There Exist Non-CM Hilbert Modular Forms of Partial Weight 1
aa r X i v : . [ m a t h . N T ] A p r THERE EXIST NON-CM HILBERT MODULAR FORMS OF PARTIALWEIGHT ONE
RICHARD A. MOY AND JOEL SPECTER
Abstract.
In this note, we prove that there exists a classical Hilbert modular cusp form over Q ( √
5) of partial weight one which does not arise from the induction of a Gr¨ossencharacterfrom a CM extension of Q ( √ . Introduction
It is a well-established “folklore question” whether there exists a totally real field F and aclassical Hilbert modular form f of partial weight one which does not arise from the inductionof a Gr¨ossencharacter from some CM extension of F . This note answers the question in theaffirmative (see Theorem 1.1). If, in addition, all the weights of f have the same parity,then, assuming local-global compatibility, there exists a compatible family of representations( L, { ρ λ } ) with the following intriguing property:Let ℓ be a prime in O F not dividing the level of f and totally split in F . If λ is a prime in O L above ℓ , then the corresponding representation, ρ λ : G F → GL ( O λ )will be geometric, have Zariski dense image, and yet be unramified for at leastone v | ℓ .Many cases of local-global compatibility are now known [7]. Although such a beast seemssomewhat peculiar, there is no obvious a priori reasons why it should not exist. On theother hand, there does not seem to be any obvious way (even conjecturally) to produce sucha modular form, either by automorphic or motivic methods. Hence, to answer the question,we must find such a form, which we do. Although (in principle) the method of computationused in this paper applies to general totally real fields, we shall restrict to real quadratic fields F with narrow class number one for convenience. Indeed, all of our computations took placewith Hilbert modular forms for the field F = Q ( √ Mathematics Subject Classification. We originally learnt of this problem through Fred Diamond. In conversations with Kevin Buzzard, DonBlasius, and Fred Diamond, it became clear that the question of whether such forms existed was apparent tothe authors of [1] in the ’80s (and may well have occurred to others before then). The question gained someurgency with the advent of Fraser Jarvis’ construction of Galois representations for partial weight one forms[6] in the mid-’90s, since, if the only such forms were CM, then [6] would be a trivial consequence of Class fieldtheory. We have heard several reports of the question being raised again at this time. In light of these stories,we feel safe in calling the problem a “well-known folklore question.”
The Computation.
Our search for partial weight one Hilbert modular forms is premisedon the philosophy that finite dimensional spaces of meromorphic modular forms which arestable under the action of the Hecke algebra ought to be modular. In the case of classicalmodular forms, this idea has been formalized by George Schaeffer. Let V be a finite dimen-sional space of meromorphic modular forms on Γ ( N ) of weight k and nebentypus χ whichare holomorphic at infinity. In his thesis [9], Schaeffer proves that if V is stable under theaction of a Hecke operator T p for p ∤ N, then V ⊆ M k (Γ ( N ) , χ, C ). As a corollary, oneobserves that for any such V containing M k (Γ ( N ) , χ, C ) , the chain (for p N ) V ⊇ V ∩ T p V ⊇ V ∩ T p V ∩ T p V ⊇ ... stabilizes to M k (Γ ( N ) , χ, C ) in less than dim C V steps [9, 10].Schaeffer’s principal application of this theorem is the effective computation of the space M (Γ ( N ) , χ, C ) of classical weight one modular forms. Suppose one wishes to compute thisspace. To begin, simply take any