aa r X i v : . [ m a t h . N T ] J u l FOURIER EXPANSIONS AT CUSPS
FRANC¸ OIS BRUNAULT AND MICHAEL NEURURER
Abstract.
In this article we study the number fields generated by the Fourier coefficientsof modular forms at arbitrary cusps. We give upper bounds for these number fields which arecyclotomic extensions of the field generated by the Fourier coefficients at ∞ , and determinethem explicitly for newforms with trivial Nebentypus. The main tool is an extension of aresult of Shimura on the compatibility between the actions of SL ( Z ) and Aut( C ) on thespace of modular forms. We give two new proofs of this result: one based on products ofEisenstein series, and the other using the theory of algebraic modular forms. Introduction
In this article we study the number fields generated by the Fourier coefficients of modularforms at the cusps of X ( N ). To do this we study the connections between two actionson spaces of modular forms: the action of GL +2 ( Q ) via the slash-operator and the actionof Aut( C ) on the Fourier coefficients of a modular form. A detailed study of these actionswas conducted by Shimura in [16], where he proved a formula for the action of Aut( C ) on f | g for modular forms of even weight. In Theorem 3.3 we give an extension of his result tomodular forms of any integral weight and provide two new proofs of it: one using a theoremof Khuri-Makdisi [11] on products of Eisenstein series, and the other using Katz’s theory ofalgebraic modular forms [10].We use this theorem to bound the fields generated by the Fourier coefficients of modularforms at the cusps. Let us assume for simplicity that f is a modular form in M k (Γ ( N )),and let g = ( A BC D ) ∈ SL ( Z ). We show in Theorem 4.1 that the coefficients of f | g lie in thecyclotomic extension K f ( ζ N ′ ), where K f is the number field generated by the coefficients of f , and N ′ = N/ gcd( CD, N ). In the case f has non-trivial Nebentypus, we show in Theorem4.4 that the coefficients of f | g belong to a 1-dimensional K f ( ζ N ′ )-vector space, which is itselfcontained in an explicit cyclotomic extension K f ( ζ M ).We apply these results in Section 5 to find number fields that contain the Atkin–Lehnerpseudo-eigenvalues of a newform, recovering a result of Cohen in [4].In Section 6 we discuss how to choose g among the matrices in SL ( Z ) that map ∞ toa given cusp α so that N ′ (or M ) becomes minimal. Assuming that f ∈ M k (Γ ( N )) isan eigenfunction of the Atkin–Lehner operators, we describe how to further reduce N ′ bypotentially replacing α with its image under a suitable Atkin–Lehner operator. The Fourierexpansion of f | g can then easily be obtained from another Fourier expansion f | g ′ which hascoefficients in the field K f ( ζ gcd( δ,N/δ ) ), where δ = gcd( C, N ) is the denominator of the cusp α = A/C . Note that Q ( ζ gcd( δ,N/δ ) ) is the field of definition of the cusp α in the canonicalmodel of X ( N ) over Q . The second author was partially funded by the DFG-Forschergruppe 1920 and the LOEWE research unit“Uniformized Structures in Arithmetic and Geometry”. n the last section, we prove in Theorem 7.6 that if f is a newform for Γ ( N ) then the numberfield provided by Theorem 4.1 is the best possible, in the sense that it is the number fieldgenerated by the coefficients of f | g .Recently three algorithms for the computation of the Fourier expansion of f | g have appeared:two algorithms in Sagemath, one by Dan Collins [5] and another by Martin Dickson and thesecond author [8]. The third algorithm was implemented in PARI/GP by Karim Belabasand Henri Cohen [4]. While the first algorithm only uses numerical approximations of theFourier coefficients in order to compute Petersson inner products, the latter two calculatethe Fourier coefficients as algebraic numbers. The knowledge of the number field (or vectorspace) generated by the Fourier coefficients of f | g could provide a significant speed-up forthese calculations. Acknowledgements:
We thank Henri Cohen for encouraging us to write this article.We are grateful to Abhishek Saha for pointing out an alternative approach to the Theoremsin Section 4, sketched in Remark 4.8.
Notations.
For any integer k ∈ Z , we define the weight k action of GL +2 ( R ) on functions f : H → C by f | k g ( τ ) = det( g ) k/ ( cτ + d ) k f (cid:16) aτ + bcτ + d (cid:17) (cid:18) g = (cid:18) a bc d (cid:19) ∈ GL +2 ( R ) (cid:19) . We will usually omit k from the notation and just write f | g for f | k g .The automorphism group Aut( C ) acts on spaces of modular forms as follows: for any modularform f ( τ ) = P n a n e πinτ/w , we let f σ ( τ ) = X n σ ( a n ) e πinτ/w ( σ ∈ Aut( C )) . For any integer N ≥
1, we denote ζ N = e πi/N ∈ C .2. Eisenstein series
Definitions.
We refer the reader to [9, §
3] for more details on Eisenstein series.For integers k ≥ N ≥ a, b ∈ Z /N Z , define the series E ( k ) a,b ( τ ) = ( k − − πi ) k X ω ∈ Z τ + Z ω = − (˜ aτ +˜ b ) /N ω + ˜ aτ +˜ bN ) k | ω + ˜ aτ +˜ bN | s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s =0 where ˜ a, ˜ b denote any representatives of a, b in Z , and ·| s =0 denotes analytic continuation to s = 0 (this is needed only when k ∈ { , } ). It follows from the definition that the weight k action of SL ( Z ) on these series is given by E ( k ) a,b | g = E ( k )( a,b ) g for every matrix g ∈ SL ( Z ). Inparticular, the function E ( k ) a,b is modular of weight k with respect to the principal congruencesubgroup Γ( N ). If k = 2, then E ( k ) a,b is a holomorphic Eisenstein series of weight k withrespect to Γ( N ). If k = 2, then ˜ E (2) a,b := E (2) a,b − E (2)0 , is a holomorphic Eisenstein series ofweight 2 with respect to Γ( N ). .2. Fourier expansions.
