Magnetic (quasi-)modular forms

A (folklore?) conjecture states that no holomorphic modular form $F(\tau)=\sum_{n=1}^\infty a_nq^n\in q\mathbb Z[[q]]$ exists, where $q=e^{2\pi i\tau}$, such that its anti-derivative $\sum_{n=1}^\infty a_nq^n/n$ has integral coefficients in the $q$-expansion. A recent observation of Broadhurst and Zudilin, rigorously accomplished by Li and Neururer, led to examples of meromorphic modular forms possessing the integrality property. In this note we investigate the arithmetic phenomenon from a systematic perspective and discuss related transcendental extensions of the differentially closed ring of quasi-modular forms.


Introduction
One of the arithmetic features of modular and quasi-modular forms is integrality of the coefficients in their Fourier expansions. This is trivially seen on the generators E 2 (τ ) = 1−24 ∞ n=1 nq n 1 − q n , E 4 (τ ) = 1+240 ∞ n=1 n 3 q n 1 − q n , E 6 (τ ) = 1−504 ∞ n=1 n 5 q n 1 − q n (1) of the ring of quasi-modular forms, as well as on the 'discriminant' cusp form where q = q(τ ) = e 2πiτ for τ from the upper half-plane Im τ > 0. All q-expansions above converge for q inside the unit disk, and in fact have polynomial growth of the coefficients. A more suprising fact, brought to the mathematical community by Ramanujan [21] more than 100 years ago, is that the three Eisenstein series in (1) satisfy the algebraic system of differential equations where δ = 1 2πi d dτ = q d dq .
Ramanujan's notation for the Eisenstein series (1) was P (q), Q(q), R(q), respectively, as he mainly viewed them as functions of the q-nome. Since the functions E 2 , E 4 , E 6 are algebraically independent over C, and even over C(q) and over C(τ, q) [19,22], this fine structure gives rise to remarkable applications in transcendental number theory to the values of quasi-modular forms. One particular notable example in this direction is a famous theorem of Nesterenko [20], which states that, given a complex number q with 0 < |q| < 1, at least three of the four quantities q, P (q), Q(q), R(q) are algebraically independent over Q.
Establishing transformation properties of a double integral, which characterises the output voltage of a Hall plate affected by the shape of the plates and sizes of the contacts and which is -for this reason -dubbed magnetic, in the work [5] Broadhurst and the second author came across a meromorphic modular form (on a congruence subgroup), whose anti-derivative had integral coefficients in its q-expansion and was not a modular object itself. This arithmetic observation was subsequently proven by Li and Neururer in [17] who also noticed that the formal anti-derivativẽ dq q of the meromorphic modular form F 4a (τ ) = ∆/E 2 4 has integer coefficients in its qexpansion. (They proved a slightly weaker version about the integrality of the antiderivative of 64∆/E 2 4 .) The function F 4a (τ ) has weight 4 and possesses the double pole at τ = ρ = e 2πi/3 in the fundamental domain, and a simple analysis reveals that it is not the image under δ of an element from the (differentially closed) field C(q, E 2 , E 4 , E 6 ). This implies that the anti-derivativeF 4a = δ −1 F 4a is transcendental over the field, hence the addition ofF 4a to the latter increases the transcendence degree by 1. Following the background in [5], Li and Neururer coined the name 'magnetic modular form' to a meromorphic modular form like F 4a . A principal goal of this note is to investigate the 'magnetic modular' phenomenon further and to give more examples of those. Theorem 1. The meromorphic modular forms F 4a (τ ) = ∆/E 2 4 and F 4b (τ ) = E 4 ∆/E 2 6 of weight 4 are magnetic. In other words, their anti-derivatives δ −1 F 4a and δ −1 F 4b have integral q-expansions.
There are other instances in the literature of related integrality phenomena; however the existing methods of proofs seem to be quite different from what we use below. Investigating the solution space of the linear differential equation in [13] Honda and Kaneko found that, when k = 4, it is spanned by E 4 and They numerically observed and proved some related results about the p-integrality ofẼ 4 for primes p ≡ 1 mod 3. This theme was later analysed and generalised in [2,11,12]. Bringing some parallel to that investigations, it is easy to check that the functions E 4 and E 4 δ −1 (∆/E 2 4 ) (both with integer coefficients in their q-expansions!) span the solution space of the differential equation Df = 0, where At the same time, the only quasi-modular solutions of D 5 y = 0 are spanned by δE 4 (see [15,Theorem 2]). A somewhat different account of strong divisibility of the coefficients of modular forms shows up in the context of arithmetic properties of traces of singular moduli initiated in Zagier's work [23]. As this topic remains quite popular, we only list a selection of contributions [1,3,8,9,10,14]. The methods involved make use of the Shimura correspondence, which is also the main ingredient of our proof of Theorems 1 and 2.

