are isogenous to A or its Galois conjugate. Example . Let N ⊆ Z F be such that N ( N ) ≤ N (cid:45)
61, then dim S ( N ) new ≤
2: each newform f of conductor N has rational Fouriercoefficients and is either the base change of a classical modular form over Q or associated toan elliptic curve which is uniformized by a Shimura curve.We compute each space in turn using the definite method. Those forms f which have theproperty that a p ( f ) = a p ( f ), where denotes the Galois involution, are candidates to arisefrom base change over Q . For each of these forms f , we find a candidate classical modularform g using the tables of Cremona [9]: such a form will have conductor supported at 5 andthe primes dividing N ( N ). Each curve comes with a Weierstrass equation, and we find aquadratic twist of the base change of this curve to F which has conductor N . Since everyelliptic curve over Q is modular, the base change and its quadratic twist are also modular,and since we exhaust the space this way we are done.For example, when N = (8) we have dim S (8) new = 1 and find the form f with Heckeeigenvalues a ( w +2) = − a (3) = 2, a ( w +3) = a ( w +7) = − a ( w − = a ( w +4) = 4, and so on. Inthe tables of Cremona, we locate the form g ( q ) = q − q − q + q − q − q − q + · · · ∈ S (200)associated to an elliptic curve A g with label and equation y = x + x − x −
2. Wefind that the quadratic twist of (the base change of) A g by − ( w + 2) = −√ quation y = x + ( w − x − wx has conductor (8), so we conclude that f is the basechange of g . Note that this example is not covered by the known cases of the Eichler-Shimuraconstruction, since the level (8) = (2) is squarefull. Example . Now consider the prime level N = (3 w + 7) |
61, then S ( N ) new is a 2-dimensional irreducible Hecke module arising from the Jacobian of the Shimura curve X B (1)where B is the quaternion algebra ramified at one of the infinite places and N . This followsfrom Examples 4.4 and 5.4.One is naturally led to ask, can one identify the genus 2 Shimura curve X = X B (1) with B of discriminant N ?Consider the hyperelliptic curve C : y + Q ( x ) y = P ( x ) over F with P ( x ) = − wx + ( w − x + (5 w + 4) x + (6 w + 4) x + 2 w + 1 Q ( x ) = x + ( w − x + wx + 1 . This curve was obtained via specialization of the Brumer-Hashimoto family [4, 31] of curves(see also Wilson [73]) whose Jacobian has real multiplication by Q ( √ J = Jac( C ) of C . The discriminant of C is disc( C ) = N . One can showthat J is modular using the theorem of Skinner and Wiles [62, Theorem A] and the fact that J has a torsion point of order 31. Since C is hyperelliptic of genus 2, we compute that theconductor of (the abelian surface) J is N and that its level is N . Since J is modular of level N , it corresponds to the unique Hecke constituent of that level. It follows then by Faltings’isogeny theorem that J and the Jacobian of the Shimura curve X are isogenous.These examples exhaust all modular abelian varieties of GL -type over Q ( √
5) with level N of norm N ( N ) ≤
100 (up to isogeny).We conclude with some (further) cases where Conjectures 6.1 and 6.2 are not known.
Example . The first case of squarefull level with a form that is not a base change is level N = ( w + 36) = ( w + 3) so N ( N ) = 121. We have dim S ( w + 36) = 1 and the correspondingnewform f has the following eigenvalues (where t = 3): p (2) ( w + 2) (3) ( w + 3) ( w −
4) ( w −
5) ( w + 4) ( w + 5) ( w − N p a p ( f ) t − t − − t t − − t − F = Q ( √
5) with good reduction away from p = ( w + 3) and with real multipli-cation by Q ( √ f is not a base change from Q since a p ( f ) (cid:54) = a p ( f ) in general.Can one find an explicit genus 2 curve over F , analogous to the previous example, with the L -function of its Jacobian given by the above Frobenius data? Example . We next consider a situation when we are successful in establishing the corre-spondence in a case which is not covered by known results. Consider N = (17 w −
8) = ( w +4) so N ( N ) = 361. There exist forms f, g ∈ S ( N ) with Hecke eigenvalues as follows: p (2) ( w + 2) (3) ( w + 3) ( w −
4) ( w −
5) ( w + 4) ( w + 5) ( w − N p a p ( f ) 2 − − − a p ( g ) − − − − − − he form g is a quadratic twist of f by w ( w + 4), and the forms f, g are not base changesfrom Q .We use the method of Cremona and Lingham [10]—without an attempt to be exhaustive—to find an elliptic curve E over F with good reduction away from p = ( w + 4). We find thecurve E : y + ( w + 1) y = x + wx − x − w (which could also be found by a naive search) with conductor N . We verify that E ( F p ) = N p + 1 − a p ( f ) for all primes p up to the limit of the computation. We prove that E ismodular using the fact that E has a 3-isogeny, with kernel defined over F (cid:48) = F ( (cid:112) w ( w + 4))generated by the points with x = w −
1, so we can apply the theorem of Skinner and Wiles.Therefore indeed E corresponds to f .We now turn to the existence of elliptic curves with everywhere good reduction over totallyreal fields. The Shimura-Taniyama conjecture predicts that such curves arise from cusp formsof level (1). Example . Let F = Q ( λ ) = Q ( ζ ) + be the totally real subfield of the cyclotomic field Q ( ζ ) with λ = ζ + ζ − .The space S (1) over F has dimension 1, represented by the cusp form f . Conjecture 6.1predicts the existence of an elliptic curve A f over F with everywhere good reduction. Wenote that A f must be a Q -curve since the twist f σ of f by an element σ ∈ Gal( F/ Q ) isagain a form of level 1 and so (since dim S (1) = 1) f σ = f and consequently A f must beisogenous its Galois conjugates.Indeed, let A : y + a xy + a y = x + a x + a x + a be the elliptic curve with the following coefficients: a = λ + λ + 1 ,a = − λ + λ + λ,a = λ + λ + λ,a = λ − λ − λ − λ + 3 λ,a = 13 λ − λ − λ + 52 λ + 33 λ − . We verify that A has everywhere good reduction and that A ( F ) tors = 19 with A ( F ) tors generated by the point( λ − λ − λ + 3 λ + 2 λ − , − λ + λ + λ ) . Therefore, by Skinner and Wiles [61, Theorem 5] applied to the 3-adic representation at-tached to A or [62, Theorem A] to the 19-adic one, we conclude that A is modular, and thus A is our abelian variety A f .The elliptic curve A was obtained from a curve over Q ( √ ⊆ F as follows. There is a2-dimensional Hecke constituent g in the space of classical modular forms S (169) new over Q given by g ( q ) = q + √ q + 2 q + q − √ q + . . . ; et A g be the associated abelian surface, defined over Q . Roberts and Washington [49] showedthat A g twisted by √
13 is isomorphic over F to E × E σ , where E : y = x + 3483 √ − √ − , a Q -curve of conductor (13) and σ is the nontrivial automorphism of Q ( √ A g is related to the elliptic curve A as follows. Roberts and Washingtonshow that J (13) and A g are isogenous over Q . The abelian surface J (13) is isogenous over F to A × A , since J (13) obtains good reduction over F (see Mazur and Wiles [46, Chapter3.2, Proposition 2] and Schoof [52]). Indeed, Serre showed that the abelian surface J (13)has a rational point of order 19 and Mazur and Tate showed that J (13) twisted by Q ( √ A by showing thatit is a quadratic twist of E by some unit in F . We would like to thank Elkies and Watkinsfor pointing us to the rich history of this curve.The next example gives the list of all modular elliptic curves (up to isogeny) with every-where good reduction over a real quadratic field F of discriminant ≤ Example . Let 0 ≤ D ≤ F = Q ( √ D )has strict class number one. We have Z F = Z [ w ] with w = √ D, (1 + √ D ) /
2, accordingas D ≡ , A be a modular elliptic curve over F with everywhere goodreduction. Then A corresponds to a cusp form f of level 1 with rational Fourier coefficients.Computing the spaces S (1), we find that either A is a Q -curve (so A is isogenous to itsGalois conjugate), or D ∈ { , , , } and the isogeny class of A is represented bythe curve with coefficients as below, up to Galois conjugacy: D a a a a a − w − w
162 + 3 w
71 + 34 w w − − w − w − w w − w w − − w − − w