We refer the reader to [10, §
3] and [15, Chap. VII] for proofs ofthe following facts.If k = 2, then the Fourier expansion of E ( k ) a,b is given by E ( k ) a,b ( τ ) = a ( E ( k ) a,b ) + X m,n ≥ m ≡ a ( N ) ζ bnN n k − q mn/N + ( − k X m,n ≥ m ≡− a ( N ) ζ − bnN n k − q mn/N ( q = e πiτ ) . If k = 2, then the Fourier expansion of ˜ E (2) a,b is given by˜ E (2) a,b ( τ ) = a ( ˜ E (2) a,b ) + X m,n ≥ m ≡ a ( N ) ζ bnN nq mn/N + X m,n ≥ m ≡− a ( N ) ζ − bnN nq mn/N − X m,n ≥ nq mn . The constant terms a ( E ( k ) a,b ) and a ( ˜ E (2) a,b ) are elements of Q ( ζ N ) and are given in [9, 3.10] and[3, § Proposition 2.1.
Let g = ( A BC D ) ∈ SL ( Z ) and σ ∈ Aut( C ) such that σ ( ζ N ) = ζ λN with λ ∈ ( Z /N Z ) × . If k = 2, then ( E ( k ) a,b | g ) σ = ( E ( k ) a,b ) σ | g λ , where g λ is any lift in SL ( Z ) of the matrix (cid:0) A λBλ − C D (cid:1) ∈ SL ( Z /N Z ). If k = 2, then thesame statement holds with E (2) a,b replaced by ˜ E (2) a,b . Proof.
Note that ( E ( k ) a,b ) σ = E ( k ) a,λb , so that( E ( k ) a,b | g ) σ = E ( k ) aA + bC,λ ( aB + bD ) = E ( k ) a,λb (cid:12)(cid:12)(cid:12)(cid:18) A λBλ − C D (cid:19) = ( E ( k ) a,b ) σ | g λ . The argument for ˜ E (2) a,b is similar. (cid:3) The actions of SL ( Z ) and Aut( C ) on modular forms In this section we investigate the connection between the natural actions of SL ( Z ) andAut( C ) on modular forms.Let us first recall Khuri-Makdisi’s result [11] giving generators of the graded algebra ofmodular forms. Let R N be the subalgebra of M ∗ (Γ( N )) = L k ≥ M k (Γ( N )) generated bythe Eisenstein series E (1) a,b with a, b ∈ Z /N Z . Theorem 3.1. [11] If N ≥
3, then R N contains all modular forms on Γ( N ) of weight 2 andabove. In other words, R N misses only the cusp forms of weight 1 on Γ( N ). Proof.
This follows from combining [11, Theorem 3.5, Remark 3.14, Theorem 5.1]. The linkbetween our notations and Khuri-Makdisi’s notations is E (1) a,b ( τ ) = − πi G (cid:16) τ, aτ + bN (cid:17) = 12 πi λ ( aτ + b ) /N , see [11, Definition 2.1 and Corollary 3.13]. (cid:3) emark . If N = 2, then the algebra M ∗ (Γ(2)) is generated by the weight 2 Eisensteinseries ˜ E (2)1 , , ˜ E (2)0 , and ˜ E (2)1 , , the only relation being ˜ E (2)1 , + ˜ E (2)0 , + ˜ E (2)1 , = 0, see [11, Remark3.6]. Of course, if N = 1 then M ∗ (SL ( Z )) is freely generated by the usual Eisenstein seriesof weight 4 and 6.The following theorem is an extension of Shimura’s result [16, Theorem 8] which deals withthe case of modular forms of even weight. We note however, that Shimura works with amuch wider class of functions, including Hilbert modular forms and also certain derivativesof them. His methods can be extended to provide a full proof of Theorem 3.3, but we willgive two new proofs of it, the first using Khuri-Makdisi’s Theorem 3.1. The second proofuses Katz’s theory of algebraic modular forms and is given in the appendix along with abrief introduction to the theory of algebraic modular forms. Theorem 3.3.
Let f ∈ M k (Γ( N )) be a modular form of weight k ≥ N ). Let g = ( A BC D ) ∈ SL ( Z ) and σ ∈ Aut( C ) such that σ ( ζ N ) = ζ λN with λ ∈ ( Z /N Z ) × . Then( f | g ) σ = f σ | g λ , where g λ is any lift in SL ( Z ) of the matrix (cid:0) A λBλ − C D (cid:1) ∈ SL ( Z /N Z ). First proof.
Let us assume N ≥
3, and let f ∈ R N . The maps h ( h | g ) σ and h h σ | g λ are both σ -linear ring homomorphisms. We are thus reduced to the case f = E (1) a,b , whichfollows from Proposition 2.1. If f ∈ S (Γ( N )), then f and f are in R N , so Theorem 3.3holds for them. Using f = f /f , we get ( f | g ) σ = f σ | g λ . In the case N = 2, we proceedsimilarly by applying Proposition 2.1 to ˜ E (2) a,b . Finally, the case N = 1 is trivial. (cid:3) Remark . If we restrict to modular forms on Γ ( N ), then Theorem 3.3 also follows fromthe result of Borisov and Gunnells [2, Thm 5.15] that all modular forms of sufficiently largeweight are toric.By Theorem 3.3, the space M k (Γ( N ); Q ( ζ N )) of modular forms with coefficients in Q ( ζ N )is stable under the weight k action of SL ( Z ). It is thus endowed with a right action ofSL ( Z /N Z ). The Galois group Gal( Q ( ζ N ) / Q ) also acts on M k (Γ( N ); Q ( ζ N )), by means ofthe usual action on the Fourier expansion. But more is true: for any f ∈ M k (Γ( N ); Q ( ζ N )),let us define f (cid:12)(cid:12)(cid:12)(cid:18) λ (cid:19) = f σ λ ( λ ∈ ( Z /N Z ) × ) , where σ λ ∈ Gal( Q ( ζ N ) / Q ) is the automorphism defined by σ λ ( ζ N ) = ζ λN . Then the aboveactions of SL ( Z /N Z ) and ( Z /N Z ) × combine to give a right action of GL ( Z /N Z ) on M k (Γ( N ); Q ( ζ N )). Indeed(1) g (cid:18) λ (cid:19) = (cid:18) A λBC λD (cid:19) = (cid:18) λ (cid:19) (cid:18) A λBλ − C D (cid:19) = (cid:18) λ (cid:19) g λ and Theorem 3.3 says precisely that both sides of this equality act in the same way on M k (Γ( N ); Q ( ζ N )). Note also that with this definition, the identities E ( k ) a,b | g = E ( k )( a,b ) g for k = 2 and ˜ E (2) a,b | g = ˜ E (2)( a,b ) g are true for any g ∈ GL ( Z /N Z ). Remark . Let ˜ Y ( N ) be the model of Γ( N ) \H over Q ( ζ N ) constructed in [17, Chapter 6].The automorphism group of the Q -scheme ˜ Y ( N ) is GL ( Z /N Z ). This gives a right action of L ( Z /N Z ) on the function field Q ( ζ N )( ˜ Y ( N )), which is a subfield of Q ( ζ N )(( q /N )). Theaction of SL ( Z /N Z ) is the slash action of weight 0, and the action of the diagonal matrix( λ ) with λ ∈ ( Z /N Z ) × coincides with the natural action of σ λ . Shimura’s original proof ofTheorem 3.3 relies on this fact. Corollary 3.6.
