A conjectural extension of Hecke's converse theorem

We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.


Introduction
Let f ∈ M k (Γ 0 (N), ξ) be a classical holomorphic modular form of weight k, level N and nebentypus character ξ, and define Nz .
Let f n and g n denote the Fourier coefficients of f and g, respectively, and define for ℜ(s) > k+1 2 , where Γ C (s) := 2(2π) −s Γ(s). Then Λ f (s) and Λ g (s) continue to entire functions of finite order, apart from at most simple poles at s = 1±k 2 , and satisfy the functional equation (1.3) Λ f (s) = i k N 1 2 −s Λ g (1 − s). Conversely, when N ≤ 4, Hecke [13,14] (see also [1]) showed that the modular forms of level N are characterized by these properties. Precisely, given sequences {f n } ∞ n=1 , {g n } ∞ n=1 of at most polynomial growth, if the functions Λ f (s) and Λ g (s) defined by (1.2) continue to entire functions of finite order and satisfy (1.3) then f n and g n are the Fourier coefficients of modular forms of level N and weight k, related by (1.1). When N ≥ 5, Hecke's proof no longer goes through, and in fact the vector space of sequences {f n } ∞ n=1 , {g n } ∞ n=1 satisfying the above conditions is infinite dimensional. Weil [22] showed that one can recover the converse statement by assuming additional functional equations for twisted L-functions for primitive characters χ of conductor coprime to N. On the other hand, it has been conjectured (see [8,Conjecture 1.2]) that if Λ f (s) and Λ g (s) have Euler product expansions 1 of the shape satisfied by primitive Hecke eigenforms then the single functional equation (1.3) should suffice to imply modularity, without the need for character twists. Some partial progress on this problem was made by Conrey and Farmer [3] (see also [4]), who proved the conjecture for some values of N exceeding 4. One drawback of assuming an Euler product is that it imposes a nonlinear constraint on the Fourier coefficients f n , g n , so the solutions to (1.3) no longer form a vector space. In turn, it is unclear how to make use of this constraint to extend Hecke's proof to higher level. In this paper we propose a replacement for the Euler product that, we conjecture, characterizes the modular forms of any level N, yet retains the linearity of (1. for ℜ(s) > σ + 1 − k−1 2 . For every q coprime to N, suppose that Λ f (s, c q ) and Λ g (s, c q ) continue to entire functions of finite order and satisfy the functional equation Then f (z) := ∞ n=1 f n e(nz) is an element of M k (Γ 0 (N), ξ). To understand the motivation behind this conjecture, we first consider a more general family of twists. Let χ (mod q) be a Dirichlet character, not necessarily primitive, and define χ(a)e an q , Note that when χ is the trivial character mod q, c χ reduces to the Ramanujan sum, c q . In Lemma 4.10, we show that if we start from a pair of modular forms f, g satisfying (1.1), then Λ f (s, c χ ) and Λ g (s, c χ ) satisfy the functional equation 1 We regard the factors of Γ C (s + k−1 2 ) in (1.2) as Euler factors for the archimedean place.
Given any Q ∈ N and q | Q, we can view c χ for χ (mod q) as a function on Z/QZ. One can show that as χ ranges over all characters of modulus dividing Q, the functions c χ form an orthogonal basis for the space of functions on Z/QZ. Thus, any twist of f with periodic coefficients and period coprime to N is a linear combination of the twists by c χ . In this sense, (1.8) is the most general functional equation (from twists with period coprime to the level) that one can expect. Conjecture 1.1 arises from the speculation that any constraints on the solutions to (1.3) imposed by the assumption of an Euler product are already implied by the extra functional equations (1.8) that one obtains from taking χ equal to the trivial character mod q. In Section 2, we prove five theorems that lend some support to the conjecture: (1) Theorem 2. To set these results in context, we note that one reason why Hecke's argument fails for N ≥ 5 is that there are counterexamples arising from more general kinds of modular forms. If one believes that a twistless converse theorem is possible assuming an Euler product, then it is reasonable to ask how these counterexamples are eliminated by the Euler product. Points (2) and (3) above address two such generalizations of modular forms, namely forms for noncongruence groups and forms for more general weight-k multiplier systems (not necessarily of finite order). Concerning point (5), Diaconu, Perelli and Zaharescu [6] showed that if Λ f (s) is given by an Euler product, then there exists a prime q (depending on N) such that the analytic properties and functional equations of the character twists (1.4) for all primitive χ of conductor dividing q suffice to imply modularity. On the other hand, again under the assumption of an Euler product, it follows from a theorem of Piatetski-Shapiro [19] that it suffices to assume the expected properties of (1.4) for all primitive χ (mod p j ) for any fixed prime p and all j ≥ 0. Point (5) can be seen as a complement to both of these results. We conjecture that the proof of Theorem 2.5 can be extended to all sufficiently large primes q, and we study this problem in detail in Section 3.