997 0 w − − w − − w The Q -curves over F are listed in Cremona [8]: for example, the curve y + xy + uy = x with u = (5 + √ / D = 509 was discovered by Pinch [52, p. 415]: it was the firstexample of an elliptic curve having everywhere good reduction which is not a Q -curve, andit was proven to be modular by Socrates and Whitehouse [60] using the method of Faltingsand Serre. The curve for D = 853 was discovered by Cremona and Watkins independently;the one for D = 929 by Elkies; and the one for D = 997 by Cremona. Their modularitycan be established by applying the theorem of Skinner and Wiles [61, Theorem 5] to theassociated 3-adic representation.As a third and final application, we consider the existence of number fields with smallramification—these are linked to Hilbert modular forms via their associated Galois repre-sentations.In the late 1990s, Gross proposed the following conjecture. onjecture 6.10. For every prime p , there exists a nonsolvable Galois number field K ramified only at p . Conjecture 6.10 is true for p ≥
11 by an argument due to Serre and Swinnerton-Dyer[53]. For k = 12 , , , ,
22 or 26, let ∆ k ∈ S k (1) be the unique newform of level 1 andweight k . For each such k and every prime p , there is an associated Galois representation ρ k,p : Gal( Q / Q ) → GL ( F p ) ramified only at p . In particular, the field K fixed by the kernelof ρ k,p is a number field ramified only at p with Galois group Gal( K/ Q ) = img ρ k,p .We say that the prime p is exceptional for k if ρ k is not surjective. The “open image”theorem of Serre [55] implies that the set of primes where ρ k is exceptional is finite. Serreand Swinnerton-Dyer produced the following table of exceptional primes: k Exceptional p
12 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , p ≥
11 that Conjecture 6.10 is true.Conversely, it is a consequence of the proof of Serre’s conjecture by Khare and Winten-berger [39] together with standard level lowering arguments that if p ≤ Q / Q ) to GL ( F p ) which is ramified only at p is necessarily reducible (and thus solvable).To address Gross’ conjecture, we instead look at the residual representations associatedto Hilbert cusp forms of parallel weight 2 and level 1 over fields F ramified only at a prime p ≤
7. This idea was first pursued by the first author [16] following a suggestion of Gross,who found such a form with F = Q ( ζ ) + , the totally real subfield of the cyclotomic field Q ( ζ ), settling the case p = 2. This line of inquiry was followed further by the authors withMatthew Greenberg [17], settling the cases p = 3 ,
5. (The case p = 7 was recently settled byDieulefait [20] by considering the mod 7 Galois representation attached to a genus 2 Siegelcusp form of level 1 and weight 28. Therefore Gross’ conjecture is now a theorem.)We sketch below the resolution of the case p = 5. We take the base field F = Q ( b ) to bethe subfield of Q ( ζ ) of degree 5, where b + 5 b − b − b − F has strictclass number 1. Let E be the elliptic curve over F with j -invariant given by(5 · ) j = − b − b + 11247914660553215 b + 464399360515483572 b + 353505866738383680and minimal conductor N . Then N = p p , where p = (( − b − b + 31 b + 25) /
7) is theunique prime above 5 and p = (( − b − b + 8 b + 53 b + 6) /
7) is one of the five primesabove 7. Roberts [50] showed that the mod 5 Galois representation ρ E, : Gal( F /F ) → End( E [5]) ∼ = GL ( F p ) = GL ( F )has projective image PGL ( F ) and is ramified only at the prime p (and not p ). (We willverify below that E has indeed the right conductor.) The representation ρ E, and its Galois onjugates gives an extension K of Q with Galois groupGal( K/ Q ) ∼ = PSL ( F ) (cid:111) Z / Z . By work of Skinner and Wiles (as in Example 6.3), we prove that the 3-adic representation ρ E, associated to E is modular, and hence E itself is modular. Since E is modular and F has odd degree, E is uniformized by a Shimura curve. Namely, let B be the quaternionalgebra over F ramified at four of the five real places. Let X B ( p p ) be the Shimura curveassociated to an Eichler order of level p p contained in a maximal order O , and let J B ( p p )be the Jacobian of X B ( p p ). We compute that J B ( p p ) new has dimension 203. We thenfind the unique Hilbert newform f E of parallel weight 2 and level p p with integer Fouriercoefficients which corresponds to E .The elliptic curve E found by Roberts [50] was obtained from our computations of Hilbertmodular forms at level p over F . The space S ( p ) new has 2 Hecke constituents of dimension10 and 20, respectively. Let f be a newform in the 20-dimensional constituent. Let T f bethe restriction of the Hecke algebra T = Z [ T p ] p to this constituent: this is the constituentwhich yields the Galois representation obtained in the 5-torsion of Roberts’ curve. Let E f = T f ⊗ Q = Q ( a p ( f )) be the field of Fourier coefficients of f and let ∆ = Aut( E f ). Bydirect calculations, we see that K f = E ∆ f is the (totally real) quartic field generated by aroot of the polynomial x + 2 x − x − x + 816.The Galois group Gal( F/ Q ) acts on T via its action on ideals of Z F , namely σ ( T p ) = T σ ( p ) .This action preserves the decomposition of T into Hecke-irreducible components. By workof Shimura, the action of ∆ on Fourier expansions preserves the Hecke constituents. Inparticular, both these actions preserve T f and hence K f and must be compatible. Therefore,for each σ ∈ Gal( F/ Q ), there is a unique τ = τ ( σ ) ∈ ∆ such that, for all prime ideals p ⊂ Z F ,we have a σ ( p ) ( f ) = τ ( a p ( f )) . The map τ thus yields a homomorphism(6.11) Gal( F/ Q ) → ∆ σ (cid:55)→ τ ( σ )By direct calculation, we show that this map is an isomorphism. Since Gal( F/ Q ) is abelian,the field E f must be a ray class field over K f ; in fact, we compute that it has conductor equalto a prime above 71 which splits completely in E f . The ideal 5 Z K f factors as 5 Z K f = P P (cid:48) .The prime P splits completely in E f , and the primes above it in E f are permuted byGal( E f /K f ) = Aut( E f ); the prime P (cid:48) is inert.We have a Galois representation ρ f : Gal( F /F ) → GL ( T f ⊗ F )with Tr( ρ f (Frob p )) ≡ a p ( f ) (mod 5) and det( ρ f (Frob p )) ≡ N p (mod 5) for all p (cid:45)
5. Let m ( i ) f , i = 1 , . . . , P . They give rise to the 5residual representations ρ ( i ) f : Gal( F /F ) → GL ( F ) . Let L i be the fixed field of ker( ρ ( i ) f ), and L the compositum of the L i . Then L is a Galoisextension of F and is ramified only at 5. Since F is Galois and ramified only at 5, and by theabove Gal( F/ Q ) permutes the fields L i so preserves L , we see that L is a Galois extension f Q ramified only at 5. By a result of Shepherd-Barron and Taylor [56], each extension L i can be realized in the 5-torsion of an elliptic curve E i /F .Recall the projective representation P ρ E, from Roberts’ elliptic curve E is surjective,ramified at p but unramified at p . Therefore, the level p p is a nonoptimal level for ρ E, ;thus, by Mazur’s Principle [37], we have ρ E, ∼ = ρ ( i ) f for some 1 ≤ i ≤
5. In other words, theextension constructed by Roberts is isomorphic to our field L .Roberts [50] has given an explicit equation for the number field L (obtained from the5-division polynomial of the elliptic curve E ): the field L is the the splitting field of thepolynomial x − x + 25 x + 110 x − x + 1250 x − x + 21750 x − x + 112500 x − x + 448125 x − x + 1744825 x − x − x + 4269125 x − x + 949625 x − x + 1303750 x − x + 291625 x − . To conclude, we consider a question that touches on each of the above three subjects. Wereconsider Gross’ conjecture (Conjecture 6.10) in the case p = 2 [16]. The nonsolvable fieldwhich is ramified only at 2 arises from the Galois representation associated to a constituenteigenform f ∈ S (1) in a 16-dimensional subspace of the space of Hilbert cusp forms ofparallel weight 2 and level 1 over F = Q ( ζ ) + . Let E f = Q ( a p ) be the field of Fouriercoefficients of f and let ∆ = Aut( E f ). Let K f be the fixed field of ∆ so that Gal( E f /K f ) =∆. The map (6.11) in this context is again an isomorphism, and so E f is abelian over K f and 8 = [ F : Q ] divides [ E f : Q ] = 16. By direct calculations, we show that K f = Q ( √ A f defined over F with everywhere good reduction and real multiplicationby E f associated to f . More should be true, as communicated to us by Gross (privatecommunication). In fact, A f should descend to an abelian variety of dimension 16 over Q ,and we should have L ( A f , s ) = L ( f, s ) L ( f σ , s ), where σ ∈ Gal( E f / Q ) is any element thatrestricts to the nontrivial element of K f = Q ( √ A f over Q shouldbe the ring of integers of K f , and over Q , the variety A f would have bad reduction only atthe prime 2; the nonsolvable extension would then arise as its 2-division field. The conductor N = 2 = d of A f over Q can be computed from the functional equation of L ( f, s ), wherewe note d = 2 is the discriminant of F . We note that A f is not of GL -type over Q itselfas it would be modular by the proof of Serre’s conjecture.Although one knows that the associated Galois representation exists by work of Taylor[65], as Gross says, “With such nice properties, it’s a shame that we can’t even prove thatthe abelian variety A f exists! That’s an advantage you have when F has odd degree.”7. Adelic quaternionic modular forms