Eisenstein series E ∈ M (Γ ( N ) , χ − , C ) and let V be thespace of ratios of forms in M (Γ ( N ) , C ) and E. Then V ⊇ M (Γ ( N ) , χ, C ) . It then sufficesto compute the intersection of V with its Hecke translates. One can reduce this computationto one in linear algebra by passing to Fourier expansions. The Fourier expansions of formsin M (Γ ( N ) , C ) are easily calculated to any bound via modular symbols and the Fourierexpansion of E has a simple formula. Hence, the Fourier expansion of any form in V is easilycalculated to any bound. The operator T p acts on Fourier expansions formally via a wellknown formula. What makes the method effective is that it requires only an explict finitenumber of Fourier coefficients for a basis of the space V to calculate M (Γ ( N ) , χ, C ) . Thenumber of coefficients required is determined by the Sturm bounds.Schaeffer’s method generalizes nicely to the case of Hilbert modular forms. Let n be a mod-ulus of Q ( √
5) and χ a ray class character of conductor n ; we are interested in calculating thespace S [ m, (Γ ( n ) , χ, C ) of Hilbert cusp forms of partial weight one. As in the case of classicalmodular forms, there exists an Eisenstein series E ,χ − ∈ M [1 , (Γ ( n ) , χ − , C ) and one canconsider the space V of ratios with numerators in S [ m +1 , (Γ ( n ) , C ) and denominator E ,χ − . This is a finite dimensional space of meromorphic forms which contains S [ m, (Γ ( n ) , χ, C )as its maximal holomorphic subspace. Assuming n is square free, one can use Dembele’salgorithm [4] as implemented in magma [2], to produce the Fourier expansions of a basis forthe space S [ m +1 , (Γ ( n ) , Q ( √ E ,χ − is given by an explicit formula. Hence, the Fourier expansion of the meromorphicforms in V can be calculated to any desired degree of accuracy. For a prime p of O F , theHecke operator T p acts on the Fourier expansions of the meromorphic forms in V formallyvia an explicit formula. So, as in the case of classical forms, one may hope to calculate the T p stable subspace of V via techniques in linear algebra.Unfortunately, this direct generalization of Schaeffer’s method is impractical from a com-putational prospective. In comparison with the case of classical modular forms, the numberof Fourier coefficients needed to prove equality of two modular forms and the amount of com-putation needed to calculate those Fourier coefficients is much greater. For this reason, westructure our search method so that it requires as few Fourier coefficients as possible.For the details of our search, we refer the reader to Section 2.6. But the idea is as fol-lows; we calculate the Fourier expansions of the forms in S [ m +1 , (Γ ( n ) , /E ,χ − truncatedto some chosen bound. We calculate the intersection of these spaces of truncated formal HERE EXIST NON-CM HILBERT MODULAR FORMS OF PARTIAL WEIGHT ONE 3
Fourier expansions using linear algebra. If the dimension of the intersection coincides withthe dimension of the subspace of forms with complex multiplication (CM), then every formin S [ n, (Γ ( n ) , χ, Q ( √ f ∈ V in such a candidate space is holomorphic by checking if there exists aform g ∈ S [ dn,d ] (Γ ( n ) , χ d , Q ( √ f d = g . Our search yielded the existence ofa nonparallel weight one Hilbert modular form without CM. Main Theorem.
Let n = (14) ⊂ O Q ( √ and let χ be the degree ray class character ofconductor (7) · ∞ ∞ such that χ (2) = − √− . The space of cusp forms S [5 , (Γ ( n ) , χ, C ) is -dimensional, and has a basis with coefficients in H = Q ( √ , χ ) . This space has a basisover L = H ( √− consisting of two conjugate eigenforms, neither of which admit complexmultiplication. Remark 1.1.