Acknowledgements. This paper grew out of a focused research workshop on the Sarnak rigidity conjecture at the Heilbronn Institute for Mathematical Research. We thank the Institute for their support, which made this work possible.

Main results
Let H = {z ∈ C : ℑ(z) > 0} denote the upper half-plane. For any function h : H → C and any matrix γ = ( a b c d ) ∈ GL + 2 (R) = {M ∈ GL 2 (R) : det M > 0}, define where k ∈ N is the integer appearing in Conjecture 1.1. (We assume that k is fixed from now on and suppress it from the notation.) Note that this defines a right action, i.e. h|(γ 1 γ 2 ) = (h|γ 1 )|γ 2 for any γ 1 , γ 2 ∈ GL + 2 (R). We extend the action linearly to the group algebra Let f be as in Conjecture 1.1, and define g(z) = ∞ n=1 g n e(nz). Then, by Hecke's argument [17,Theorem 4.3.5], the fact that Λ f (s, c 1 ) and Λ g (s, c 1 ) continue to entire functions of finite order and satisfy (1.5) for q = 1 is equivalent to the identity f | ( −1 , since f and g are given by Fourier series, we have , and thus f |γ = ξ(γ)f for every γ ∈ −I, T, W . To prove that f ∈ M k (Γ 0 (N), ξ), it suffices to verify this equality for every γ ∈ Γ 0 (N), since the holomorphy of f at cusps follows from modularity and the growth estimate f n = O(n σ ).
Note that if γ, γ ′ ∈ Γ 0 (N) have the same top row then γ ′ γ −1 is a power of W , so that f |γ ′ = f |γ. Thus, f |γ depends only on the top row of γ. With this in mind, we will write γ q,a to denote any element of Γ 0 (N) with top row ( q −a ). Proof. The following table shows, for each N in the statement of the theorem, minimal generating sets for Γ 0 (N), verified with Sage [5]: In particular, for N ≤ 4, Γ 0 (N) is generated by −I, T and W , so there is nothing to prove. For all other levels we apply the methods of Conrey and Farmer [3], in the form of For N ∈ {8, 11, 15, 17, 23} we obtain values of q ∈ {3, 4, 6} for which f |γ q,1 = ξ(q)f from Lemma 4.3. For N ∈ {8, 11, 17} these are sufficient to establish the claim.
Proof. We may assume without loss of generality that H contains T and W . By Lemma 4.1, for any prime q ∤ N, : H], and let g 1 , . . . , g h ∈ Γ 0 (N) be coset representatives for H\Γ 0 (N). Replacing g i by W g i if necessary, we may assume without loss of generality that g i is not upper triangular. For each γ q,a ∈ Γ 0 (N), there exists i ∈ {1, . . . , h} such that γ q,a ∈ Hg i , so that f |γ q,a = f |g i . Rearranging (2.1), we get has a Fourier expansion: i.e., for n ∈ Z, Fix n ∈ Z \ {0}. By Dirichlet's theorem, we can choose distinct primes q 1 , . . . , q h ∤ mnN and integers a 1 , . . . , a h such that γ q i ,a i ∈ T g i ⊆ Hg i for each i. Thus, from (2.3) for q ∈ {q 1 , . . . , q h }, we obtain a system of linear equations of the shape This concludes the proof. Theorem 2.3. Assume the hypotheses of Conjecture 1.1, and suppose that there is a con- Proof. If f = 0 then the conclusion is trivially true, so from now on assume f = 0. Let M denote the level of H, so that H ⊇ Γ(M). Since f |T = f |W = f and Γ 1 (N) is generated by {T, W }∪Γ(M), we may assume without loss of generality that H ⊇ Γ 1 (N). By Theorem 3.2, there exists a prime q ≡ 1 (mod N) such that Γ 1 (N) is generated by {T, W, γ q,a : 1 ≤ a < q}. By Lemma 4.6, there exists m ∈ N such that q | m and {f m , g m } = {0}. Since ( −1 N ) normalizes Γ 1 (N), we may swap the roles of f and g if necessary, so as to assume that f m = 0.