In this section, we begin again, and we revisit the definition of Hilbert and quaternionicmodular forms allowing F to have arbitrary class number; we refer to Hida [34] as a referencefor this section. e renew our notation. Let F be a totally real field of degree n = [ F : Q ] with ring ofintegers Z F . Let B be a quaternion algebra over F of discriminant D . Let v , . . . , v n be thereal places of F (abbreviating as before x i = v i ( x ) for x ∈ F ), and suppose that B is splitat v , . . . , v r and ramified at v r +1 , . . . , v n , i.e.(7.1) B (cid:44) → B ∞ = B ⊗ Q R ∼ −→ M ( R ) r × H n − r . Let ι i denote the i th projection B → M ( R ) and ι = ( ι , . . . , ι r ). Let F × + = { x ∈ F : x i > i } be the group of totally positive elements of F and let Z × F, + = Z × F ∩ F × + .Let O (1) ⊆ B be a maximal order. Let N be an ideal of Z F coprime to D and let O = O ( N ) ⊆ O (1) be an Eichler order of level N .With a view towards generalizations, rather than viewing modular forms as functions on(a Cartesian power of) the upper half-plane which transform in a certain way, we now viewthem instead more canonically as functions on B ×∞ . Let H ± = C \ R be the union of theupper and lower half-planes. Via the embeddings v , . . . , v r , corresponding to the first r factors in (7.1), the group B ×∞ acts on ( H ± ) r on the right transitively with the stabilizer of( √− , . . . , √− ∈ H r being K ∞ = ( R × SO ( R )) r × ( H × ) n − r . Therefore we can identify(7.2) B ×∞ /K ∞ → ( H ± ) r g (cid:55)→ z = g ( √− , . . . , √− . From this perspective, it is natural to consider the other (nonarchimedean) places of F atthe same time. Let (cid:98) Z = lim ←− n Z /n Z = (cid:89) (cid:48) p Z p (where (cid:48) denotes the restricted direct product) and let (cid:98) denote tensor with (cid:98) Z over Z . Wewill define modular forms on B as analytic functions on B ×∞ × (cid:98) B × which are invariant onthe left by B × and transform by K ∞ × (cid:98) O × in a specified way.We must first define the codomain of these functions to obtain forms of arbitrary weight.Let k = ( k , . . . , k n ) ∈ ( Z ≥ ) n and suppose that the k i have the same parity; we call k a weight . Let k = max i k i , m i = ( k − k i ) / , and w i = k i − . For an integer w ≥
0, let P w = P w ( C ) be the subspace of C [ x, y ] consisting of homogeneouspolynomials of degree w . For γ ∈ GL ( C ), let γ be the adjoint of γ , so that γγ = γγ = det γ .Define a right action of GL ( C ) on P w ( C ) by( q · γ )( x, y ) = q (( x y )¯ γ ) = q ( dx − cy, − bx + ay )for γ = (cid:18) a bc d (cid:19) ∈ GL ( C ) and q ∈ P w ( C ). For m ∈ Z , GL ( C ) also acts on P w ( C ) viathe character γ (cid:55)→ (det γ ) m . By twisting the above action by this character, we get a rightGL ( C )-module denoted by P w ( m )( C ). Define the right GL ( C ) n − r -module(7.3) W k ( C ) = W ( C ) = P w r +1 ( m r +1 )( C ) ⊗ · · · ⊗ P w n ( m n )( C ) . By convention, if r = n then we set W k ( C ) = C .) For the ramified real places v r +1 , . . . , v n of F , we choose splittings ι i : B (cid:44) → B ⊗ F C ∼ = M ( C ) . We abbreviate as above γ i = ι i ( γ ) for γ ∈ B . Then W k ( C ) becomes a right B × -module via γ (cid:55)→ ( γ r +1 , . . . , γ n ) ∈ GL ( C ) n − r . From now on, W k ( C ) will be endowed with this action,which we denote by x (cid:55)→ x γ for x ∈ W k ( C ) and γ ∈ B × . One may identify W k ( C ) with thesubspace of the algebra C [ x r +1 , y r +1 , . . . , x n , y n ] consisting of those polynomials q which arehomogeneous in ( x i , y i ) of degree w i but with a twisted action.We consider the space of functions φ : B ×∞ × (cid:98) B × → W k ( C ), with a right action of K ×∞ × (cid:98) B × defined by(7.4) ( φ | k ( κ, (cid:98) β ))( g, (cid:98) α ) = (cid:32) r (cid:89) i =1 j ( κ i , √− k i (det κ i ) m i + k i − (cid:33) φ ( gκ − , (cid:98) α (cid:98) β − ) κ . where recall j ( κ i , √−
1) = c i √− d i ∈ C if κ i = (cid:18) a i b i c i d i (cid:19) . (The presence of the inverses isforced as we want a right action on functions via multiplication of the argument on the right.This almost extends to a right action of B ×∞ × (cid:98) B × , except that j ( gh, z ) = j ( g, hz ) j ( h, z ) (cid:54) = j ( g, z ) j ( h, z ) unless h fixes z .) Definition 7.5. A (quaternionic) modular form of weight k and level N for B is an analyticfunction φ : B ×∞ × (cid:98) B × → W k ( C ) such that for all ( g, (cid:98) α ) ∈ B ×∞ × (cid:98) B × we have: (i) ( φ | k ( κ, (cid:98) u ))( g, (cid:98) α ) = φ ( g, (cid:98) α ) for all κ ∈ K ∞ and (cid:98) u ∈ (cid:98) O × ; and (ii) φ ( γg, γ (cid:98) α ) = φ ( g, (cid:98) α ) for all γ ∈ B × . In other words, Definition 7.5 says that a quaternionic modular form of weight k and level N is an analytic function which is B × -invariant on the left and ( K ∞ × (cid:98) O × )-equivariant onthe right under the action (7.4). In particular, we have(7.6) φ ( g, (cid:98) α (cid:98) u ) = ( φ | k (1 , (cid:98) u ))( g, (cid:98) α (cid:98) u ) = φ ( g, (cid:98) α )for all (cid:98) u ∈ (cid:98) O × and(7.7) φ ( gκ, (cid:98) α ) = ( φ | k ( κ, gκ, (cid:98) α ) = (cid:32) r (cid:89) i =1 j ( κ i , √− k i (det κ i ) m i + k i − (cid:33) φ ( g, (cid:98) α ) κ . We denote by M Bk ( N ) the space of quaternionic modular forms of weight k and level N for B .Let φ be a quaternionic modular form. For ( z, (cid:98) α (cid:98) O × ) ∈ ( H ± ) r × (cid:98) B × / (cid:98) O × , choose g ∈ B ×∞ such that g ( √− , . . . , √−
1) = z and define(7.8) f : ( H ± ) r × (cid:98) B × / (cid:98) O × → W k ( C ) f ( z, (cid:98) α (cid:98) O × ) = (cid:32) r (cid:89) i =1 (det g i ) m i + k i − j ( g i , √− k i (cid:33) φ ( g, (cid:98) α ) g − . he map f in (7.8) is well-defined by (7.6) and (7.7): we have (cid:32) r (cid:89) i =1 (det g i κ i ) m i + k i − j ( g i κ i , √− k i (cid:33) φ ( gκ, (cid:98) α (cid:98) u ) ( gκ ) − = (cid:32) r (cid:89) i =1 (det g i ) m i + k i − j ( g i , √− k i (cid:33) φ ( g, (cid:98) α ) g − using the fact that j ( g i κ i , √−
1) = j ( g i , √− j ( κ i , √− γ ∈ B × . Then g ( √− , . . . , √−
1) = z isequivalent to ( γg )( √− , . . . , √−
1) = γz , so(7.9) f ( γz, γ (cid:98) α (cid:98) O × ) = (cid:32) r (cid:89) i =1 j ( γ i g i , √− k i (det γ i g i ) m i + k i − (cid:33) φ ( γg, γ (cid:98) α ) ( γg ) − = (cid:32) r (cid:89) i =1 j ( γ i , z i ) k i (det γ i ) m i + k i − (cid:33) (cid:32) r (cid:89) i =1 j ( g i , √− k i (det g i ) m i + k i − (cid:33) φ ( g, (cid:98) α ) g − γ − = (cid:32) r (cid:89) i =1 j ( γ i , z i ) k i (det γ i ) m i + k i − (cid:33) f ( z, (cid:98) α (cid:98) O × ) γ − where now j ( γ i , z i ) = c i z i + d i ∈ C if γ i = (cid:18) a i b i c i d i (cid:19) for all γ ∈ B × and we have the relation j ( γδ, z ) = j ( γ, δz ) j ( δ, z ) for all z ∈ H and g, h ∈ B × . Accordingly, we define a right actionof B × on the space of functions in (7.8) by(7.10) ( f | k γ )( z, (cid:98) α (cid:98) O × ) = (cid:32) r (cid:89) i =1 (det γ i ) m i + k i − j ( γ i , z i ) k i (cid:33) f ( γz, γ (cid:98) α (cid:98) O × ) γ . Then ( f | k γ )( z, (cid:98) α (cid:98) O × ) = f ( z, (cid:98) α (cid:98) O × ) . Note that the central subgroup F × ⊆ B × acts by ( f | k a )( z, (cid:98) α (cid:98) O × ) = N F/ Q ( a ) k − f ( z, a (cid:98) α (cid:98) O × )for a ∈ F × .The C -vector space of modular forms of weight k and level N on B is finite-dimensionaland is denoted M Bk ( N ). Lemma 7.11.