Let π be the automorphic representation of GL ( A ∞ F ) associated to either ofthese newforms. Since the character χ has conductor prime to 2 and the level at 2 is Γ (2),the local component π is Steinberg (up to an unramified quadratic twist). In particular,this implies that local-global compatibility results of [7, 8] could not be proved directly usingcongruence methods to higher weight, which would only be sufficient for proving compatibilityup to N semi-simplification. 2. Hilbert Modular Forms
In this section, we state some basic definitions and results on classical Hilbert modularforms. Let F be a real quadratic field of narrow class number one. We fix an ordering on thetwo embeddings of F into R and denote, for a ∈ F, the image of a under the i -th embeddingby a i . We say an element a ∈ F is totally positive if a i > i and denote the ring of allsuch elements by O + F . Similarly, we have two natural embeddings of the matrix ring M ( F )into the matrix ring M ( R ). If γ ∈ M ( F ), let γ and γ denote the image of γ under the i -th embedding. Let d F = ( δ ) be the different of F/ Q where δ ∈ O + F . For an integral ideal n of F, we defineΓ ( n ) := (cid:26)(cid:18) a bc d (cid:19) ∈ GL +2 ( F ) : a, d ∈ O F , c ∈ nd , b ∈ d − , ad − bc ∈ O × F (cid:27) where GL +2 ( F ) is the subgroup of GL ( F ) composed of matrices with totally positive deter-minant. If H is the complex upper half-plane, the group Γ ( n ) acts on H × H via fractionallinear transformations by the rule (cid:18) a bc d (cid:19) . ( z , z ) = (cid:18) a z + b c z + d , a z + b c z + d (cid:19) . Let k := [ k , k ] be an ordered pair of nonnegative integers. For γ = (cid:18) a bc d (cid:19) ∈ GL +2 ( F )and z ∈ H × H set j ( γ, z ) k := det( γ ) − k det( γ ) − k ( c z + d ) k ( c z + d ) k . RICHARD A. MOY AND JOEL SPECTER If f : H × H → C and γ ∈ GL +2 ( F ) , we write f | γ to mean the function f | γ : H × H → C given by f | γ ( z ) = j ( γ, z ) − k f ( γz ) . Consider a numerical character χ : ( O F / n ) × → C × which satisfies χ ( u ) = (cid:16) u | u | (cid:17) − k (cid:16) u | u | (cid:17) − k for all u ∈ O × F . A Hilbert modular form of weight k, level n , and character χ is a holomorphicfunction f : H × H → C such that for all γ ∈ Γ ( n ),(2.1) f | γ ( z ) = χ ( d ) f ( z ) . We denote the C -vector space of all such functions by M k (Γ ( n ) , χ, C ) and by M k (Γ ( n ) , C )when χ is the trivial character. As in the case of classical modular forms, we can computeFourier expansions of Hilbert modular forms.2.1. Fourier Expansions. If f ∈ M k (Γ ( n ) , χ, C ) , then for all d ∈ d − F f ( z ) = f ( z + d )by the transformation rule (2.1) since (cid:0) d (cid:1) ∈ Γ ( n ). It follows from Fourier analysis that theform f is given by the series f ( z ) = X α ∈O F c α ( f ) e πi ( α z + α z ) in a neighborhood of the cusp ( ∞ , ∞ ) . The Koecher Principle [5, §
1] states that c α ( f ) = 0unless α is totally positive or α = 0 . If the constant term of the Fourier expansion of f | γ is zero for all γ ∈ GL +2 ( F ), then we call f a cusp form and denote the space of such forms S k (Γ ( n ) , χ, C ) . We denote the space of cusp forms of level n , weight k , and trivial characterby S k (Γ ( n ) , C ).Besides the Koecher Principle, the Fourier expansions of Hilbert modular forms have ad-ditional structure. Let f ∈ S k (Γ ( n ) , χ, C ) . For any totally positive unit η in O F , one cancheck that the coefficient c α ( f ) satisfies the identity:(2.2) c ηα ( f ) = η k / · η k / · c α ( f ) = η ( k − k ) / · c α ( f )by using the transformation rule (2.1) with (cid:0) η
00 1 (cid:1) ∈ Γ ( n ) and equating Fourier expansions.If desired, we can create a formal Fourier expansion indexed over the ideals of F rather thanindexed over elements of O F . In particular, for an ideal a = ( α ), we can set(2.3) c ( a , f ) := N ( a ) ( k − k ) / · c α ( f ) /α ( k − k ) / = c α ( f ) · α k − k ) / , and one can easily check that this is independent of the choice of totally positive generator α of a by using (2.2) above.2.2. Hecke Operators.