Proof. Let H be the smallest subgroup of Γ 0 (N) containing T , W and all commutators Then H is a normal subgroup with abelian quotient H\Γ 0 (N), and f |γ = f for all γ ∈ H. If N ∈ {2, 3} then H, −I = Γ 0 (N) and there is nothing to prove, so we assume henceforth that N ≥ 5.
Let R = {r ∈ Z : 2 ≤ |r| < 1 2 N}, and for each r ∈ R, fix a matrix γ r,1 with top row ( r −1 ). Then, by Lemma 4.7, for any prime q ∤ N and a coprime to q, we have Since H\Γ 0 (N) is abelian, we are free to permute the τ i without changing the coset H τ i . Hence, since H contains T, W , we may write for some ǫ ∈ {0, 1} and non-negative integers e r (depending on q and a), satisfying r∈R e r ≤ log 2 q. Now, fix s ∈ R, n ∈ Z \ {0} and X ∈ N, and let Q = Q(s, n, X) denote the set of primes q satisfying qs ≡ 1 (mod N), q ∤ n and q ≤ X. As in the proof of Theorem 2.2, we consider (2.1) for all primes q ∈ Q. Let g 1 , . . . , g h be a minimal set of representatives for the cosets Hγ q,a of all matrices occurring there. By the above, we may take each g i of the form (−I) ǫ r∈R γ −er r,1 with ǫ ∈ {0, 1}, e r ≥ 0 and r∈R e r ≤ log 2 X. In particular, Hγ −1 s,1 = Hγ q,−1 for every q ∈ Q, so we may take g 1 = γ −1 s,1 . By Dirichlet's theorem, we have #Q ≫ X/ log X, and thus h ≤ 2(1 + log 2 X) N −3 ≤ #Q for all sufficiently large X. For has a Fourier expansion as in (2.2), with m = 1. In turn, this leads to the system of linear equations (2.4), where we take {q j } to be any subset of Q of cardinality h. Applying Lemma 4.8, by appropriate permutation of the rows and columns we can select a square subsystem for which the diagonal entries are non-zero. Since the coset Hg 1 occurs in every row, the column i = 1 is necessarily one of the variables in the subsystem.
Theorem 2.5. Assume the hypotheses of Conjecture 1.1. There is a set Q of prime numbers such that (i) Q has density 1 in the set of all primes, and (ii) if there exists q ∈ Q such that the multiplicative twists Λ f (s, χ) and Λ g (s, χ), for all primitive characters χ (mod q), continue to entire functions of finite order and satisfy the functional equation In particular, for each N in the following table, the set Q contains every prime q ∤ N in the indicated interval. N q N q 10 (11, 10 9 ) 18 (53, 10 9 ) 12 (35, 10 9 ) 19 (37, 10 9 ) 13 (5, 10 9 ) 20 (79, 10 9 ) 14 (43, 10 9 ) 21 (83, 10 9 ) 16 (47, 10 9 ) 22 (43, 10 9 ) Proof. Let Q be the set of primes q ∤ N such that H q ⊇ Γ 1 (N), in the notation of Section 3. By Theorem 3.2, Q has density 1 in the set of all primes, so (i) holds, and the fact that Q contains the numbers indicated in the table is the content of Theorem 3.3.
Let q ∈ Q. Then by [17, Lemmas 4.3.9 and 4.3.13], the assumed analytic properties of Λ f (s, χ) and Λ g (s, χ) described in (ii), together with the functional equation (2.5) for all primitive χ (mod q), imply the equality for any integers a, b coprime to q. By Lemma 4.1, it follows that f |γ q,a = ξ(q)f for every a coprime to q. By the definition of Q, we thus have f |γ = ξ(γ)f for every γ ∈ H q ⊇ Γ 1 (N). Applying Theorem 2.2 with H = Γ 1 (N), we conclude that f ∈ M k (Γ 0 (N), ξ).

Generating Γ 1 (N)
In this section, we consider the question of when the elements of Γ 0 (N) with a fixed upper-left entry generate a subgroup containing Γ 1 (N). By the proof of Theorem 2.5, any such upper-left entry gives sufficient conditions to imply modularity using twists of a single modulus.
For any q ∈ N coprime to N, let H q denote the subgroup of Γ 0 (N) generated by the matrices Conjecture 3.1. There exists q 0 = q 0 (N) such that H q ⊇ Γ 1 (N) for every q ≥ q 0 coprime to N.
holds for almost all q ∈ N coprime to N and for almost all primes q ∤ N, i.e.