There is a bijection between M Bk ( N ) and the space of functions f : ( H ± ) r × (cid:98) B × / (cid:98) O × → W k ( C ) that are holomorphic in the first variable and locally constant in the secondone and such that f | k γ = f for all γ ∈ B × . From now on, we will only work with modular forms f as presented in Lemma 7.11.We define the quaternionic Shimura variety of level N associated to B as the double coset X B ( N )( C ) = B × \ ( B ×∞ /K ∞ × (cid:98) B × / (cid:98) O × ) = B × \ (( H ± ) r × (cid:98) B × / (cid:98) O × );the set X B ( N )( C ) can be equipped the structure of a complex (possibly disconnected) Rie-mannian manifold of dimension r . Example . We recover first the definition of classical modular forms when F = Q and B = M ( Q ). For simplicity, we take N = 1. In this case, r = 1 so W k ( C ) = C and m = 0. he action (7.10) is simply( f | k γ )( z, (cid:98) α (cid:98) O × ) = (det γ ) k − j ( γ, z ) k f ( γz, γ (cid:98) α (cid:98) O × ) . We take the definition (7.8) as our starting point. The element (cid:18) − (cid:19) ∈ B × = GL ( Q )identifies the upper and lower half-planes, so a modular form f : H ± × GL ( (cid:98) Q ) / GL ( (cid:98) Z ) → C is determined by its restriction f : H × GL ( (cid:98) Q ) / (cid:98) Γ ( N ) → C and the subgroup of GL ( Q )which preserves H is exactly GL +2 ( Q ).We wish to recover the classical action using this new action of GL +2 ( Q ), so we are ledto consider the double coset GL +2 ( Q ) \ GL ( (cid:98) Q ) / GL ( (cid:98) Z ). An element of this double coset isspecified by an element (cid:98) α ∈ GL ( (cid:98) Q ) up to right-multiplication by GL ( (cid:98) Z ), i.e. a (cid:98) Z -latticein (cid:98) Q , i.e. (cid:98) Λ ∈ Lat( (cid:98) Q ), specified by the rows of (cid:98) α . But the map Lat( Q ) → Lat( (cid:98) Q ) byΛ (cid:55)→ (cid:98) Λ is a bijection, with inverse (cid:98) Λ (cid:55)→ (cid:98) Λ ∩ Q . And since GL +2 ( Q ) acts transitively on theleft on the set of lattices Lat( Q ), we conclude that +2 ( Q ) \ GL ( (cid:98) Q ) / GL ( (cid:98) Z ) = 1.It follows that f is uniquely specified by the function f ( z, (cid:98) O × ) for z ∈ H , which by abuseof notation we write simply f : H → C . The stabilizer of GL +2 ( Q ) acting on GL ( (cid:98) Z ) bymultiplication on the left is GL +2 ( Z ), so we recover the condition( f | k γ )( z ) = (det γ ) k − j ( γ, z ) k f ( γz )for all γ ∈ GL +2 ( Z ), which is exactly the definition given in Section 1.The interested reader can modify this argument for N >
1; alternatively, we give a generalderivation below.We now define cusp forms. If B ∼ = M ( F ), then we are in the situation of Hilbert modularforms (but over a field with arbitrary class number): so we define a cusp form to be a form f such that f ( z ) → z tends to a cusp P ( F ) (cid:44) → P ( R ) n . Otherwise, the Shimuravariety X B ( N )( C ) is compact. If B is indefinite, so 0 < r ≤ n , then there are no cusps, andwe define the space of cusp forms to be S Bk ( N ) = M Bk ( N ). Finally, suppose B is definite;then r = 0. If k = (2 , . . . , E Bk ( N ) to be the space of those f ∈ M Bk ( N )such that f factors through nrd : (cid:98) B × → (cid:98) F × ; otherwise, we set E Bk ( N ) = 0. Then there is anorthogonal decomposition M Bk ( N ) = S Bk ( N ) ⊕ E Bk ( N ) and we call S Bk ( N ) the space of cuspforms for B .The spaces M Bk ( N ) and S Bk ( N ) come equipped with an action of pairwise commutingdiagonalizable Hecke operators T n indexed by the nonzero ideals n of Z F , defined as follows.Given f ∈ S Bk ( N ) and (cid:98) π ∈ (cid:98) B × , we define a Hecke operator associated to (cid:98) π as follows: wewrite(7.13) (cid:98) O × (cid:98) π (cid:98) O × = (cid:71) i (cid:98) O × (cid:98) π i and let(7.14) ( T π f )( z, (cid:98) α (cid:98) O × ) = (cid:88) i f ( z, (cid:98) α (cid:98) π − i (cid:98) O × ) . Again, although it may seem unnatural, the choice of inverse here is to make the definitionsagree with the classical case.)For a prime p of Z F with p (cid:45) DN , we denote by T p the Hecke operator T (cid:98) π where (cid:98) π ∈ (cid:98) B × is such that (cid:98) π v = 1 for v (cid:54) = p and (cid:98) π p = (cid:18) p
00 1 (cid:19) ∈ O ⊗ Z F Z F, p ∼ = M ( Z F, p ) where p ∈ Z F, p isa uniformizer at p .Equivalently, for a prime p and (cid:98) p ∈ (cid:98) Z F such that (cid:98) p (cid:98) Z F ∩ Z F = p , we defineΘ( p ) = (cid:98) O × \{ (cid:98) π ∈ (cid:98) O : nrd( (cid:98) π ) = (cid:98) p } . a set of cardinality N p + 1, and define(7.15) ( T p f )( z, (cid:98) α (cid:98) O × ) = (cid:88) (cid:98) π ∈ Θ( p ) f ( z, (cid:98) α (cid:98) π − (cid:98) O × )where we have implicitly chosen representatives (cid:98) π ∈ (cid:98) O for the orbits in Θ( p ). For an ideal n of Z F , the operator T n is defined analogously.We say that a cusp form f is a newform if it is an eigenvector of the Hecke operatorswhich does not belong to M k ( M ) for M | N .To unpack this definition further, and to relate this definition with the definitions givenpreviously, we investigate the structure of the Shimura variety X B ( N )( C ) = B × \ ( B ×∞ /K ∞ × (cid:98) B × / (cid:98) O × ) = B × \ (( H ± ) r × (cid:98) B × / (cid:98) O × ) . By Eichler’s theorem of norms [66, Theor`eme III.4.1], we havenrd( B × ) = F × (+) = { a ∈ F × : v i ( a ) > i = r + 1 , . . . , n } , i.e. the norms from B × consists of the subgroup of elements of F which are positive at allreal places which are ramified in B . In particular, B × /B × + ∼ = ( Z / Z ) r , where B × + = { γ ∈ B : nrd( γ ) ∈ F × + } is the subgroup of B × whose elements have totally positive reduced norm.The group B × + acts on H r , therefore we may identify X B ( N )( C ) = B × + \ ( H r × (cid:98) B × / (cid:98) O × )and a modular form on ( H ± ) r × (cid:98) B × / (cid:98) O × can be uniquely recovered from its restriction to H r × (cid:98) B × / (cid:98) O × . Now we have a natural (continuous) projection map X B ( N )( C ) → B × + \ (cid:98) B × / (cid:98) O × . The reduced norm gives a surjective map(7.16) nrd : B × + \ (cid:98) B × / (cid:98) O × → F × + \ (cid:98) F × / (cid:98) Z × F ∼ = Cl + Z F . where Cl + Z F denotes the strict class group of Z F , i.e. the ray class group of Z F with modulusequal to the product of all real (infinite) places of F . The theorem of strong approximation[66, Th´eor`eme III.4.3] implies that (7.16) is a bijection if B is indefinite. So our descriptionwill accordingly depend on whether B is indefinite or definite.First, suppose that B is indefinite. Then space X B ( N )( C ) is the disjoint union of connectedRiemannian manifolds indexed by Cl + Z F , which we identify explicitly as follows. Let the deals a ⊆ Z F form a set of representatives for Cl + Z F , and let (cid:98) a ∈ (cid:98) Z F be such that (cid:98) a (cid:98) Z F ∩ Z F = a . (For the trivial class a = Z F , we choose (cid:98) a = (cid:98) (cid:98) α ∈ (cid:98) B × suchthat nrd( (cid:98) α ) = (cid:98) a . We let O a = (cid:98) α (cid:98) O (cid:98) α − ∩ B so that O (1) = O , and we put Γ a = O × a , + = (cid:98) O × a ∩ B × + .Then we have(7.17) X B ( N )( C ) = (cid:71) [ a ] ∈ Cl + ( Z F ) B × + ( H r × (cid:98) α (cid:98) O × ) ∼ −→ (cid:71) [ a ] ∈ Cl + ( Z F ) Γ a \H r , where the last identification is obtained via the bijection(7.18) B × + \ ( H r × (cid:98) α (cid:98) O × ) ∼ −→ Γ a \H r B × + ( z, (cid:98) α (cid:98) O × ) (cid:55)→ z Now let f ∈ M Bk ( N ), so that f : ( H ± ) r × (cid:98) B × / (cid:98) O × → W k ( C ) satisfies f | k γ = f for all γ ∈ B × . Let M Bk ( N , a ) be the space of functions f a : H r → W k ( C ) such that f a | k γ = f a forall γ ∈ Γ a , where we define( f a | γ )( z ) = (cid:32) r (cid:89) i =1 (det γ i ) m i + k i − j ( γ i , z ) k i (cid:33) f a ( γz ) γ for γ ∈ B × . Then by (7.18), the map(7.19) M Bk ( N ) → (cid:77) [ a ] ∈ Cl + ( Z F ) M Bk ( N , a ) f (cid:55)→ ( f a )where f a : H r → W k ( C ) f a ( z ) = f ( z, (cid:98) α (cid:98) O × )is an isomorphism.We now explain how the Hecke module structure on the left-hand side of (7.19), defined in(7.14)–(7.15), is carried over to the right-hand side. We follow Shimura [57, Section 2]. Weconsider the action on the summand corresponding to [ a ] ∈ Cl + ( Z F ). Extending the notationabove, among the representatives chosen, let b be such that [ b ] = [ ap − ], let (cid:98) b (cid:98) Z F ∩ Z F = b ,and let (cid:98) β ∈ (cid:98) B × be such that nrd( (cid:98) β ) = (cid:98) b .By definition, ( T p f ) a ( z ) = ( T p f )( z, (cid:98) α (cid:98) O × ) = (cid:88) (cid:98) π ∈ Θ( p ) f ( z, (cid:98) α (cid:98) π − (cid:98) O × ) . Let (cid:98) π ∈ Θ( p ). Then by strong approximation, we have(7.20) (cid:98) α (cid:98) π − (cid:98) O (cid:98) β − ∩ B = (cid:36) − O b with (cid:36) ∈ B × , since this lattice has reduced norm [ ap − ( ap − ) − ] = [(1)]. Therefore, thereexists (cid:98) u ∈ (cid:98) O such that (cid:98) α (cid:98) π − (cid:98) u (cid:98) β − = (cid:36) − whence(7.21) ( T p f ) a ( z ) (cid:88) (cid:98) π ∈ Θ( p ) f ( z, (cid:98) α (cid:98) π − (cid:98) O × ) = (cid:88) (cid:36) f ( z, (cid:36) − (cid:98) β (cid:98) O × ); he second sum runs over a choice of (cid:36) as in equation (7.20) corresponding to each (cid:98) π ∈ Θ( p ).This latter sum can be identified with a sum over values of f b as follows. We have f ( z, (cid:36) − (cid:98) β (cid:98) O × ) = ( f | k (cid:36) )( z, (cid:36) − (cid:98) β (cid:98) O × )= (cid:32) r (cid:89) i =1 (det (cid:36) i ) m i + k i − j ( (cid:36) i , z i ) k i (cid:33) f ( (cid:36)z, (cid:98) β (cid:98) O × ) (cid:36) = ( f b | k (cid:36) )( z ) . The first equality follows from the B × -invariance of f and the others by definition of theslash operators. Putting these together, we have(7.22) ( T p f ) a ( z ) = (cid:88) (cid:36) ( f b | k (cid:36) )( z ) . (The naturality of this definition explains the choice of inverses above.)This adelic calculation can be made global as follows. Let I a = (cid:98) α (cid:98) O ∩ B and I b = (cid:98) β (cid:98) O ∩ B .For (cid:98) π ∈ Θ( p ), we have (cid:98) α (cid:98) π − (cid:98) O (cid:98) β − ∩ B = (cid:36) − O b hence(7.23) O b (cid:36) = (cid:98) β (cid:98) O (cid:98) π (cid:98) α − = ( (cid:98) β (cid:98) O ) (cid:98) α − ( (cid:98) α (cid:98) O (cid:98) α − ) (cid:98) α (cid:98) π (cid:98) α − ∩ B = I b I − a J. The elements (cid:36) thus obtained are characterized by their norms (in the right lattice), as withthe Hecke operators defined previously (5.1): we analogously define(7.24) Θ( p ) a , b = Γ b \ { (cid:36) ∈ I b I − a ∩ B × + : nrd( I b I − a ) p = (nrd( (cid:36) )) } = Γ b \ { (cid:36) ∈ I b I − a ∩ B × + : nrd( (cid:36) ) b = ap } . Then for f b ∈ M Bk ( N , b ), we have T p f b ∈ M Bk ( N , a ) and T p f b = (cid:88) (cid:36) ∈ Θ( p ) a , b f b | k (cid:36) where [ b ] = [ ap − ]. Example . If F has strict class number 1, then I a = I b = O = O a so O b (cid:36) a = O π a as inSection 5.We note that the isomorphism (7.19) preserves the subspace of cusp forms in a way thatis compatible with the Hecke action, so we have a decomposition S Bk ( N ) ∼ −→ (cid:77) [ a ] ∈ Cl + ( Z F ) S Bk ( N , a ) . Example . Let B = M ( F ), and let O = O ( N ) ⊂ O (1) = M ( Z F ). Then we may take (cid:98) α = (cid:18)(cid:98) a
00 1 (cid:19) ∈ GL ( (cid:98) F ), and so we find simply that O a = (cid:98) α M ( Z F ) (cid:98) α − ∩ B = (cid:18) Z F aNa − Z F (cid:19) = O ( N , a )Let Γ ( N , a ) = O ( N , a ) × + = (cid:26) γ = (cid:18) a bc d (cid:19) ∈ O ( N , a ) : det γ ∈ Z × F, + (cid:27) . hen X B ( N )( C ) = (cid:71) [ a ] ∈ Cl + ( Z F ) Γ ( N , a ) \H n is a disjoint union.A Hilbert modular form of weight k and level N is a tuple ( f a ) of holomorphic functions f a : H n → C , indexed by Cl + Z F , such that for all a we have( f a | k γ )( z ) = f a ( z ) for all γ ∈ Γ ( N , a )(with the extra assumption that f is holomorphic at the cusps if F = Q ). Or, put anotherway, let M k ( N , a ) be the set of holomorphic functions H n → C such that ( f | k γ )( z ) = f ( z )for all γ ∈ Γ ( N , a ); then M k ( N ) = (cid:77) [ a ] M k ( N , a ) . In particular, we recover the definitions in Section 2 when F has strict class number 1.A modular form f ∈ M k ( N , a ) admits a Fourier expansion f ( z ) = a + (cid:88) µ ∈ ( ad − ) + a µ e πi Tr( µz ) analogous to (2.5). We say that f ∈ M k ( N , a ) is a cusp form if f ( z ) → z tends to anycusp. Letting S k ( N , a ) be the space of such cusp forms, we have S k ( N ) = (cid:77) [ a ] S k ( N , a ) . Let f = ( f a ) ∈ S k ( N ) be a Hilbert cusp form and let n ⊆ Z F be an ideal. Suppose that[ n ] = [ ad − ] amongst the representatives chosen for Cl + ( Z F ), and let ν ∈ Z F be such that n = ν ad − . We define a n = ν m a ν ( f a ); the transformation rule implies that a n only dependson n and we call a n the Fourier coefficient of f at n .Now suppose that B is definite. Then the Shimura variety is simply X B ( N )( C ) = B × \ (cid:98) B × / (cid:98) O × = Cl O and so is canonically identified with the set of right ideal classes of O . Note that the reducednorm map (7.16) here is the map nrd : Cl O → Cl + Z F which is surjective but not a bijection,in general. A modular form f ∈ M Bk ( N ) is then just a map f : (cid:98) B × / (cid:98) O × → W k ( C ) such that f | k γ = f for all γ ∈ B × . Such a function is completely determined by its values on a setof representatives of Cl O ; moreover, given any right ideal I = (cid:98) α (cid:98) O ∩ B , the stabilizer of B × acting on (cid:98) α (cid:98) O by left multiplication is O L ( I ) × = (cid:98) α (cid:98) O × (cid:98) α − ∩ B × . Therefore, there is anisomorphism of complex vector spaces given by(7.27) M Bk ( N ) → (cid:77) [ I ] ∈ Cl( O ) I = (cid:98) α (cid:98) O∩ B W k ( C ) Γ( I ) f (cid:55)→ ( f ( (cid:98) α )) , where Γ( I ) = (cid:98) α (cid:98) O × (cid:98) α − ∩ B × = O L ( I ) × and W k ( C ) Γ( I ) is the Γ( I )-invariant subspace of W k ( C ). aving now discussed both the definite and indefinite cases in turn, we return to a generalquaternion algebra B . Let f ∈ S Bk ( N ) new be a newform. A theorem of Shimura states thatthe coefficients a n are algebraic integers and E f = Q ( { a n } ) is a number field. The Heckeeigenvalues a n determine the L -series L ( f, s ) = (cid:88) n ⊆ Z F a n N n s = (cid:89) p (cid:45) N (cid:18) − a p N p s + 1 N p s +1 − k (cid:19) − (cid:89) p | N (cid:18) − a p N p s (cid:19) − associated to f (defined for Re s > f is a Galois representation:for l a prime of Z E f and E f, l the completion of E f at l , there is an absolutely irreducible,totally odd Galois representation ρ f, l : Gal( F /F ) → GL ( E f, l )such that, for any prime p (cid:45) lN , we haveTr( ρ f, l (Frob p )) = a p ( f ) and det( ρ f, l (Frob p )) = N p k − . The existence of this representation is due to work of Blasius-Rogawski [2], Carayol [6],Deligne [13], Saito [51], Taylor [65], and Wiles [72].The statement of the Jacquet-Langlands correspondence (3.9) reads the same in this moregeneral context.