For an integral ideal n of O F , letΓ ( n ) = (cid:26)(cid:18) a bc d (cid:19) ∈ GL +2 ( F ) : a ∈ O F , b ∈ d − , c ∈ nd , d − ∈ n (cid:27) . If q is an integral ideal of O F , we may choose a totally positive generator π of q and writethe disjoint union Γ ( n ) (cid:18) π (cid:19) Γ ( n ) = a j Γ ( n ) γ j HERE EXIST NON-CM HILBERT MODULAR FORMS OF PARTIAL WEIGHT ONE 5 where the γ j are a finite set of right coset representatives. We define the q th Hecke operator to be(2.4) T q f := X j f | γ j . If q = ( π ) is a prime ideal relatively prime to n , then we may choose the following cosetrepresentatives for the γ j : γ β := (cid:18) ǫδ − π (cid:19) and γ ∞ := (cid:18) α βδ − δν π (cid:19) (cid:18) π
00 1 (cid:19) where ǫ runs through a complete set of representatives for O F / n , δ is a totally positivegenerator for the different d , ν is a totally positive generator for n , and α, β ∈ O F suchthat απ − νβ = 1. If we normalize our Hecke operator by multiplying it by the constant π k / − π k / − , then it has the following effect on the Fourier expansion of a modular form f ∈ M k (Γ ( n ) , χ, C ): c α ( T q f ) = c απ + π k − π k − χ ( q ) c α/π = c απ + π k − k N ( q ) k − χ ( q ) c α/π where N ( q ) denotes the numerical norm of the ideal q . On the other hand, if q is prime andexactly divides n , then c α ( T π f ) = c απ . Basis For S k (Γ ( n ) , χ, C ) . In general, there will not be a basis of eigenforms for S k (Γ ( n ) , χ, C ).Rather, there will be a new-space S new k (Γ ( n ) , χ, C ) which will be generated by eigenformswhich we now describe.Let m be a divisor of n , and let b be a divisor of n / m . Then there is a map V m , b : S k (Γ ( m ) , χ, C ) → S k (Γ ( n ) , χ, C )given by X α ∈O F c α q α X α ∈O F c α q bα where b = ( b ) and b ∈ O + F . This map only depends on b up to a scalar which one caneasily verify from (2.2). Let S old k (Γ ( n ) , χ, C ) be the subspace of S k (Γ ( n ) , χ, C ) spanned by V m , b ( f ) for all f ∈ S k (Γ ( m ) , χ, C ) and all ( m , b ) with m | n where m = n and b | ( n / m ). Theorthogonal complement of S old k (Γ ( n ) , C ), under the Petersson inner product, is the space S new k (Γ ( n ) , χ, C ); it has a basis of eigenforms which we will refer to as newforms. Dembele’s algorithm computes the space of newforms S new k (Γ ( n ) , χ, C ) by using the factthat they are in bijection, via the Jacquet-Langlands correspondence, with a certain space ofautomorphic forms on a quaternion algebra. We then exploit the fact, special to GL(2), thatthe Fourier expansion of a newform can be recovered from its Hecke eigenvalues. Let q be anon-zero integral prime ideal, and write q = ( π ) for some totally positive π . There is a Heckeoperator T q which acts on the space of cusp forms S k (Γ ( n ) , χ, C ) as defined in (2.4). Withthe identities T q n = T q n − T q − χ ( q ) π k − π k − T q n − , for ( q , n ) = 1, T q n = T n q RICHARD A. MOY AND JOEL SPECTER for q | n , and T rs = T r T s for ( r , s ) = 1, one can compute the Fourier expansions of the newforms in S new k (Γ ( n ) , χ, C ).One can easily calculate the effect of the Hecke operators on formal Fourier expansions indexedover ideals of O F by using (2.3).2.4. Eisenstein Series of Weight One.
