On the canonical decomposition of generalized modular functions
aa r X i v : . [ m a t h . N T ] M a r On the canonical decomposition of generalized modularfunctions
Geoffrey Mason ∗ Department of Mathematics,University of California Santa Cruz,CA 95064, U.S.A.Winfried KohnenMathematisches Institut der Universit¨at HeidelbergINF 288, D-69120 HeidelbergGermany
Abstract
The authors have conjectured ([KoM]) that if a normalized generalized modular function(GMF) f , defined on a congruence subgroup Γ, has integral Fourier coefficients, then f isclassical in the sense that some power f m is a modular function on Γ. A strengthened form ofthis conjecture was proved (loc cit) in case the divisor of f is empty . In the present paper westudy the canonical decomposition of a normalized parabolic GMF f = f f into a productof normalized parabolic GMFs f , f such that f has unitary character and f has emptydivisor . We show that the strengthened form of the conjecture holds if the first ”few” Fouriercoefficients of f are algebraic. We deduce proofs of several new cases of the conjecture, inparticular if either f = 1 or if the divisor of f is concentrated at the cusps of Γ. Mathematics Subject Classification 2000 : 11F03, 11F99, 17B69
Let Γ ⊂ Γ := SL ( Z ) be a congruence subgroup and let H be the complex upper half-plane.We consider generalized modular functions of weight zero (GMF’s) on Γ. These are holomorphicfunctions f : H → C which satisfy f ( γ ◦ z ) = χ ( γ ) f ( z ) ( ∀ γ ∈ Γ)where χ : Γ → C ∗ is a (not necessarily unitary) character, and which are meromorphic at the cusps.We call f a parabolic generalized modular form (PGMF) if it also satisfies χ ( γ ) = 1 for all parabolic ∗ Supported by the NSF, NSA, and the Committee on Research at the University of California, Santa Cruz γ ∈ Γ of trace 2. If f is a GMF then some power f m of f is a PGMF. For further detailswe refer to [KM1]. In this paper we deal mainly with PGMFs.At the cusp infinity, a PGMF f has a Fourier expansion f ( z ) = X n ≥ h a ( n ) q nN (0 < | q N | < ǫ )for appropriate h ∈ Z , N ∈ N and where q N := e πiz/N ( z ∈ H ). We shall call f normalized if a ( h ) = 1. According to [KM2], each normalized PGMF f on Γ has a canonical decomposition (1) f = f f where f and f are normalized PGMF’s on Γ, f has unitary character, and the divisor of f is empty . The canonical decomposition is indeed unique, which amounts to the assertion (loc cit) thata PGMF with empty divisor and unitary character is constant. Note that it follows from theseconditions and our assumptions that f = q hN + . . . , f = 1 + . . . . It was conjectured in [KoM], Sect. 1, that if a normalized PGMF f on the Hecke congruence sub-group Γ ( N ) of level N has integral Fourier coefficients, then it must be classical, i.e. the character χ is of finite order. A proof of this conjecture would have some very important consequences inrational conformal field theory, as explained in [KoM].In the present paper we shall show that the conjecture is equivalent to requiring that the first“few” Fourier coefficients of the function f in the decomposition (1) are algebraic numbers. Wewill actually prove slightly stronger statements regarding the hypothesis on the Fourier coefficientsof f , requiring only that they are rational and p -integral for almost all primes p . This result impliesseveral new cases of the conjecture: if χ is unitary (i.e. f = 1), or if the divisor div ( f ) of f issupported at the cusps of Γ. Some special cases of the second assertion, including the case when div ( f ) is empty (i.e. f = 1), were established in [KoM]. Our present results are valid for anarbitrary congruence subgroup Γ.Apart from the decomposition (1), there are two main ingredients to the proof of our results: thefirst is a theorem of Scholl-Waldschmidt ([Sc], [W]) on the transcendence of canonical differentialsof the third kind on modular curves; the second one is the result on PGMF’s with empty divisors[KoM] already mentioned above, whose proof largely depends on the analytic theory of Dirichletseries.The paper is organized as follows. We give the proof of the main results, Theorem 1 andCorollary 2, in Section 3. In Section 4 we consider the action of complex conjugation on PGMFsand their characters. For example, we show that if the Fourier coefficients of a PGMF f are realthen so are those of f and f . Acknowledgements.