Theorem 7.28 (Jacquet-Langlands) . There is an injective map of Hecke modules S Bk ( N ) (cid:44) → S k ( DN ) whose image consists of those forms which are new at all primes dividing D . We are now ready to state the main general result of this article, generalizing the resultof Theorem 2.8 to arbitrary class number and arbitrary weight.
Theorem 7.29 (Demb´el´e-Donnelly [18], Voight [69]) . There exists an algorithm which, givena totally real field F , a nonzero ideal N ⊆ Z F , and a weight k ∈ ( Z ≥ ) [ F : Q ] , computes thespace S k ( N ) of Hilbert cusp forms of level N over F as a Hecke module. The proof of this theorem is discussed in the next two sections. It falls again naturallyinto two methods, definite and indefinite, which overlap just as in Remark 3.10.8.
Definite method, arbitrary class number
In this section, we return to the totally definite case but allow arbitrary class number. Asexplained above, the space M Bk ( N ) of modular forms of level N and weight k on B is thespace of functions f : (cid:98) B × / (cid:98) O × → W k ( C ) such that f | k γ = f for all γ ∈ B × .We can use the identification (7.27) to compute the space S Bk ( N ) as in the direct approachof Section 4, with the appropriate modifications. Let I , . . . , I H be a set of representativesfor Cl O such that nrd( I i ) is coprime to DN for all i . Let (cid:98) α i ∈ (cid:98) O be such that (cid:98) α i (cid:98) O ∩ O = I i ,and let Γ i = O L ( I i ) × . Then dualizing the isomorphism (7.27), we have M Bk ( N ) ∼ = H (cid:77) i =1 W k ( C ) Γ i . The Hecke module structure on this space is defined similarly as in Section 4, as the followingexample illustrates. xample . Consider the totally real quartic field F = Q ( w ) where w − w − w + 1 = 0.Then F has discriminant 5744 = 2
359 and Galois group S . We have Cl + Z F = 2 (butCl Z F = 1).The quaternion algebra B = (cid:18) − , − F (cid:19) is unramified at all finite places (and ramified atall real places). We compute a maximal order O and find that O = 4. We computethe action of the Hecke operators as in (3.8): we identify the isomorphism classes of the N p + 1 right ideals of norm p inside each right ideal I in a set of representatives for Cl O .We compute, for example, that T ( w − w − = where N ( w − w −
1) = 4; note this matrix has a block form, corresponding to the fact that( w − w −
1) represents the nontrivial class in Cl + Z F . Correspondingly, T ( w − w − = with N ( w − w −
4) = 13 is a block scalar matrix, as ( w − w −
4) is trivial in Cl + Z F . Inthis case, the space E (1) of functions that factor through the reduced norm has dimensiondim E (1) = 2, so dim S (1) = 2, and we find that this space is irreducible as a Hecke moduleand so has a unique constituent f .We obtain the following table of Hecke eigenvalues: p ( w − w −
1) ( w −
1) ( w − w −
2) ( w −
3) ( w − w −
4) ( w − N p a p ( f ) 0 t − t − t t Here t satisfies the polynomial t − F with real multiplication by Q ( √
6) and everywhere good reduction.As in Section 4, the disadvantage of the approach used in Example 8.1 is that for eachlevel N , one must compute the set of ideal classes Cl O = Cl O ( N ). By working with amore complicated coefficient module we can work with ideal classes only with the maximalorder O (1), as follows.Changing notation, now let I , . . . , I h be representatives for Cl O (1), with h = O (1),and let I i = (cid:98) α i O (1) ∩ B . By strong approximation, we may assume that each nrd( I i ) iscoprime to DN : indeed, we may assume each nrd( I i ) is supported in any set S of primesthat generate Cl + Z F . Let O (1) i = O L ( I i ) = (cid:98) α i (cid:98) O (1) (cid:98) α − i ∩ B be the left order of I i . Then O (1) i ⊗ Z F Z F, N ∼ = O (1) ⊗ Z F Z F, N .Let (cid:98) β a for a ∈ P ( Z F / N ) represent the O (1) , O -connecting ideals of norm N : that is, (cid:98) β a ∈ (cid:98) O (1) and if J a = (cid:98) O (1) (cid:98) β a ∩ B then O R ( J a ) = O . Then the set { I i J a } i,a , where I i J a = (cid:98) α i (cid:98) β a (cid:98) O ∩ B , covers all isomorphism classes of right O -ideals, but not necessarilyuniquely: two such ideals I i J a and I j J b are isomorphic if and only if i = j and there exists ∈ O (1) × i such that γJ a = J b , comparing the elements (cid:98) α i (cid:98) β a , (cid:98) α j (cid:98) β b ∈ (cid:98) O (1). The action of O (1) × i can be equivalently given on the set of indices a ∈ P ( Z F / N ): via the (reduction ofa) splitting map(8.2) ι N : O (1) (cid:44) → O (1) ⊗ Z F Z F, N ∼ = M ( Z F, N ) , each O (1) × i acts on the left on P ( Z F / N ), and we have (cid:98) O (1) × i / (cid:98) O × i ∼ −→ P ( Z F / N ).We conclude that M Bk ( N ) ∼ = M Bk ( N ) = (cid:76) hi =1 M Bk ( N ) i , where M Bk ( N ) i = (cid:8) f : P ( Z F / N ) → W k ( C ) : f | k γ = f for all γ ∈ O (1) × i (cid:9) . In this presentation, the Hecke operators act as follows. For a prime p , letΘ( p ) i,j = O (1) × i \ (cid:8) x ∈ I i I − j : nrd( xI i I − j ) = p (cid:9) . We then define the linear map T p : M Bk ( N ) → M Bk ( N ) on each component by the rule(8.3) ( T p ) i,j : M Bk ( N ) i → M Bk ( N ) j f (cid:55)→ (cid:88) γ ∈ Θ( p ) i,j f | k γ. This is indeed an isomorphism of Hecke modules. For further details, see work of the firstauthor [15, Theorem 2] which traces these maps under the assumption that F has narrowclass number one, but this assumption can be easily removed.Put another way, by the decomposition (cid:98) B × = h (cid:71) i =1 B × (cid:98) α i (cid:98) O (1) × , we decompose the set X B ( N ) as X B ( N ) = B × \ (cid:98) B × / (cid:98) O × = h (cid:71) i =1 B × \ (cid:16) B × (cid:98) α i (cid:98) O (1) × (cid:17) / (cid:98) O × ∼ −→ h (cid:71) i =1 O (1) × i \ (cid:98) O (1) × i / (cid:98) O × i , where the last identification is obtained by sending γ (cid:98) α i u to (cid:98) α i u (cid:98) α − i . Thus, analogouslyto (7.17), we get a decomposition(8.4) X B ( N ) = h (cid:71) i =1 X B ( N ) i = h (cid:71) i =1 Γ i \ P ( Z F / N ) . In particular, this gives X B (1) = Cl O (1). From this, we get an identification(8.5) M Bk ( N ) → h (cid:77) i =1 M Bk ( N ) i f (cid:55)→ ( f i ) i , where we set f i ( x ) = f ( (cid:98) α (cid:98) α i ) after choosing (cid:98) α ∈ (cid:98) O (1) × i such that x = (cid:98) α · ∞ i . Again,the decomposition (8.5) is analogous to (7.19), and one shows that it is a Hecke moduleisomorphism by arguing similarly.Now f ∈ M Bk ( N ) is by definition a map f : (cid:98) B × / (cid:98) O × → W k ( C ) such that f | k γ = f forall γ ∈ B × . Associated to such a map, via the identification (8.4), such a map is uniquely efined by a tuple of maps ( f i ) i with f i : (cid:98) O (1) × i / (cid:98) O × i → W k ( C ) such that f | k γ = f for all γ ∈ Γ i = O (1) × i . In other words, M Bk ( N ) ∼ −→ h (cid:77) i =1 H (Γ i , Hom( (cid:98) O (1) × i / (cid:98) O × i , W k ( C ))) ∼ = h (cid:77) i =1 H (cid:0) Γ i , Coind (cid:98) O (1) × i (cid:98) O × i W k ( C ) (cid:1) . But then as in (8.2), we have H (cid:0) Γ i , Coind (cid:98) O (1) × i (cid:98) O × i W k ( C ) (cid:1) ∼ = M Bk ( N ) i . Example . The real quadratic field F = Q ( √ S (1) of Hilbert cusp forms of level 1 and parallelweight 2 has dimension 50. It decomposes into four Hecke constituents of dimension 1, sixof dimension 2, two of dimension 4 and one of dimension 26. The table below contains thefirst few Hecke eigenvalues of the one-dimensional constituents. p (2 , w ) (3 , w + 1) (3 , w −
1) (5 , w + 1) (5 , w −
1) (3 w + 31) (3 w − N p a p ( f ) − − − − a p ( f ) − − − − a p ( f ) 1 2 − − − a p ( f ) 1 − − − f and f (resp. f and f ) are interchanged by the action of Gal( F/ Q ) (on theideals p ). The forms f and f (resp. f and f ) are interchanged by the action of Cl + Z F ,so these forms are twists via the strict class character of Gal( F + /F ), where F + denotes thestrict class field of F .Elkies has found a curve E which gives rise to the above data: E : y − wxy − wy = x + ( − − w ) x + ( − w )+ ( − w ) . The curve E has j -invariant j ( E ) = 264235 + 25777 w and has everywhere good reduction.We conclude that E is modular using Kisin [42, Theorem 2.2.18] (see also Kisin [43, Theorem3.5.5]): we need to verify that 3 is split in F , that E has no CM, and that the representation ρ : Gal( F /F ) → GL ( Z ) has surjective reduction ρ : Gal( F /F ) → GL ( F ) which issolvable hence modular.We find that E matches the form f ; so its conjugate by Gal( F/ Q ) corresponds to f andthe quadratic twist of E by the fundamental unit 4005 − w (of norm −
1) corresponds to f (and its conjugate to f ).The input of our algorithm is a totally real number field F of degree n , a totally definitequaternion algebra B with discriminant D , an integral ideal N ⊆ Z F which is coprime with D , a weight k ∈ Z n such that k i ≥ k i ≡ k j (mod 2), and a prime p (cid:45) D . The outputis then a matrix giving the action of T p in a basis of M Bk ( N ) = (cid:76) hi =1 M Bk ( N ) i which isindependent of p . By computing enough T p and simultaneously diagonalising, one obtainsall Hecke constituents corresponding to Hilbert newforms of level N and weight k . he algorithm starts by finding a maximal order O (1), then computes a set of repre-sentatives Cl O (1) of the right ideal classes of O (1) whose norms generate Cl + ( Z F ) andare supported outside DN . This part of the algorithm uses work of the second author andKirschmer [41], it is the most time consuming part but can be seen as a precomputation.Next, the algorithm finds a fundamental domain for the action of each Γ i on P ( Z F / N ), andcomputes M Bk ( N ) as the direct sum of the M Bk ( N ) i = (cid:77) [ x ] ∈ X B ( N ) i W k ( C ) Γ x , where Γ x is the stabilizer of x in Γ i . From this, one obtains a basis of M Bk ( N ). Finally, thealgorithm computes the sets Θ( p ) i,j , and then the block matrices which give the action of T p in this basis. We refer to [15] and [18] for further details on the implementation.9. Indefinite method, arbitrary class number
In this section, we generalize the indefinite method to arbitrary class number. We carryover the notation from Section 7, and now take the quaternion algebra B to be ramified atall but one real place.In this case, from (7.17)–(7.18), the space X ( C ) = X B ( N )( C ) = B × + \ ( H × (cid:98) B × / (cid:98) O × ) is thedisjoint union of Riemann surfaces indexed by Cl + Z F . Let { a } be a set of representativesfor Cl + Z F and let (cid:98) a ∈ (cid:98) Z F be such that (cid:98) a (cid:98) Z F ∩ Z F = a for each a . Then(9.1) X ( C ) = (cid:71) [ a ] ∈ Cl + ( Z F ) Γ a \H = (cid:71) [ a ] ∈ Cl + ( Z F ) X a ( C )where O a = (cid:98) α (cid:98) O (cid:98) α − ∩ B and Γ a = O × a , + .Therefore, a modular form of weight k and level N is a tuple ( f a ) of functions f a : H → W k ( C ), indexed by [ a ] ∈ Cl + Z F , such that for all a , we have( f a | k γ )( z ) = f a ( z )for all γ ∈ Γ a and all z ∈ H . In particular, if k = (2 , . . . ,
2) is parallel weight 2, then ( f a )corresponds to a tuple of holomorphic 1-forms ((2 πi ) f a ( z ) dz ) a , one for each curve X a ( C ).We compute with this space of functions by relating them to cohomology, and for thatwe must modify the coefficient module. Define the right GL ( C ) n = GL ( C ) × GL ( C ) n − -module V k ( C ) = n (cid:79) i =1 P w i ( m i )( C ) = P w ( m )( C ) ⊗ W k ( C ) . The group B × acts on V k ( C ) via the composite splitting B × (cid:44) → GL ( C ) n given by γ (cid:55)→ ( γ i ) i . The Eichler-Shimura theorem, combined with the isomorphism (7.19), applied to eachcomponent X a ( C ) of X ( C ) in (9.1), gives the isomorphism of Hecke modules(9.2) S Bk ( N ) ∼ −→ (cid:77) [ a ] H (Γ a , V k ( C )) + , where + denotes the +1-eigenspace for complex conjugation.In the description (9.2), the Hecke operators act on (cid:76) H (Γ b , V k ( C )) in the following way;we follow their definition in (7.24). Let p be a prime ideal of Z F with p (cid:45) DN . We consider he [ b ]-summand, and given f ∈ H (Γ b , V k ( C )) we will define T p f ∈ H (Γ a , V k ( C )), where[ b ] = [ p − a ]. Let I a = (cid:98) α (cid:98) O ∩ B and I b = (cid:98) β (cid:98) O ∩ B so that nrd( I b ) = b and nrd( I a ) = a , and letΘ( p ) a , b = Γ b \ (cid:8) (cid:36) ∈ B × + ∩ I b I − a : nrd( I b I − a ) p = (nrd( (cid:36) )) (cid:9) = Γ b \ (cid:8) (cid:36) ∈ B × + ∩ I b I − a : nrd( (cid:36) ) a = pb (cid:9) , where Γ b = O × b , + acts by multiplication on the left. Let γ ∈ Γ a , so that γI a = I a . Then themap (cid:36) (cid:55)→ (cid:36)γ on B × induces a bijection (of the equivalence classes) of Θ( p ) a , b . Therefore,for every (cid:36) ∈ Θ( p ) a , b , there exists δ (cid:36) ∈ Γ a and (cid:36) γ ∈ Θ( p ) a , b such that (cid:36)γ = δ (cid:36) (cid:36) γ . From(7.22) and the Eichler-Shimura theorem, we have(9.3) ( T p f )( γ ) = (cid:88) (cid:36) ∈ Θ( p ) a , b f ( δ (cid:36) ) (cid:36) . One can similarly define the Atkin-Lehner involutions.Admittedly, this description is complicated, but it can be summarized simply: a Heckeoperator T p permutes the summands (9.2) in accordance with translation by [ p ] in Cl + Z F ,and adjusting for this factor one can principalize as before (when the strict class number was1). The resulting Hecke matrices are consequently block matrices.We illustrate this with an example; we then give a few more details on the algorithm. Example . Let F = Q ( w ) be the (totally real) cubic field of prime discriminant 257, with w − w − w + 3 = 0. Then F has Galois group S and Z F = Z [ w ]. The field F has classnumber 1 but strict class number 2: the unit ( w − w −