In [11], Shimura gives a prescription which attachesto any pair of narrow class characters of F an Eisenstein series of parallel weight k. The Fourierexpansions of these Eisenstein series are calculated in [3], and we recall this result here. As weonly make use of Eisenstein series of parallel weight k = [1 ,
1] associated to pairs consisting ofa trivial and nontrivial character, we include only the details which are relevant to this case.In the classical setting, the Eisenstein series are defined as sums over a lattice, and ananalogous construction is used in the case of Hilbert Modular Forms. Let ψ be a totally oddcharacter of the narrow ray class group modulo n and let U = { u ∈ O × F : Nm( u ) = 1 , u ≡ n } . For z ∈ H , s ∈ C with Re(2 s + 1) >
2, and e F ( x ) = exp(2 πi · Tr F/ Q ( x )), define f ( z, s ) := C · n ) X a ∈O F , b ∈ d − ( a,b ) mod U, ( a,b ) =(0 , az + b ) | az + b | s × X c ∈O F / n sgn( c ) [1 , ψ ( c ) e F ( − bc ) where C := √ d F [ O × F : U ]Nm( d )( − πi ) and sgn( c ) r := sgn( c ) r sgn( c ) r and r = [ r , r ] ∈ ( Z / Z ) .Observe that the above sum for f ( z, s ) is over pairs ( a, b ) of nonzero elements of theproduct O f × d − modulo the action of U (which is diagonal multiplication) as well as overthe representatives c for O F / n .For fixed z , f ( z, s ) has meromorphic continuation in s to the entire complex plane. Set E ,ψ ( z ) := f ( z, . In [3], the authors compute the Fourier series of the above Eisenstein series, E ,ψ . Their resultis summarized in the following proposition. Proposition 2.1.
Let n be an integral ideal of F and let ψ be a totally odd character of thenarrow ray class group modulo n . Then there exists an element E ,ψ ∈ M [1 , (Γ ( n ) , ψ, C ) suchthat c ( a , E ,ψ ) = P m | a ψ ( m ) for all nonzero ideals a of O and c (0 , E ,ψ ) = L ( ψ, . Explicitly, E ,ψ = L ( ψ, X b ∈O + F X m | ( b ) ψ ( m ) · e F ( bz ) HERE EXIST NON-CM HILBERT MODULAR FORMS OF PARTIAL WEIGHT ONE 7
CM Forms.
While in general spaces of Hilbert modular forms of partial weight oneare mysterious, we do have one source to reliably produce such forms; we can obtain themvia automorphic induction from certain Gr¨ossencharacters. Specifically, let K be a totallyimaginary quadratic extension of F and A K be the adeles of K. Consider a Gr¨ossencharacter ψ : GL ( K ) \ GL ( A K ) → C × such that the local components of ψ at the infinite places are ψ ∞ ( z ) = z k − and ψ ∞ ( z ) = | z | k − ∞ . Then, by a theorem of Yoshida [12], there exists a unique Hilbert modular eigenform f ψ ofweight [ k,
1] such that the L -function of f ψ is equal to the L -function of ψ. A Hilbert modular eigenform f is said to have CM if its primitive form is equal to f ψ for some character ψ . From the equality of L -functions, one observes that if p is a prime of F which is inert in K, then the normalized Hecke eigenvalue c ( p , f ψ ) = 0 . Conversely, thisproperty classifies CM Hilbert modular forms. That is, if f is a Hilbert modular form of level c and K is a totally imaginary extension of F such that c ( p , f ) = 0 for all primes p ∤ c whichare inert in K, the primitive form of f is f ψ for some Gr¨ossencharacter ψ of K. By class fieldtheory, one can restate this fact as follows.
Theorem 2.2.