Proof. For q ∈ N coprime to N, set Then Γ q is a group satisfying Γ 1 (N) ∪ H q ⊆ Γ q ⊆ Γ 0 (N), and we have Consider a fixed q 0 ∈ N coprime to N, and letq 0 be a multiplicative inverse of q 0 (mod N). Then, for any q ≡ q 0 (mod N), so that H q and Γ q = Γ q 0 contain T, W . Let be a fixed generating set for Γ q 0 , with γ 1 = q 0 1 q 0q0 −1q 0 . For i ≥ 2, replacing γ i by γ n i 1 γ i for a suitable n i , we may assume that A i ≡ q 0 (mod N). Also, we may assume that A i = 0, since otherwise N = 1 and γ i is contained in T, W .
Next, we modify γ 1 , . . . , γ h by multiplying by powers of T and W . First, multiplying by W m i on the left leaves A i unchanged and replaces C i by C i + m i A i . Hence, by Dirichlet's theorem, we may take C 1 , . . . , C h to be distinct primes not dividing N. Second, by the Chinese remainder theorem, we can choose q 1 ∈ N satisfying q 1 ≡ q 0 (mod N) and q 1 ≡ A i (mod C i ) for every i. Multiplying on the left by T (q 1 −A i )/(N C i ) replaces each A i by q 1 . Now, let q ∈ N with q ≡ q 0 (mod N). Suppose that the divisors of q − q 1 represent all invertible residue classes modulo Nq 1 , i.e.
For the relatively small number of values of q remaining, we computed the expansions of every element γ q,a for 1 ≤ a ≤ q in terms of the generators S = ( −1 1 ) and T = ( 1 1 1 ) of SL 2 (Z), and presented SL 2 (Z) ∼ = S, T : S 4 = S 2 (ST ) 3 = 1 as an abstract group to GAP [9]. We then used GAP's implementation of the Todd-Coxeter algorithm [21] to attempt to compute the index [SL 2 (Z) : H q ]. When this terminated with a number equal to the expected index [SL 2 (Z) : Γ q ], we obtained the claim for q.
The first strategy tends to work better at finding prime values of q, which explains the discrepancy in the sizes of the intervals for larger values of N, where there are eventually too many exceptions to test by the second method in a reasonable amount of time.
For some q (those for which the Todd-Coxeter algorithm appeared not to terminate), our results were inconclusive, though we expect that H q ⊇ Γ 1 (N) in those cases. In a very small number of cases, H q has finite index in SL 2 (Z) but is not the full group Γ q .

Lemmas
Lemma 4.1. Let q ∈ N with (q, N) = 1. The assumptions of Conjecture 1.1 imply the relation where γ q,a is any element of Γ 0 (N) with top row ( q −a ).
Proof. From Hecke [17,Theorem 4.3.5] we know that the functional equation in Conjecture 1.1 is equivalent to the equation In particular we find for q = 1, that . Now we shall note that (4.2) may be rewritten as Combining this with the matrix identity where a = a(c) is chosen so that Nca ≡ −1 (mod q) and s = (Nac + 1)/q, we derive Here the summation over c may be replaced by the summation over a (mod q), (a, q) = 1, by choosing appropriate representatives, thereby proving the lemma. Proof. This is an extension of Weil's Lemma [2, Lemma 1.5.1], which is the special case ζ = 1. It can be proven by the same method or, alternatively, derived as a consequence, as follows. Suppose that ζ has order n, and let M = ( a b c d ). Then we have Applying Weil's Lemma to h n (and the weight-kn slash operator), we conclude that h n = 0, whence h = 0. Proof. Note that ϕ(q) = ϕ(s) = 2, and we have γ q,±1 = γ −1 s,∓1 = q ∓1 ∓N s . Hence, applying Lemma 4.1 to both q and s, we obtain Note that | tr M| < 2 and tr M / ∈ Z, so M is elliptic of infinite order. Applying Lemma 4.
Proof. Replacing (m, n) by (m/ gcd(m, n), n/ gcd(m, n)) if necessary, we may assume without loss of generality that (m, n) = 1. We prove the claim by induction on h.
Suppose h ≥ 2 and expand det[S i,j ] with respect to the first line. We get an expression of the form P (ζ mq 1 ) for some polynomial P ∈ Q(ζ mq 2 , . . . , ζ mq h )[x]. We claim that P is not constant. To see this, let a ∈ s 1,1 (such a exists because s 1,1 = ∅). Then a / ∈ s i,1 for any i = 1, since s i 1 ,1 ∩ s i 2 ,1 = ∅ for i 1 = i 2 . Thus, the coefficient of x a in P (x) is the determinant of the cofactor matrix for S 1,1 . This determinant satisfies all hypotheses of the lemma for h − 1 and primes q 2 , . . . , q h ; hence it is nonzero by the inductive hypothesis.