2) generates the group Z × F, + / Z × F of totally positive units modulo squares.Let B = (cid:18) − , w − F (cid:19) be the quaternion algebra with i = − j = w −
1, and ji = − ij .Then B has discriminant D = (1) and is ramified at two of the three real places andunramified at the place with w (cid:55)→ . . . . , corresponding to ι ∞ : B (cid:44) → M ( R ). Theorder O = Z F ⊕ ( w + w − i Z F ⊕ (( w + w ) − i + j ) / Z F ⊕ (( w + w − i + ij ) / Z F is an Eichler order of level N = ( w ) where N ( w ) = 3.A fundamental domain for the action of Γ = ι ∞ ( O × + ) on H is as follows. The ideals (1) and a = ( w + 1) Z F represent the classes in the strict class group Cl + Z F .The ideal J a = 2 O + (( w + w + 2) / − i + (1 / j ) O has nrd( J a ) = a . The left order of J a is O L ( J a ) = O a where O a = Z F ⊕ ( w − w − i Z F ⊕ (cid:0) ( w + w ) / − i + (1 / j (cid:1) Z F ⊕ (1 / (cid:0) (174 w − w − i + ( w − w − j + ( − w + 2 w + 2) ij (cid:1) Z F . A fundamental domain for the action of Γ a = ι ∞ ( O × a , + ) on H is as follows. The orders O and O a are not isomorphic since the connecting ideal I a (with left order O a and right order O ) is not principal. This implies that the groups Γ and Γ a are not conjugateas subgroups of PSL ( R ) but nevertheless are isomorphic as abstract groups: they both havesignature (1; 2 , , , ∼ = Γ a ∼ = (cid:104) γ, γ (cid:48) , δ , . . . , δ : δ = · · · = δ = [ γ, γ (cid:48) ] δ · · · δ = 1 (cid:105) . In particular, both X (1) ( C ) and X a ( C ) have genus 1, sodim H ( X ( C ) , C ) = dim H ( X (1) ( C ) , C ) + dim H ( X a ( C ) , C ) = 4 = 2 dim S ( N ) . We choose a basis of characteristic functions on γ, γ (cid:48) as a basis for H ( X (1) ( C ) , C ) andsimilarly for H ( X a ( C ) , C ).We now compute Hecke operators following the above. Let H = H ( X ( C ) , C ). Wecompute that complex conjugation acts on H by the matrix H | W ∞ = − − − . Note that W ∞ in this case preserves each factor. Now consider the Hecke operator T p where p = (2 w −
1) and N ( p ) = 7. Then p represents the nontrivial class in Cl + Z F . We computethat H | T p = − − − − − − nd restricting we get H + | T p = (cid:18) − − (cid:19) . Therefore there are two eigenspaces for T p with eigenvalues 4 , −
4. By contrast, the Heckeoperator T (2) acts by the scalar matrix 3 on H , preserving each component.Continuing in this way, we find the following table of eigenvalues: N p a p ( f ) − − − − − − − a p ( g ) − − − − − − −
12 8 2 − g is the quadratic twist of the form f by the nontrivial character of thestrict class group Gal( F + /F ), where F + is the strict class field of F . Note also that theseforms do not arise from base change from Q , since a p has three different values for the primes p of norm 61.We are then led to search for elliptic curves of conductor N = ( w ) , and we find two: E f : y + ( w + 1) xy = x − x + ( − w + 51 w − x + ( − w + 557 w − E g : y + ( w + w + 1) xy + y = x + ( w − w − x + (4 w + 11 w − x + (4 w + w − Z / Z -torsion over F , so as above they are modular andwe match Hecke eigenvalues to find that E f corresponds to f and E g corresponds to g .In this situation, although by the theory of canonical models due to Deligne we know thatthe variety X ( C ) = X (1) ( C ) (cid:116) X a ( C ) has a model X F over F , the curves themselves arenot defined over F —they are interchanged by the action of Gal( F + /F ). Nevertheless, theJacobian of X F is an abelian variety of dimension 2 defined over F which is isogenous to E f × E g —we characterize in this way isogeny classes, not isomorphism classes.As in the case of class number 1, the application of Shapiro’s lemma allows us always towork with the group associated to a maximal order, as follows. Let O (1) ⊇ O be a maximalorder containing O , and for each ideal a , let O (1) a = (cid:98) α (cid:98) O (1) (cid:98) α − ∩ B be the maximal ordercontaining O a , and let Γ(1) a = ι ∞ ( O × + ). Further, define V k ( C ) a = Coind Γ(1) a Γ a V k ( C )for each a . Then Shapiro’s lemma implies that H (Γ a , V k ( C )) ∼ = H (Γ(1) a , V k ( C ) a )and so S Bk ( N ) ∼ −→ (cid:77) [ a ] H (Γ(1) a , V k ( C ) a ) + . Our algorithm takes as input a totally real field F of degree [ F : Q ] = n , a quaternionalgebra B over F split at a unique real place, an ideal N ⊂ Z F coprime to the discriminant D of B , a vector k ∈ (2 Z > ) n , and a prime p (cid:45) DN , and outputs the matrix of the Heckeoperator T p acting on the space H = (cid:76) b H (cid:0) Γ(1) b , V k ( C ) b (cid:1) + with respect to some fixedbasis which does not depend on p . From these matrices, we decompose the space H intoHecke-irreducible subspaces by linear algebra. We give a short overview of this algorithm.First, some precomputation. We precompute a set of representatives [ a ] for the strict classgroup Cl + Z F with each a coprime to pDN . For each representative ideal a , precompute right O (1)-ideal I a such that nrd( I a ) = a and let O (1) a = O L ( I a ) be the left order of I a . Next, we compute for each a a finite presentation for Γ(1) a consisting of a (minimal)set of generators G a and relations R a together with a solution to the word problem for thecomputed presentation [67]. Then using standard linear algebra techniques, we compute abasis for the space (cid:76) [ a ] H (Γ(1) a , V k ( C ) a ).The main issue then is to make the description (9.3) amenable to explicit computation.First, compute a splitting ι p : O (1) (cid:44) → M ( Z F, p ). Then for each ideal a , perform thefollowing steps.First, compute the ideal b with ideal class [ b ] = [ p − a ]. Compute the left ideals J a = O a ι − p (cid:18) x y (cid:19) + O a p indexed by the elements a = ( x : y ) ∈ P ( F p ). Then compute the left O b -ideals I b I − a J a andcompute totally positive generators (cid:36) a ∈ O a ∩ B × + corresponding to O b (cid:36) a = I b I − a J a [41].Now, for each γ in a set of generators G a for Γ a , compute the permutation γ ∗ of P ( F p )[30, Algorithm 5.8] and then the elements δ a = (cid:36) a γ(cid:36) − γ ∗ a for a ∈ P ( F p ); write each suchelement δ a as a word in G b and then apply the formula( T p f a )( γ ) = (cid:88) a ∈ P ( F p ) f b ( δ a ) (cid:36) a . The algorithm in its full detail is rather complicated to describe; we refer the reader towork of the second author [69] for the details.
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Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
E-mail address : [email protected] Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave,Burlington, VT 05401, USA
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