Let f be a Hilbert modular eigenform of level c . Then f has CM if andonly if there exists a totally odd quadratic Hecke character ǫ of F of conductor f such that c ( p , f ) ǫ ( p ) = c ( p , f ) for all p ∤ cf . In this case, we say f has CM by ǫ. If f ψ is a newform arising from the character ψ, then the level of f is equal to ∆ K/F N K/F ( f ( ψ ))where f ( ψ ) is the conductor of ψ . It follows that if f is CM form of level Γ ( c ) , then f hasCM by some Hecke character of conductor dividing c . There are only finitely many such Heckecharacters, and so one can verify by calculating finitely many Hecke eigenvalues of f that f does not have CM.2.6. The Algorithm.
In this section, we outline the algorithm used to search for non-CMmodular forms of weight [ k, O F . Fix a field H and consider the ring of formal Fourier expansions over H (coefficients indexed by the totallynonnegative elements of O F ). For any pair of integers B := ( b , b ) there is an ideal of thisring consisting of all formal Fourier series whose Fourier coefficient c α = 0 if | α | ∞ < b and | α | ∞ < b . The ring of formal Fourier expansions (over H ) truncated to bound B is definedto be the quotient of the ring of formal Fourier expansions by this ideal. Algorithm 1.
The following is a procedure to search for weight [ k, modular forms. Whichon input ( k, n , χ, B ) consisting of(1) k = [ k, a pair of odd integers,(2) n a square free integral ideal of F, (3) χ a totally odd ray class character of F of conductor dividing n · ∞ ∞ ,(4) B = ( b , b ) a pair of positive integers,outputs candidate non-CM weight k , level n , character χ modular forms or finds that noneexist. RICHARD A. MOY AND JOEL SPECTER (1) Using Demb´el´e’s algorithm [4] (see section 2.3), compute, for each m | n , a basis for theimage of S new[ k +1 , (Γ ( m ) , F ) in the ring of formal Fourier expansions over F truncatedto bound B. (2) Using the spaces calculated in step 1 and following the procedure described in Section2.3, compute a basis for the image S [ k +1 , (Γ ( m ) , F ) in the ring of formal Fourierexpansions over F truncated to bound N F/ Q ( q ) · B where q is the small prime fromStep 4.(3) Divide each of the truncated Fourier expansions calculated in step 2 by the Fourierexpansion for E ,χ − described in Section 2.4. Call the space spanned by the resultingtruncated Fourier expansions V ( B ) . (4) Choose a small prime q , which was the principal ideal (2) in our case, and compute T q f for each basis element f of V ( B ) from the previous step.(5) One has now computed two spaces of Fourier expansions, V ( B ) and T q ( V ( B )), eachof which are dimension D = dim S [ k +1 , (Γ ( n )). If the the dimension of V ( B ) is lessthan D , increase B . Compute the intersection V (2) ( B ) := V ( B ) ∩ T q ( V ( B )).(6) Compute the dimension of the subspace in S [ k +1 , (Γ ( n ) , χ, C ) spanned by CM formsusing class field theory. Denote this dimension by h. (7) If dim( V ( B ) ∩ T q ( V ( B ))) = h , then all the forms are CM and the algorithm returnsthe empty set. Otherwise the algorithm returns V (2) ( B ) . When the algorithm returns a nonempty output one increases the bound B and reruns thealgorithm. If the dimension stabilizes at some value greater than the dimension of the spaceof CM forms after several increases in precision, one has found a candidate for a non-CMweight k form.All of our calculations were made for F = Q ( √ . We first used the algorithm to calculatethe dimensions of the spaces M [3 , (Γ ( n ) , χ ) where n is a square-free ideal of O F and χ is atotally odd character modulo n . We restricted ourselves to the case where n is square-free,because the magma package used only worked in this case. Our program searched through allsquare-free n of norm less than 500 and quadratic χ , but we did not find any non-CM Hilbertmodular forms. (In fact, our calculations show that none exist in the spaces we computed.)We next used our algorithm to calculate dimensions of M [5 , (Γ ( n ) , χ ) for all square-freeideals n of norm less than 300. The only candidate space our algorithm found is describedbelow in Section 3. In all other spaces of modular forms, our algorithm found that all formswere CM. 3. A non CM form
Let F = Q ( √ . We order the infinite places of F such that |√ | ∞ > . The ray classgroup of conductor (7) · ∞ ∞ is isomorphic to Z / Z . Let χ be the order 6 character suchthat χ (2) = − √− . The character χ is totally odd. Theorem 3.1.