Proof. Suppose that the conclusion is false for some prime q ∤ N, so that f n = g n = 0 for every n divisible by q. Then we have f n c q (n) = −f n and g n c q (n) = −g n for every n, so that On the other hand, (1.5) applied to c 1 and c q shows that so ξ(q)q 1−2s = 1. Since q > 1, this is a contradiction.
Lemma 4.7. Let N be a prime, and for each r ∈ Z with 2 ≤ |r| < 1 2 N, let γ r,1 ∈ Γ 0 (N) be a matrix with top row ( r −1 ). Then any matrix ( A B CN D ) ∈ Γ 0 (N) may be written in the form ±τ 1 τ 2 · · · τ l with τ i ∈ {T, T −1 , W, W −1 , γ −1 r,1 : 2 ≤ |r| < 1 2 N} for each i = 1, . . . , l, in such a way that #{i : τ i ∈ {γ −1 r,1 }} ≤ log 2 (|A|). Proof. If C = 0 then ( A B CN D ) = ±T α for some choice of sign and α ∈ Z. In the general case we may multiply on the left by a power of T to replace A by any integer A ′ such that A ′ ≡ A (mod CN). Choosing A ′ such that |A ′ | ≤ 1 2 |CN|, we also have |A ′ | ≤ |A|. Similarly we may multiply on the left by W and replace C by any integer C ′ ≡ C (mod A ′ ) with |C ′ | ≤ 1 2 |A ′ |. Repeating this process will either lead to C = 0 or will eventually stagnate. Thus we may assume now that |A| ≤ 1 2 |CN| and 0 < |C| ≤ 1 2 |A|. In particular, this implies that N ≥ 4, so N is an odd prime. Let r be the nearest integer to the fraction CN/A (note that A = 0 since (A, N) = 1), rounded toward 0 in the case of a tie. We have 2 ≤ |CN/A| ≤ 1 2 N, and thus 2 ≤ |r| < 1 2 N. Multiplying on the left by γ r,1 , the new top-left corner is rA−CN = A(r− CN A ), which does not exceed 1 2 |A| in absolute value. Thus, by repeating this process we eventually end up in the case C = 0, having used at most log 2 (|A|) matrices γ r,1 . Note that for S = {1, . . . , n} we have m S ≤ #S. Hence, there is a minimal non-empty set R ⊆ {1, . . . , n} satisfying m R ≤ #R. Since A has non-zero rows, we have m S > 0 whenever S = ∅. From this and the minimality of R it follows that m R = #R. Moreover, for any S ⊆ R we have m S ≥ #S. By Hall's marriage theorem [11], it follows that there is a subset C ⊆ {1, . . . , n} and a bijection i : C → R such that a i(j)j = 0 for every j ∈ C. Writing m = #C = #R and replacing A by P AQ for appropriate permutation matrices P and Q, we may assume that C = R = {1, . . . , m} and i(j) = j. The block form of A then follows from the definition of m S . Lemma 4.9. Given p 0 , a, q ∈ Z with p 0 = 0 and (a, q) = 1, define P (p 0 ; a, q) = {p prime : ∃d ∈ N such that d ≡ a (mod q) and p ≡ p 0 (mod d)} and P (p 0 ; q) = 1≤a≤q (a,q)=1 P (p 0 ; a, q).
Proof. This is proven for p 0 = 1 in [12], uniformly for q ≤ 2 (1−ε) log log x . One can generalize the proof to all p 0 = 0, and if one is not concerned with the uniformity in q a simpler proof suffices. For completeness we give the argument here. For a character χ modulo q and a ∈ Z with (a, q) = 1 let  f n c χ (n)e(nz), and similarly for g χ . Then provided that uvN ≡ −1 (mod q), we have The conclusion now follows by Hecke's argument [17,Theorem 4.3.5].
Note that T, W has some outer automorphisms that preserve the height function. Specifically, conjugating an element γ = τ 1 · · · τ ℓ by ( 1 −1 ) leaves ht(γ) unchanged and swaps every occurrence of T with T −1 and W with W −1 . Similarly, conjugating by ( −1 N ) swaps T with W −1 and W with T −1 . Thus, applying an appropriate outer automorphism, we may assume without loss of generality that τ ℓ = T .