Let f ∈ M k (Γ ( N ) , χ ) be a nonzero modular form of weight k ≥
1, level N ≥ χ . Then the field K f generated by the Fourier coefficientsof f contains the field Q ( χ ) generated by the values of χ . Proof.
We have to prove that every σ ∈ Aut( C /K f ) fixes Q ( χ ). Let g ∈ Γ ( N ). Then f | g = χ ( g ) f . Applying σ , we get ( f | g ) σ = χ ( g ) σ f . But Theorem 3.3 implies( f | g ) σ = f σ | g λ = f | g λ = χ ( g λ ) f = χ ( g ) f, so that χ = χ σ . (cid:3) Remark . Corollary 3.6 can also be proved using Katz’s theory of algebraic modular forms(see the Appendix), noting that the diamond operators h δ i , δ ∈ ( Z /N Z ) × are defined over Q , hence leave stable the space M k (Γ ( N ); K ) of modular forms with coefficients in a fixedsubfield K of C . 4. Bounding the coefficient field of f | g Theorem 4.1.
Let f ∈ M k (Γ ( N )) be a modular form of integral weight k ≥ ( N ).Let K f be the subfield of C generated by the Fourier coefficients a n ( f ), n ≥
1. Let g =( A BC D ) ∈ SL ( Z ).(1) The modular form f | k g has coefficients in K f ( ζ M ) with M = N/ gcd( C, N ).(2) If f ∈ M k (Γ ( N )) then f | k g has coefficients in K f ( ζ N ′ ) with N ′ = N/ gcd( CD, N ). Proof.
Let σ ∈ Aut( C ). By Theorem 3.3, a sufficient condition for f | g being fixed by σ isgiven by f σ = f = f | g λ g − , where σ ( ζ N ) = ζ λN . We have(2) g λ g − ≡ (cid:18) AD − λBC AB ( λ − CD ( λ − − AD − λ − BC (cid:19) mod N. We see that g λ g − ∈ Γ ( N ) if and only if λ ≡ N ′ . If f ∈ M k (Γ ( N )), then f | g is fixedby every σ ∈ Aut( C /K f ( ζ N ′ )), hence has coefficients in K f ( ζ N ′ ), which proves (2).Furthermore AD − λBC = 1 + BC (1 − λ ) so that g λ g − ∈ Γ ( N ) if and only if λ ≡ N ′ and λ ≡ N/ gcd( BC, N ). Since B and D are coprime, the conjunction of theseconditions is equivalent to λ ≡ N/ gcd( C, N ). This proves (1). (cid:3)
We now turn to modular forms with characters. We will actually bound not only the fieldof coefficients of f | g , but also the vector space generated by the coefficients of f | g .In order to state our results, we need some more notation. Let f ∈ M k (Γ ( N ) , χ ), where χ is a Dirichlet character of conductor m dividing N , and let g = ( A BC D ) ∈ SL ( Z ). Put N ′ = N/ gcd( CD, N ), m ′ = m/ gcd( BC, m ) and M = lcm( N ′ , m ′ ). Let K = K f ( ζ N ′ )and L = K f ( ζ M ). Since L = K ( ζ m ′ ), the extension L/K is abelian and its Galois group G = Gal( L/K ) identifies with a subgroup G ′ of ( Z /m ′ Z ) × by means of the cyclotomiccharacter λ : G → ( Z /m ′ Z ) × . Since Gal( L/K ) ∼ = Gal( Q ( ζ m ′ ) /K ′ ) with K ′ = K ∩ Q ( ζ m ′ ),the subgroup G ′ ⊂ ( Z /m ′ Z ) × corresponds to the subfield K ′ ⊂ Q ( ζ m ′ ). emma 4.2. The map χ g : G → C × defined by χ g ( σ ) = χ ( AD − λ ( σ ) − BC ) ( σ ∈ G )is a group homomorphism. Proof.
Let σ ∈ G . Note that m ′ BC is divisible by m , so that AD − λ ( σ ) − BC is well-defined in Z /m Z . Let λ N ( σ ) ∈ ( Z /N Z ) × denote the cyclotomic character modulo N . Since λ N ( σ ) ≡ N ′ , the identity (2) shows that g λ N ( σ ) g − is upper-triangular modulo N . Itfollows that AD − λ N ( σ ) − BC ∈ ( Z /N Z ) × and thus AD − λ ( σ ) − BC ∈ ( Z /m Z ) × . Thereforethe map χ g is well-defined.Let us show that χ g is a group homomorphism. We may write χ g as the composition G λ −→ G ′ ψ −→ ( Z /m Z ) × χ −→ C × where ψ is defined by ψ ( µ ) = AD − µ − BC . Using the relation (1), we get the followingidentity in GL ( Z /N Z ) g µµ ′ g − = (cid:18) µ − (cid:19) g µ ′ g − (cid:18) µ (cid:19) g µ g − ( µ, µ ′ ∈ ( Z /N Z ) × ) . Specialising to the case µ, µ ′ ≡ N ′ and comparing the bottom-right entries, we deducethat ψ is a group homomorphism. (cid:3) Note that the character χ g takes values in Q ( χ ) × , which is contained in K × f by Corollary3.6. By the normal basis theorem, L is a free K [ G ]-module of rank 1. Since K f is containedin K , the character χ g cuts out a K -line L χ g in L , namely(3) L χ g = { x ∈ L : ∀ σ ∈ G, σ ( x ) = χ g ( σ ) x } . We are now ready to state our result.