The authors thank Henri Darmon and Jan H. Bruinier for useful conversations.The results of the present paper emerged from discussions during the stimulating Workshop onnoncongruence modular forms, organized by Ling Long and Winnie Li in August 2009, at theAmerican Institute of Mathematics in Palo Alto.2
Statement of results
We define κ Γ := [ 16 [ P Γ : P Γ]] + 1 − c Γ . Here P Γ := P SL ( Z ) and P Γ denotes the image of Γ under the natural projection Γ → P Γ .Furthermore [ x ] ( x ∈ R ) denotes the greatest integer function and c Γ is the number of cusps of Γ.The main result of the paper is the following: Theorem 1
Let f be a normalized PGMF on Γ whose Fourier coefficients a ( n ) are rational for all n and are p -integral for all but a finite number of primes p . Let (1) be the canonical decompositionof f . Then the character χ of f is of finite order if, and only if, the Fourier coefficients a ( n ) of f are algebraic for h ≤ n ≤ κ Γ + h . Theorem 1 has the following consequence.
Corollary 2
Let f be a PGMF on Γ and let χ be the character of f . Assume that the Fouriercoefficients a ( n ) are rational for all n and are p -integral for all but a finite number of primes p .Then χ has finite order if either of the following conditions hold: ( a ) χ is unitary , ( b ) The divisor of f is concentrated at the cusps of Γ . We point out ([KM1]) that the condition on the divisor in part (b) is equivalent to the assumptionthat the logarithmic derivative f ′ /f , which is generally a meromorphic modular form of weight 2on Γ, is in fact holomorphic . Without loss of generality we may assume that Γ = Γ( N ) is the principle congruence subgroup ofΓ of level N .In one direction the conclusion is easy. Indeed, assume that χ has finite order m . Then f m hastrivial character, so that f m = f m · f m = f m · f m . Therefore f and f differ by an m -th root of unity, andsince they are both normalized then they are equal and all Fourier coefficients of f are algebraic,indeed rational.Now supppose that a ( n ) is algebraic for h ≤ n ≤ κ + h , where we have abbreviated κ := κ Γ .Solving recursively in (1) for the Fourier coefficients a ( n ) of f , we see that our assumption impliesthat each a ( n ) (0 ≤ n ≤ κ ) is also algebraic. 3et(2) 2 πiN g := f ′ f be the logarithmic derivative of f and write g = X n ≥ b ( n ) q nN . Then we see that each b ( n ) (1 ≤ n ≤ κ ) is algebraic. Note that g is a cusp form of weight 2 on Γwith trivial character, since div ( f ) = ∅ [KM1].We now assert Lemma 3
All Fourier coefficients b ( n ) ( n ≥ are contained in a finite extension K/ Q . Proof:
Our claim is essentially well-known, and follows from linear algebra combined with thevalence formula and the fact that Γ = Γ( N ). However, for the reader’s convenience we will give adetailed proof.First recall that the valence formula says that the sum of the orders (measured in the appropriatelocal variables) of a non-zero cusp form of weight 2 on Γ on the complete modular curve X Γ := Γ \H is equal to [ [ P Γ : P Γ]].Since Γ = Γ( N ), it is well-known that the space S (Γ) of cusp forms of weight 2 for Γ has abasis { g , . . . , g d } of functions with rational (in fact integral) Fourier coefficients ([Sh], Thm. 3.52).Note that the valence formula implies that d ≤ κ .We write g as a linear combination of the g ν (1 ≤ ν ≤ d ). Bearing in mind that the first κ Fourier coefficients of g are algebraic, our claim will follow if we can show that the κ × d matrix A whose columns consists of the first κ Fourier coefficients of g , . . . , g d has maximal rank.To show this we argue as follows. Let h , i denote the usual inner product on S (Γ). Let P n ( n ≥
1) be the n -th “Poincar´e series” in S (Γ) with respect to h , i , i.e. P n is the dual of thefunctional that sends a cusp form g ∈ S (Γ) to its n -th Fourier coefficient a g ( n ). By the valenceformula, { P , . . . , P κ } generates S (Γ). Hence, there exists a basis { P n , . . . , P n d } where 1 ≤ n ν ≤ κ for all ν .On the other hand, let ℓ ∈ S ∗ (Γ) be any functional. Then by standard duality, there exists L ∈ S (Γ) such that h g, L i = ℓ ( g ) for all g . Writing L in terms of the basis { P n , . . . , P n d } , we seethat ℓ is a linear combination of a ( n ) , . . . , a ( n d ), hence the latter functionals form a basis of S ∗ (Γ).From the above it follows that the κ × d matrix B whose columns are the first κ Fourier coefficientsof P n , . . . , P n d has maximal rank (the rows with indices n , . . . , n d are linearly independent). Hencethe same is true for A , since A is obtained from B by multiplying with an invertible d × d matrix.This completes the proof of the Lemma. ✷ In (2) we now solve recursively for the a ( n ). Using Lemma 1, we see that a ( n ) ∈ K for all n .Therefore by (1), each a ( n ) also lies in K . 