The space of cusp forms S [5 , (Γ (14) , χ, C ) is -dimensional and has a basiswith coefficients in H . This space has a basis over L = H ( √− consisting of two conjugateeigenforms, neither of which admit complex multiplication. HERE EXIST NON-CM HILBERT MODULAR FORMS OF PARTIAL WEIGHT ONE 9
Proof.
For n a positive integer, we define b ( n ) := n − √ n , n + √ n ! . Applying Algorithm 1 with input ( k, n , χ, B ) = ([5 , , O , χ, B ) with B = b (24) , b (26)and b (28) , respectively, one finds that for each value V (2) ( B ) is two dimensional. Table 1lists the initial normalized Fourier coefficients of one of the truncated forms in V (2) ( B ) . Let f ∈ S [6 , (Γ (14) , , H ) /E ,χ − be a meromorphic modular form whose Fourier expan-sion truncated to b (28) is found in Table 1 . We show f ∈ S [5 , (Γ (14) , χ, H ) , by showing f ∈ S [15 , (Γ (14) , χ , H ) . This is done in two steps.(1) First we show the map taking a form in S [18 , (Γ (14) , H ) to its Fourier expansiontruncated to bound b (28) is an injection.(2) Next we find a form g ∈ S [15 , (Γ (14) , χ , H ) such that the Fourier expansions of g and f are equivalent when truncated to bound b (28) . Noting that ( f − g ) E ,χ ∈ S [18 , (Γ (14) , H ) , it follows from (1) and (2) that f and g areequal.The proofs of facts (1) and (2) are both computational. Using the magma package, one com-putes that the space of cusp forms S [18 , (Γ (14) , H ) has dimension 356 . Then one computesexplicitly the Fourier expansions truncated to bound b (28) for a basis of S [18 , (Γ (14) , H )and shows that the resulting set of truncated formal Fourier series span a space of the samedimension. This proves (1) . To prove (2), one must construct an element S [15 , (Γ (14) , χ ) with a desired property.Unfortunately, the creation of spaces of Hilbert modular forms with nontrivial nebentypusand the computation of their Fourier expansions has not yet been implemented in the magma package. To skirt this issue, we instead use the magma package to compute the truncated tobound b (56) Fourier expansions of the 56 dimensional space S [14 , (Γ (14) , χ , H ) . One thenobtains the Fourier expansions for the forms in the subspace E ,χ .S [14 , (Γ (14) , χ , H ) + T ( E ,χ .S [14 , (Γ (14) , χ , H )) ⊆ S [15 , (Γ (14) , χ )truncated to bound b (28) , in which, following a calculation in linear algebra, one finds a form g as desired in (2) . It follows S [5 , (Γ (14) , χ, C ) is 2-dimensional and has a basis with elementsin H. We now demonstrate the second claim of the proposition: that S [5 , (Γ (14) , χ, C ) has abasis over L = H ( √−
19) consisting of two conjugate eigenforms, neither of which admit com-plex multiplication. Utilizing Algorithm 1, one computes that V (2) ([5 , , O , χ, b (28)) = 0 andhence S [5 , (Γ (14) , χ, C ) = S new[5 , (Γ (14) , χ, C ) . It follows S [5 , (Γ (14) , χ, C ) has a basis over C of simultaneous eigenforms for Hecke algebra. As S [5 , (Γ (14) , χ, C ) has a basis definedover H and is two dimensional, these eigenforms have as a field of definition either H or a qua-dratic extension of H. Calculating the characteristic polynomial of T on S [5 , (Γ (14) , χ, C ) , we obtain that the field of definition is H ( √− . Finally, we see that neither of the forms in S [5 , (Γ (14) , χ, C ) are CM. If this were not thecase, both forms of S [5 , (Γ (14) , χ, C ) would have CM by a quadratic character of conductor14 . The unique such character is χ . However, one observes that χ ( √ ) = − √ normalized Hecke eigenvalue does not vanish for either eigenform in S [5 , (Γ (14) , χ, C ). (cid:3) Remark 3.2.