Theorem 4.3.
The modular form f | g has coefficients in L χ g . Proof.
Let σ ∈ Aut( C /K ) with σ ( ζ N ) = ζ λN . Since λ ≡ N ′ , we have g λ g − ∈ Γ ( N ).Then(4) ( f | g ) σ = f σ | g λ = f | g λ g − g = χ ( AD − λ − BC ) f | g = χ g ( σ | L ) f | g. In particular f | g is fixed by Aut( C /L ), hence has coefficients in L . Moreover (4) shows that f | g has coefficients in L χ g . (cid:3) We summarise our result and make it slightly more precise as follows.
Theorem 4.4.
Let f ∈ M k (Γ ( N ) , χ ), where χ is a Dirichlet character of conductor m dividing N , and let g = ( A BC D ) ∈ SL ( Z ). Put N ′ = N/ gcd( CD, N ), m ′ = m/ gcd( BC, m )and M = lcm( N ′ , m ′ ). Then f | k g has coefficients in K f ( ζ M ).More precisely, let G ′ be the subgroup of ( Z /m ′ Z ) × corresponding to the abelian numberfield K f ( ζ N ′ ) ∩ Q ( ζ m ′ ), and let ζ be any m ′ -th root of unity such that c χ,g := X µ ∈ G ′ χ ( AD − µBC ) ζ µ is nonzero (such a ζ always exists). Then f | k g has coefficient in c χ,g · K f ( ζ N ′ ). roof. The fact that f | g has coefficients in L = K f ( ζ M ) was proved in Theorem 4.3. Let K = K f ( ζ N ′ ) and let π χ g : L → L χ g be the K -linear projector associated to the linearcharacter χ g : G → K × . It is given explicitly by π χ g ( x ) = 1 | G | X τ ∈ G χ g ( τ − ) τ ( x ) ( x ∈ L ) . Since L is generated as a K -vector space by the m ′ -th roots of unity, there exists ζ ∈ µ m ′ such that π χ g ( ζ ) = 0. We set c χ,g = | G | · π χ g ( ζ ), so that c χ,g = X τ ∈ G χ g ( τ − ) τ ( ζ ) = X τ ∈ G χ g ( τ − ) ζ λ ( τ ) = X µ ∈ G ′ χ ( AD − µBC ) ζ µ . By Theorem 4.3, the modular form f | g has coefficients in L χ g = c χ,g · K . (cid:3) Remark . The choice ζ = ζ m ′ does not always work. For example, take the newform f of weight k = 3, level N = 9 and character χ of conductor m = 9, with χ (4) = ζ and χ ( −
1) = −
1. We have K f = Q ( χ ) = Q ( ζ ). Taking g = ( −
11 3 ), we get N ′ = 3, m ′ = 9 and G ′ = { , , } , so that c χ,g = 0 for ζ = ζ . On the other hand, for ζ = ζ we get c χ,g = 3 ζ and f | g indeed has coefficients in ζ · Q ( ζ ) = h ζ , ζ i Q . Remark . Theorem 4.4 says that the coefficients of f | g belong to a vector space whichhas the same dimension as in the Γ ( N ) case. Remark . Theorem 4.3 also shows that the coefficients of f | g lie in the fixed field L ker χ g .Let χ ′ g : G ′ → C × be the character defined by χ ′ g ( µ ) = χ ( AD − µ − BC ) (using the notationsof the proof of Lemma 4.2, we have χ ′ g = χ ◦ ψ and χ g = χ ′ g ◦ λ ). Then the field L ker χ g isequal to the composite F · K f ( ζ N ′ ), where F is the subfield of Q ( ζ m ′ ) corresponding to thekernel of χ ′ g . Remark . An alternative approach towards proving Theorems 4.1 and 4.4 would be touse local Whittaker newforms as in [6]. In particular Proposition 3.3 in loc. cit. gives anexplicit formula for the Fourier coefficients of f | g in terms of Whittaker newforms and theGalois action on such newforms is described in the proof of Proposition 2.17.5. Atkin-Lehner operators
For a divisor Q of N with gcd( Q, N/Q ) = 1 we define the Atkin-Lehner operator on M k (Γ ( N )) as follows. Choose x, y, z, w ∈ Z with x ≡ N/Q and y ≡ Q such that the matrix W Q = (cid:0) Qx yNz Qw (cid:1) has determinant Q . Note that W Q = h Q (cid:0) Q
00 1 (cid:1) with h Q = (cid:16) x y NQ z Qw (cid:17) ∈ SL ( Z ). For a modular form f ∈ M k (Γ ( N )) we have f | k W Q ( τ ) = Q k/ ( f | k h Q ) ( Qτ ) . Therefore we can apply our previous results to find a module that contains the coefficientsof f | h Q , or equivalently the coefficients of Q − k/ f | W Q , and reprove a theorem of Cohen. Corollary 5.1 (Theorem 2.6 in [4]) . Let Q be a maximal divisor of N and let f ∈ M k (Γ ( N ))and K f be the subfield of C generated by its Fourier coefficients. Then(1) The modular form f | k W Q has coefficients in Q k/ · K f ( ζ Q ).
2) If f ∈ M k (Γ ( N ) , χ ) for a character χ of conductor m , then f | k W Q has coefficients in Q k/ G ′ ( χ Q ) · K f , where G ′ ( χ Q ) is the Gauss sum of the primitive character associatedto the Q -part of χ . Proof.