4e put(3) 2 πiN g := f ′ f . Then g also has Fourier coefficents in K , and g is a meromorphic modular form of weight 2 on Γwith trivial character. It has at worst simple poles in H with integral residues, and is holomorphicat the cusps [KM1].We let D := div ( f ) . Then degD = 0 [KM1], while div ( f ) = div ( f ) by hypothesis.The form g defined by (3) gives rise to an abelian differential of the third kind ω := 2 πiN g dz on X Γ with residue divisor D . Lemma 4
The divisor D is defined over a number field. Proof:
This essentially is well-known: indeed, the Galois group operates on meromorphic differ-entials and this operation is compatible with the formation of residue divisors. However, for theconvenience of the reader we again give a detailed proof. We shall prove somewhat more, namelythat each point P in the support of D is already fixed by Gal ( Q /L ) where L/ Q is a finite extension.The modular curve X Γ is defined over a number field F (in fact, over Q ( ζ N ) where ζ N = e πi/N ).Since the cusps of Γ are defined over a number field, we may suppose that P is contained in the“open part” Y Γ := X Γ \ { cusps } of the modular curve.The function field F X Γ of X Γ /F is a finite extension of F ( j ) where j is the classical modularinvariant. By the “theorem of the primitive element” there exists a modular function t for Γ suchthat F X Γ = F ( j, t ), and t satisfies an algebraic equation over F ( j ). The points of Y Γ then can beparametrized as ( j ( z ) , t ( z )) ( z ∈ Γ \H ).Suppose that P “corresponds” to ( j ( z ) , t ( z )) ( z ∈ H ). It is sufficient to show that j ( z ) isalgebraic over Q . Because P is not a cusp then z is necessarily a zero of f , and hence a simple poleof g . Let G := g ∆ , where ∆ is the usual discriminant function of weight 12 on Γ . Then G is a modular function onΓ with a pole at z . Because g has Fourier coefficients in K , the same is true of G . Observe thateach of the translates ( G | γ )( z ) := G ( az + bcz + d ) , γ = (cid:18) a bc d (cid:19) ∈ Γ , also have Fourier coefficients in a number field. Once again, this is a standard result. The proof isbased on the “ q -expansion principle” for congruence subgroups ([DR], VII, 4.8). (A more generalproof that works for noncongruence subgroups can be found in [KL], Proposition A1.)5e may, and shall, choose c ∈ Q such that none of the modular functions ( G | γ )( z ) − c have azero at z . Now consider the “norm” Y γ ∈ P Γ \ P Γ ( G − c ) | γ. It is a modular function on Γ with coefficients in K , and hence is a rational function A ( j ) /B ( j )with A ( j ) , B ( j ) ∈ K [ j ]. By construction there is a pole at z , so that B ( j ( z )) = 0. The algebraicityof j ( z ) follows, and the proof of the Lemma is complete. ✷ Next, observe that ω is a canonical differential on X Γ , i.e. ℜ ( Z σ ω ) = 0for all σ ∈ H ( U ; Z ) where U := X Γ \ suppD . Indeed, this is equivalent to our assumption that f has unitary character, bearing in mind the relation (3). We may now invoke an important theoremof Scholl-Waldschmidt [Sc], [W]: since g has Fourier coefficients in a number field, D has finiteorder in the divisor class group. Thus there exists m ∈ N and a modular function h on Γ such that div ( f m h ) = ∅ . Therefore f m = h , since f and h both have unitary character.We may normalize h to have Fourier coefficients in the compositum of Q ( ζ N ) and the field ofdefinition of the algebraic divisor D (cf. Lemma 4) and to have leading non-zero term equal to 1.From (1) we now obtain(4) f m = h · h , where h := f m . Since div ( h ) = ∅ and h has trivial character, it follows that (4) is the canonical decomposition of f m . Lemma 5
Let h = P b ( n ) q nN be a modular function on Γ = Γ( N ) with Fourier coefficients in anumber field, and let σ ∈ Gal ( Q / Q ) . Then h σ := P b ( n ) σ q nN is a modular function on Γ . Proof:
This is well-known. Indeed, arguing as in the proof of Lemma 4, we see that j ( z ) isalgebraic if z ∈ H is a pole of h . We therefore conclude that there exists a polynomial P withalgebraic coefficients and a large positive integer M such that H := ∆ M P ( j ) h is a cusp form of weight k ≥ H is a Q -rational linear combination of such functions (an argument similarto that used in proof of Lemma 3 with 2 replaced by k is valid). The result follows from this. ✷ F/ Q . In (4) we now take “norms” (products of Galois conjugates of σ ∈ Gal ( F/ Q )) toobtain an equation(5) f mr = H · H , where r = [ F : Q ], H and H are normalized and have rational Fourier coefficients, and (thanks toLemma 5) H has trivial character.Now notice that div ( H ) = ∅ . Indeed, if πiN w ∈ S (Γ) corresponds to the normalized PGMF v upon taking logarithmic derivatives, then the cusp form πiN w σ corresponds to v σ (where of coursethe action of Galois elements σ is defined in the same way as above).Since H has rational coefficients and Γ = Γ( N ), the coefficients of H must in fact be p -integralfor almost all primes p ([Sh]). Since H is normalized, the same holds for H − and therefore also for H by (5). So we have arrived at the situation that H is a normalized PGMF with rational Fouriercoefficients and empty divisor . By [KoM], Theorem 2, we conclude that H = 1. Thus f mr = H has trivial character, i.e. the character of f is of finite order. This concludes the proof of Theorem1. We turn to the proof of Corollary 2. Suppose first that χ is unitary . Then f = 1, so that f = f has rational Fourier coefficients by hypothesis. Therefore, χ has finite order by Theorem 1.This proves part (a) of the Corollary. As for part (b), because the divisor D = div ( f ) = div ( f ) isassumed to be concentrated at the cusps, D is defined over a number field and the Manin-Drinfeldtheorem tells us that div ( f ) has finite order in the divisor class group. Then f m has algebraicFourier coefficients for some integer m , hence f does too. Now Theorem 1 again tells us that χ hasfinite order. This completes the proof of part (b) of the Corollary. In this Section we briefly consider the action of complex conjugation on PGMFs. Recall Hecke’soperator K , defined on holomorphic functions in H as follows ([R], Section 8.6) f | K ( z ) = f ( − z ) . If f is a PGMF on Γ then so is f | K (loc. cit.), and the q -expansions at the infinite cusp are relatedas follows:(6) f ( z ) = X a ( n ) q nN , f | K ( z ) = X a ( n ) q nN . Set J = (cid:18) − (cid:19) . Note that J (cid:18) a bc d (cid:19) J − = (cid:18) a − b − c d (cid:19) .
7n the following we will need to assume that J normalizes Γ. From the last display, we see thatthis holds, for many congruence subgroups, e.g., Γ = Γ( N ), or Γ ( N ). We write γ J = J γJ − for γ ∈ Γ. A character χ of Γ may be ‘twisted’ by J to yield a second character χ J defined by χ J ( γ ) = χ ( γ J ) . Lemma 6
Assume that J normalizes Γ , and suppose that the PGMF f is associated with thecharacter χ . Then f | K is associated with the character ¯ χ J . In particular, f has real Fouriercoefficients if, and only if, χ = ¯ χ J . Proof:
For γ ∈ Γ we have f | K ( γz ) = f ( − γz ) = f ( γ J ( − z )) = ¯ χ J ( γ ) f | K ( z ) . This proves the first assertion. The second follows from (6). ✷ Suppose that f is a PGMF with canonical decomposition (1), and that χ j is the characterassociated to f j , j = 0 ,
1. We have(7) f | K = ( f | K )( f | K ) . By Lemma 6, f j | K has associated character χ j J . Because χ is unitary then so too is χ J . Moreover,it is easy to see that f | K has empty divisor. It follows from these comments that (7) is the canonicaldecomposition of f | K .If now we assume that f has real Fourier coefficients, then χ = χ J by Lemma 6. By theuniqueness of the canonical decomposition, we can conclude that χ j J = χ j for j = 0 ,
1. ApplyingLemma 6 once more, we arrive at
Lemma 7
Suppose that f has real Fourier coefficients. Then f and f have real Fourier coeffi-cients. ✷ The condition χ j J = χ j places strong restraints on the characters χ j . For example, in the caseof the unitary character χ we have χ ( γ J γ ) = 1 for γ ∈ Γ . Finally, notice that Lemma 7 applies in the context of Theorem 1. If we could replace the realfield by a number field in the statement of the Lemma, the main conjecture would follow.
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Modular Forms , Cambridge University Press, Cambs., 1977.[Sc] A. J. Scholl: Fourier coefficients of Eisenstein series on non-congruence subgroups, Math. Proc.Camb. Phil. Soc. (1986), 11-17.[Sh] G. Shimura, Introduction to the arithmetic theory of automorphic forms , Iwanami Shoten andPrinceton University Press, 1971[W] M. Waldschmidt: Nombres transcendents et groupes alg´ebraiques, Ast´erisque (1979).e-mail addresses of authors:
Winfried Kohnen [email protected]