The Galois group
Gal( L/ Q ) = ( Z / Z ) acts on the Fourier expansion asfollows. The element with fixed field H permutes the two eigenforms. The element with fixedfield Q ( √ , √− sends the eigenform to an eigenform in S [5 , (Γ (14) , χ − , C ) , where χ − is the conjugate of χ . The element with fixed field Q ( √− , √− sends the eigenform to aform in S [1 , (Γ (14) , χ, C ) . See Table 1 for the normalized coefficients c ( p ) for various prime ideals p = ( π ) of smallnorm for one of the two normalized eigenforms in S [5 , (Γ (14) , χ, C ). If c ( π ) is a coefficientin the Fourier expansion of our eigenform for a prime π , then the normalized coefficient is c ( p ) = c ( π ) π as seen in (2.3). The normalized coefficient does not depend on the choice oftotally positive generator π for the ideal p = ( π ). Table 1.
Table of Normalized Coefficients of Eigenform in S [5 , (Γ (14) , χ ) π N ( π ) c ( p ) , p = ( π )2 4 − √− √ −
45 + 15 √− √− − √ − − √− − √− (cid:18) − √− (cid:19) √ −
87 + 87 √− √ − √−
15 + 63 √− − √
57 + 24 √− − √ √ −
456 + 152 √− √ − √−
15 + 66 √− − √ − √−
95 + 39 √ √ −
162 + 4172 √ √
57 + 172 √ √ (cid:16)
49 + 12 √ (cid:17) · (cid:18) √− √− (cid:19) − √− √−
19 + 1029 √ Remark 3.3.
We checked that for N ( p ) < N ( p ) ,
14) = 1 the Satake parametersof π satisfy the Ramanujan Conjecture. Equivalently, the Hecke eigenvalues satisfy the bounds | c ( p ) | ∞ ≤ p and | c ( p ) | ∞ ≤ p . The Ramanujan conjecture would follow from Deligne’sproof of the Riemann hypothesis if one knew that π was motivic , however, the constructionof the associated Galois representations proceeds via congruences.4. Acknowledgements
The authors would like to extend special thanks to Frank Calegari for introducing thisproblem, his excellent advising, and his extensive comments on preliminary drafts of thispaper. The authors would also like to thank Kevin Buzzard, Lassina Demb´el´e, and JamesNewton for their comments on a preliminary version of this paper, and Don Blasius, KevinBuzzard, and Fred Diamond for their historical remarks.5.
Funding
The first author was supported in part by National Science Foundation Grant DMS-1404620. The second author was supported in part by National Science Foundation Grant
HERE EXIST NON-CM HILBERT MODULAR FORMS OF PARTIAL WEIGHT ONE 11
DMS-1404620 and by an National Science Foundation Graduate Research Fellowship underGrant No. DGE-1324585.
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Richard Moy, Department of Mathematics, Northwestern University, 2033 Sheridan Road,Evanston, IL 60208, United States
E-mail address : [email protected] Joel Specter, Department of Mathematics, Northwestern University, 2033 Sheridan Road,Evanston, IL 60208, United States
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