The first statement follows directly from Theorem 4.1.Now let f ∈ M k (Γ ( N ) , χ ). We will prove that the coefficients of f | h Q lie in G ′ ( χ Q ) · K f ,which is equivalent to the second statement. First we determine the character χ h Q from theprevious section. Splitting χ as a product of its Q -part and its N/Q -part we observe χ h Q ( σ ) = χ Q ( − NQ zyλ ( σ ) − ) χ N/Q ( Qwx ) = χ Q ( λ ( σ ) − ) = χ Q ( σ ) . The conclusion now follows from Theorem 4.4. Since N ′ = 1, m ′ = m Q , and for ζ = ζ m ′ wehave c χ,h Q = G ′ ( χ Q ). (cid:3) Let f ∈ S k (Γ ( N ) , χ ) be a newform. Then according to [1, §
1] there exists a newform˜ f ∈ S k (Γ ( N ) , χ Q χ N/Q ) and an algebraic number λ Q ( f ) of absolute value 1 such that(5) f | W Q = λ Q ( f ) ˜ f . The number λ Q ( f ) is called the pseudo-eigenvalue of f at Q . By looking at the first Fouriercoefficient in (5), we get the following result. Corollary 5.2. If f ∈ S k (Γ ( N ) , χ ) is a newform, then the pseudo-eigenvalue λ Q ( f ) is in Q k/ G ′ ( χ Q ) · K f and ˜ f has coefficients in K f .This should be compared to the following theorem of Atkin-Li, where an explicit formula for λ Q ( f ) is derived in a special case. Theorem 5.3. [1, Theorem 2.1] Let f = P n a n e πinτ ∈ S k (Γ ( N ) , χ ) be a newform, q be aprime dividing N and Q = N q . If a q = 0, then λ Q ( f ) = Q k/ − G ( χ Q ) a Q , where G ( χ Q ) = P u ∈ ( Z /Q Z ) × χ Q ( u ) e πiu/Q is the Gauss sum of χ Q .6. Optimising the coefficient field
We may reduce the number fields provided by Theorems 4.1 and 4.4 as follows. Let α ∈ P ( Q )be a cusp, and let g = ( A BC D ) ∈ SL ( Z ) such that g ∞ = α . In order to compute the Fourierexpansion of f at α , we may replace g by gT u with T = ( ) and u ∈ Z . Then f | gT u depends only on the class of u modulo w , where w is the width of α on the appropriatemodular curve. The following proposition gives the minimal value of the integer M fromTheorem 4.4 when u varies in Z /w Z . Proposition 6.1.
In the notation of Theorem 4.4, let M ′ be the minimal value of M for gT u as u varies. Then M ′ = N C gcd( C, N ) · m C , where N C = Q p | C p v p ( N ) is the C -part of N and m C = m/m C is the prime to C part of m .Moreover, M ′ is attained for any u such that N/N C divides uC + D . roof. Replacing g with gT u changes D to uC + D and B to uA + B . Varying u , we needto determine the minimal value of M u = lcm (cid:18) N gcd( C ( uC + D ) , N ) , m gcd( C ( uA + B ) , m ) (cid:19) . Since C and D are coprime we have gcd( C ( uC + D ) , N ) = gcd( C, N ) gcd( uC + D, N ), so N/ gcd( C ( uC + D ) , N ) is divisible by N C / gcd( C, N ). Therefore N C / gcd( C, N ) divides M u .Let p ∤ C . Since C ( uA + B ) = A ( uC + D ) −
1, we have v p (cid:18) N gcd( C ( uC + D ) , N ) (cid:19) = v p ( N ) − min( v p ( uC + D ) , v p ( N )) ,v p (cid:18) m gcd( C ( uA + B ) , m ) (cid:19) = v p ( m ) − min( v p ( A ( uC + D ) − , v p ( m )) . If v p ( uC + D ) = 0, then v p ( m/ gcd( C ( uA + B ) , m )) = v p ( m ). On the other hand, if v p ( uC + D ) = 0, then v p ( N/ gcd( C ( uC + D ) , N )) = v p ( N ) ≥ v p ( m ). In all cases, we have v p ( M u ) ≥ v p ( m ), which proves that m C divides M u . It follows that M u is always divisibleby N C / gcd( C, N ) · m C .Now choose u such that N C = N/N C divides uC + D . This is possible because C and N C are coprime. Then gcd( uC + D, N ) = N C so that N gcd( C ( uC + D ) , N ) = N gcd( C, N ) N C = N C gcd( C, N ) . Moreover gcd( C ( uA + B ) , m ) = gcd( A ( uC + D ) − , m ) is coprime to N C and thus divides m C . It follows that m C | m gcd( C ( uA + B ) , m )On the other hand, if p | C , then v p (cid:18) m gcd( C ( uA + B ) , m ) (cid:19) ≤ v p (cid:18) m gcd( C, m ) (cid:19) ≤ v p (cid:18) N C gcd( C, N ) (cid:19) . Hence M u = ( N C / gcd( C, N )) · m C . (cid:3) In practice, we may further reduce the field of coefficients as follows. Let f ∈ M k (Γ ( N )) bean eigenvector of the Atkin–Lehner operators and g = ( A BC D ) ∈ SL ( Z ). The denominator ofthe cusp α = g ∞ = A/C of X ( N ) is δ := gcd( C, N ).Now let Q be a maximal divisor of N , and let W Q be the associated Atkin-Lehner involutionof X ( N ). Using the notations of Section 5, if f is an eigenvector of W Q with eigenvalue λ Q ( f ) ∈ {± } , then we may write f | g = λ Q ( f ) f | W Q g = λ Q ( f ) f | h Q (cid:18) Q
00 1 (cid:19) (cid:18)
A BC D (cid:19) = λ Q ( f ) f | h Q AQ gcd( C,Q ) s C gcd( C,Q ) r ! (cid:18) gcd( C, Q ) rBQ − sD Q gcd( C,Q ) (cid:19) , where r, s are chosen, so that r AQ gcd( C,Q ) − s C gcd( C,Q ) = 1. he action of the upper triangular matrix (cid:16) gcd( C,Q ) rBQ − sD Q gcd( C,Q ) (cid:17) on Fourier expansions is easilycalculated. We now try to find Q such that f | g ′ has coefficients in the minimal possiblenumber field, where g ′ = h Q (cid:18) AQ gcd( C,Q ) s C gcd( C,Q ) r (cid:19) ∈ SL ( Z ).Let δ = δ Q δ Q where δ Q is the Q -part of δ . Then the cusp α ′ = g ′ ∞ = W Q α has denominator δ ′ = Qδ Q δ Q and we may choose Q such that M ′ := N δ ′ /δ ′ is minimal. Explicitly, the choice Q = Y p | N Our final goal is to determine the exact coefficient field of f | g when f is a newform. In thissection, we assume that f is a newform of (even) weight k ≥ ( N ).We will need the following theorems of Newman [12, 13] on the congruence subgroup Γ ( N ),where N is a fixed integer ≥ Theorem 7.1. [12, Theorem 3] Every intermediate subgroup between Γ ( N ) and SL ( Z ) isof the form Γ ( M ) for some positive divisor M of N .In the following, we denote by R the matrix ( ). Corollary 7.2. Let M be a positive divisor of N . The group Γ ( M ) is generated by Γ ( N )and R M = ( M ). Proof. Let Γ be the group generated by Γ ( N ) and R M . By Theorem 7.1, we have Γ = Γ ( M ′ )for some M ′ dividing N . Since R M ∈ Γ ( M ′ ), the integer M ′ divides M . Moreover Γ iscontained in Γ ( M ), so that M divides M ′ . It follows that Γ = Γ ( M ). (cid:3) Theorem 7.3. [13] The normaliser of Γ ( N ) in SL ( Z ) is equal to Γ ( N/s ), where s is thelargest divisor of 24 such that s divides N . Moreover, the quotient group Γ ( N/s ) / Γ ( N )is cyclic of order s , generated by the class of R N/s = (cid:0) N/s (cid:1) . Proof. The first assertion follows from [13, Theorem 1]. By Corollary 7.2, the group Γ ( N/s )is generated by Γ ( N ) and R N/s . It follows that the quotient group Γ ( N/s ) / Γ ( N ) isgenerated by the class of R N/s , and it is easy to see that this class has order s . (cid:3) Proposition 7.4. Let F be a nonzero element of the new subspace S new k (Γ ( N )). Then itsstabiliser Stab( F ) = { g ∈ SL ( Z ) : F | g = F } is equal to Γ ( N ). roof. Since Stab( F ) contains Γ ( N ), Theorem 7.1 implies that Stab( F ) = Γ ( M ) for somepositive divisor M of N . But F belongs to the new subspace, so we must have M = N . (cid:3) Proposition 7.5. Let f be a newform of weight k ≥ ( N ). Let g ∈ SL ( Z ) and σ ∈ Aut( C ) such that f | g = f σ . Then we have f σ = f and g ∈ Γ ( N ). Proof. By Proposition 7.4, the stabilisers of f and f σ are both equal to Γ ( N ). On theother hand Stab( f | g ) = g − Stab( f ) g , so that g normalises Γ ( N ). By Theorem 7.3, we have g ∈ Γ ( N/s ), and there exists an integer m ∈ Z such that Γ ( N ) g = Γ ( N ) R mN/s . Hence f | g = f | R mN/s .We now make use of the Atkin-Lehner involution W N = ( − N ). Let w ∈ {± } be theroot number of f , defined by f | W N = wf . Since W N is defined over Q , we also have f σ | W N = wf σ . Applying W N on both sides of the equality f | g = f σ , we get wf σ = f σ | W N = f | gW N = f | R mN/s W N = f | W N (cid:0) W − N R mN/s W N (cid:1) = wf | R ′ with R ′ = W − N R mN/s W N = (cid:0) − m/s (cid:1) . The Fourier expansion of f | R ′ is given by f | R ′ ( z ) = f ( z − m/s ) = X n ≥ a n ( f ) e − πimn/s e πinz . Comparing the first term of the Fourier expansions, we get e − πim/s = 1. This implies that s divides m , f σ = f and g ∈ Γ ( N ). (cid:3) We are now in the position to determine the exact number field of f | g . This refines Theorem4.1(2) for Γ ( N ) newforms. Theorem 7.6. Let f be a newform of weight k ≥ ( N ). Let g = ( A BC D ) ∈ SL ( Z ).Then the field generated by the Fourier coefficients of f | k g is equal to K f ( ζ N ′ ) with N ′ = N/ gcd( CD, N ). Proof. We have to show that every automorphism of C fixing f | g also fixes K f ( ζ N ′ ). Let σ ∈ Aut( C ) such that ( f | g ) σ = f | g . Define λ ∈ ( Z /N Z ) × by σ ( ζ N ) = ζ λN . By Theorem3.3, we have f σ | g λ = f | g , so that f σ = f | gg − λ . By Proposition 7.5, we have f σ = f and gg − λ ∈ Γ ( N ). It follows that σ fixes K f , and we have already seen in the proof of Theorem4.1 that the condition gg − λ ∈ Γ ( N ) is equivalent to λ ≡ N ′ . Therefore σ fixes K f ( ζ N ′ ). (cid:3) Remark . An inspection of the proofs shows that Proposition 7.5 and Theorem 7.6 arevalid for elements f = P a n q n of the new subspace of S k (Γ ( N )) that are eigenfunctions of W N and satisfy the following condition: if s denotes the largest divisor of 6 whose squaredivides N , then there exists n ∈ N which is coprime to s such that a n is a non-zero rationalnumber. One family of such forms is given by the traces P σ f σ of newforms f , where thesum is over all embeddings of K f into C . Question 7.8. What is the Q -vector space (or K f -vector space) generated by the Fouriercoefficients of f | g ? Furthermore, can we bound effectively the denominators of these coeffi-cients? Note that the q -expansion principle implies that the Fourier expansion of f | g lies in Z [[ q /N ]] ⊗ K f ( ζ N ′ ), so that the denominators of f | g are indeed bounded. In fact, by Remark8.5, we know that the denominators of f | g divide some fixed power of N . Question 7.9. It would be interesting to generalise Theorem 7.6 to newforms with non-trivial character. Is the number field provided by Remark 4.7 best possible? . Appendix: Algebraic modular forms Here we recall the theory of algebraic modular forms, in order to give a second proof ofTheorem 3.3. For more details on this theory, see [10, Chap. II] and the references therein. Definition 8.1. Let R be an arbitrary commutative ring, and let N ≥ test object of level N over R is a triple T = ( E, ω, β ) where E/R is an elliptic curve, ω ∈ Ω ( E/R ) is a nowhere vanishing invariant differential, and β is a level N structure on E/R , that is an isomorphism of R -group schemes β : ( µ N ) R × ( Z /N Z ) R ∼ = −→ E [ N ]satisfying e N ( β ( ζ , , β (1 , ζ for every ζ ∈ ( µ N ) R . Here µ N = Spec Z [ t ] / ( t N − 1) is thescheme of N -th roots of unity, and e N is the Weil pairing on E [ N ] .If φ : R → R ′ is a ring morphism, we denote by T R ′ = ( E R ′ , ω R ′ , β R ′ ) the base change of T to R ′ along φ .The isomorphism classes of test objects over C are in bijection with the set of lattices L in C endowed with a symplectic basis of N L/L [10, 2.4]. Another example is given by theTate curve Tate( q ) = G m /q Z [7, § Z (( q )) endowed with thecanonical differential ω can = dx/x and the level N structure β can ( ζ , n ) = ζ q n/N mod q Z . Thetest object (Tate( q ) , ω can , β can ) is defined over Z (( q /N )). Definition 8.2. An algebraic modular form of weight k ∈ Z and level N over R is the data,for each R -algebra R ′ , of a function F = F R ′ : { isomorphism classes of test objects of level N over R ′ } → R ′ satisfying the following properties:(1) F ( E, λ − ω, β ) = λ k F ( E, ω, β ) for every λ ∈ ( R ′ ) × ;(2) F is compatible with base change: for every morphism of R -algebras ψ : R ′ → R ′′ and for every test object T of level N over R ′ , we have F R ′′ ( T R ′′ ) = ψ ( F R ′ ( T )).We denote by M alg k (Γ( N ); R ) the R -module of algebraic modular forms of weight k and level N over R .Evaluating at the Tate curve provides an injective R -linear map M alg k (Γ( N ); R ) ֒ → Z (( q /N )) ⊗ Z R called the q -expansion map. The q -expansion principle states that if R ′ is a subring of R ,then an algebraic modular form F ∈ M alg k (Γ( N ); R ) belongs to M alg k (Γ( N ); R ′ ) if and onlyif the q -expansion of F has coefficients in R ′ .Algebraic modular forms are related to classical modular forms as follows. To any algebraicmodular form F ∈ M alg k (Γ( N ); C ), we associate the function F an : H → C defined by F an ( τ ) = F (cid:16) C πi Z + 2 πiτ Z , dz, β τ (cid:17) with β τ ( ζ mN , n ) := [2 πi ( m + nτ ) /N ]. Our definition of the Weil pairing is the reciprocal of Silverman’s definition [18, III.8]. With our definition,we have e N (1 /N, τ /N ) = e πi/N on the elliptic curve C / ( Z + τ Z ) with Im( τ ) > roposition 8.3. The map F F an induces an isomorphism between M alg k (Γ( N ); C ) andthe space M ! k (Γ( N )) of weakly holomorphic modular forms on Γ( N ) (that is, holomorphicon H and meromorphic at the cusps). Moreover, the q -expansion of F coincides with thatof F an .We now interpret the action of SL ( Z ) on modular forms in algebraic terms. Let F ∈ M alg k (Γ( N ); C ) with f = F an , and let g = ( a bc d ) ∈ SL ( Z ). A simple computation shows that(6) ( f | k g )( τ ) = F (cid:16) C πi ( Z + τ Z ) , dz, β ′ τ (cid:17) where the level N structure β ′ τ is given by(7) β ′ τ ( ζ mN , n ) = β τ ( ζ md + nbN , mc + na ) . Let ψ : ( Z /N Z ) → µ N ( C ) × Z /N Z be the isomorphism defined by ψ ( a, b ) = ( ζ bN , a ). Letus identify the level structure β τ (resp. β ′ τ ) with the map α τ = β τ ◦ ψ (resp. α ′ τ = β ′ τ ◦ ψ ).Then (7) shows that(8) α ′ τ ( a, b ) = α τ (( a, b ) g ) . What we have here is the right action of SL ( Z ) on the row space ( Z /N Z ) , which induces aleft action on the set of level N structures. As we will see, all this makes sense algebraically.For any Z [ ζ N ]-algebra R , we denote by ζ N,R the image of ζ N = e πi/N under the structuralmorphism Z [ ζ N ] → R . Lemma 8.4. If R is a Z [ ζ N , /N ]-algebra, then there is an isomorphism of R -group schemes( Z /N Z ) R ∼ = −→ ( µ N ) R sending 1 to ζ N,R . Proof. Note that ( µ N ) R = Spec R [ t ] / ( t N − 1) = Spec R [ Z /N Z ] and ( Z /N Z ) R = Spec R Z /N Z .If R = C , then C [ Z /N Z ] ∼ = C Z /N Z because all irreducible representations of Z /N Z havedimension 1. This isomorphism F C is given by the Fourier transform, and both F C and F − C have coefficients in Z [ ζ N , /N ] with respect to the natural bases. It follows that in general R [ Z /N Z ] ∼ = R Z /N Z and this isomorphism sends [1] to ( ζ aN,R ) a ∈ Z /N Z . (cid:3) Let R be a Z [ ζ N , /N ]-algebra. We have an isomorphism of R -group schemes ψ R : ( Z /N Z ) R → ( µ N ) R × ( Z /N Z ) R given by ψ R ( a, b ) = ( ζ bN,R , a ). The group SL ( Z ) acts from the right on the row space( Z /N Z ) R by R -automorphisms, and for α : ( Z /N Z ) R ∼ = −→ E [ N ] we define(9) ( g · α )( a, b ) = α (( a, b ) g ) (( a, b ) ∈ ( Z /N Z ) ) . Using ψ R , we transport this to a left action of SL ( Z ) on the set of level N structures of anelliptic curve over R . Given a test object T = ( E, ω, β ) over R , we define g · T := ( E, ω, g · β ).For any F ∈ M alg k (Γ( N ); R ), we define F | g ∈ M alg k (Γ( N ); R ) by the rule ( F | g )( T ) = F ( g · T )for any test object T over any R -algebra R ′ . The computation (6) then shows that the rightaction of SL ( Z ) on M alg k (Γ( N ); C ) corresponds to the usual slash action on M ! k (Γ( N )). Remark . The action of SL ( Z ) on algebraic modular forms over Z [ ζ N , /N ]-algebras hasthe following consequence: if a classical modular form f ∈ M ! k (Γ( N )) has Fourier coefficientsin some subring A of C , then for any g ∈ SL ( Z ), the Fourier expansion of f | g lies in Z [[ q /N ]] ⊗ A [ ζ N , /N ]. e now interpret the action of Aut( C ) in algebraic terms (see [14, p. 88]). Let σ ∈ Aut( C ).For any C -algebra R , we define R σ := R ⊗ C ,σ − C , which means that ( ax ) ⊗ x ⊗ σ − ( a )for all a ∈ C , x ∈ R . We endow R σ with the structure of a C -algebra using the map a ∈ C ⊗ a ∈ R σ . We denote by φ σ : R → R σ the map defined by φ σ ( x ) = x ⊗ φ σ is a ring isomorphism, but one should be careful that φ σ is not a morphism of C -algebras, as it is only σ − -linear. For any test object T over R , we denote by T σ its basechange to R σ using the ring morphism φ σ .Let F ∈ M alg k (Γ( N ); C ) be an algebraic modular form. For any C -algebra R , we define F σR : { isomorphism classes of test objects of level N over R } → RT φ − σ (cid:0) F R σ ( T σ ) (cid:1) . One may check that the collection of functions F σR satisfies the conditions (1) and (2) above,hence defines an algebraic modular form F σ ∈ M alg k (Γ( N ); C ). Moreover, since the Tatecurve is defined over Z (( q )), one may check that the map F F σ corresponds to the usualaction of Aut( C ) on the Fourier expansions of modular forms: for every F ∈ M alg k (Γ( N ); C )and every σ ∈ Aut( C ), we have ( F σ ) an = ( F an ) σ .We finally come to the second proof of Theorem 3.3. Proof. Let f ∈ M k (Γ( N )) with corresponding algebraic modular form F ∈ M alg k (Γ( N ); C ).Let g ∈ SL ( Z ) and σ ∈ Aut( C ). We take as test object T = (Tate( q ) , ω can , β can ) over R = Z (( q /N )) ⊗ C . Since a modular form is determined by its Fourier expansion, andunravelling the definitions of F | g and F σ , it suffices to check that the test objects g · T σ and( g λ · T ) σ over R σ are isomorphic. Since SL ( Z /N Z ) acts only on the level structures of thetest objects, we have to show that(10) g · β σ can ∼ = ( g λ · β can ) σ . For any scheme X over R , let X σ denote its base change to R σ along φ σ . Since φ σ is a ringisomorphism, the canonical projection map X σ → X is an isomorphism of schemes, and wealso denote by φ σ : X → X σ the inverse map.Put E = Tate( q ) and β = β can . Let α = β ◦ ψ R : ( Z /N Z ) R ∼ = −→ E [ N ]. By functoriality, thelevel structure β σ is given by the following commutative diagram(11) ( Z /N Z ) R ( µ N ) R × ( Z /N Z ) R E [ N ]( Z /N Z ) R σ ( µ N ) R σ × ( Z /N Z ) R σ E σ [ N ] . γ ψ R α βφ σ ∼ = φ σ ∼ = ψ Rσ α σ β σ Let us compute the dotted arrow γ . Since φ σ is σ − -linear, we have φ σ ( ζ N,R ) = ζ λ − N,R σ . Itfollows that(12) φ σ ( ψ R ( a, b )) = φ σ ( ζ bN,R , a ) = ( ζ λ − bN,R σ , a ) = ψ R σ ( a, λ − b ) o that γ ( a, b ) = ( a, λ − b ). We may thus express α σ in terms of α by(13) α σ ( a, b ) = φ σ ◦ α ◦ γ − ( a, b ) = φ σ ◦ α ( a, λb ) = φ σ ◦ α (cid:18) ( a, b ) (cid:18) λ (cid:19)(cid:19) . Let us make explicit both sides of (10). By (9) and (13), the left hand side is given by(14) ( g · α σ )( a, b ) = α σ (( a, b ) g ) = φ σ ◦ α (cid:18) ( a, b ) g (cid:18) λ (cid:19)(cid:19) . Let us now turn to the right hand side of (10). By (9), we have ( g λ · α )( a, b ) = α (( a, b ) g λ ).Applying the commutative diagram (11) with α replaced by g λ · α , we get(15) ( g λ · α ) σ ( a, b ) = φ σ ◦ ( g λ · α ) (cid:18) ( a, b ) (cid:18) λ (cid:19)(cid:19) = φ σ ◦ α (cid:18) ( a, b ) (cid:18) λ (cid:19) g λ (cid:19) . Finally, we recall equation (1), which states g ( λ ) = ( λ ) g λ . (cid:3) References [1] A. O. L. Atkin and W. C. W. Li. Twists of newforms and pseudo-eigenvalues of W -operators. Invent.Math. , 48(3):221–243, 1978.[2] Lev A. Borisov and Paul E. Gunnells. Toric modular forms of higher weight. J. Reine Angew. Math. ,560:43–64, 2003.[3] Fran¸cois Brunault. R´egulateurs modulaires explicites via la m´ethode de Rogers-Zudilin. CompositioMathematica , 153:1119–1152, 2017.[4] Henri Cohen. 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