Magnetic quasi-modular forms
In this part we formalise the notion of magnetic forms and give results, which may be thought of as generalisations of Theorems 1 and 2 but use the theorems as principal steps.
Consider the family with Nf ∈ Z[[q]] for some N ∈ Z >0 ) spanned by the q-expansions of the forms f a,b,c of weight k, that is, with 2a + 4b + 6c = k. Because the differential operator δ defines a well defined map W k → W k+2 . Clearly, the image δW k in W k+2 is a Q-subspace in Q ⊗ Z qZ[[q]]; we will call W 0 k+2 the cuspidal subspace of W k+2 , that is, the set of all elements in W k+2 with vanishing constant term in their q-expansion.
We will say that an element v ∈ W 0 k is magnetic if its formal anti-derivative We also call it strongly magnetic if δ −1 v ∈ qZ [[q]]. With the magnetic property, we can associate the equivalence relation ∼ on W k writing v ∼ w if and only if the difference v − w is in W 0 k and is magnetic. Let V k (respectively, V 0 k ) be the Q-vector subspace of W k (respectively, of W 0 k ) generated by the forms f a,b,c with a ∈ {0, 1, . . . , k − 2}. According to relation (3) this range of a makes the subspace V k stable under the δ-differentiation. Notice that δV 2 ⊆ V 0 4 . Theorem 3. Any element of V 0 4 is magnetic.
Remark 1. It seems that the elements of W 0 4 with a > 2 (that is, outside the range assumed in V 0 4 ) with the magnetic property are those that come as linear combinations of δ-derivatives of elements from W 2 . In other words, we expect that the choice of V 0 4 in the theorem as a magnetic space of weight 4 to be sharp. Derivation of Theorem 3 from Theorem 1. It follows from Theorem 1 that the forms Remark 3. In fact, it seems that the space U 0 6 possesses the strongly magnetic property: the anti-derivative of any difference of two f a,b,c from U 6 has an integral q-expansion.
We have just shown that any element in the subspace U 0 6 generated by f a,b,c with c ∈ {0, 1} does have the (strongly) magnetic property. For the rest of our theorem, we proceed by induction over c using the following consequence of equation (3) when

A magnetic extension of the field of quasi-modular forms
The functions τ, q, E 2 , E 4 , E 6 are algebraically independent over C (see [19,22]). We can identify the differential field C τ, q, E 2 , E 4 , E 6 generated by them over C with the differential field K = C τ, q, X, Y, Z equipped with the derivation Our goal is to demonstrate that the elements By [16,Lemma 3.9] applied twice, the anti-derivatives are algebraically independent over the field C τ, q, E 2 , E 4 , E 6 , the extended differential field C τ, q, E 2 , E 4 , E 6 ,Ẽ 4a ,Ẽ 4b has transcendence degree 7 over C and is a Picard-Vessiot extension of the differential field C τ, q, E 2 , E 4 , E 6 . Again, by identifying the former through the isomorphism ϕ : with the differential fieldK = C τ, q, X, Y, Z, S, T equipped with the derivation we want to demonstrate that the element Assume on the contrary that there is an element u 3 ∈K such thatDu 3 = v 3 . Notice that the functions τ , q = e 2πiτ , E 2 (τ ), E 4 (τ ) and E 6 (τ ) are all analytic at τ = ρ = e 2πi/3 , the latter three having the values With the help of Ramanujan's system (2) we find out that In turn, this implies that for some functions g 1 (τ ) and g 3 (τ ) analytic at τ = ρ, whileẼ 4b (τ ) is analytic there.
To summarise, the function is a polynomial in τ, q, X, Y, Z, T . The latter is seen to be impossible after the operatorD is applied to u and to u 3 − 2 √ 3 π S leading to a rational expression of S in terms of the other generators ofK. The contradiction we arrive at implies that the anti-derivativeẼ is transcendental over the field C τ, q, E 2 , E 4 , E 6 ,Ẽ 4a ,Ẽ 4b . On replacing the generators of the latter with the anti-derivatives of magnetic modular forms from Theorems 1 and 2 we obtain the following result.
Theorem 5. The differentially closed field generated by τ , q = e 2πiτ , the Eisenstein series (1) and the anti-derivatives with integral coefficients in their q-expansions, has transcendence degree 8 over C.
Remark 4. Another way to see that no u 3 exists inK such thatDu 3 = v 3 is by casting u 3 in the form p/q with p, q in the ring R[S], where R = C τ, q, X, Y, Z, T , and gcd(p, q) = 1. After clearing the denominators inD(p/q) = v 3 and comparing the degree in S on both sides, one concludes thatDq = uq for some u ∈ R (that is, independent of S). This leads to conclusion q ∈ R, so that u 3 is a polynomial in S. Finally, the equationDu 3 = X 2 Z/Y − Z is seen to be impossible by comparing the order in Y on both sides.
Exercise 1. We leave to the reader the exercise to prove that the anti-derivative ofF 6 (in turn, the second anti-derivative of F 6 ) is transcendental over the field in Theorem 5.

Half-integral weight weakly holomorphic modular forms
Following the ideas in [17], we will cast magnetic modular forms of weight 2k as the images of weakly holomorphic eigenforms of weight k + 1/2 under the Shimura-Borcherds lift. In our settings, an input for the lift is a form f (τ ) = n≫−∞ a(n)q n from the Kohnen plus space M !,+ k+1/2 (meaning that a(n) vanishes when (−1) k n ≡ 0, 1 mod 4); the output is the meromorphic modular form Ψ(f )(τ ) = n>0 A(n)q n with where D = D k = 1 for k even (so that the Kronecker-Jacobi symbol d D is always 1) and D = D k = −3 for k odd. In other words, and the latter expression is just F = n>0 q n d|n d k−1 a(n 2 /d 2 ) when k is even. We will also distinguish the Kohnen plus cuspidal space S !,+ k+1/2 in M !,+ k+1/2 by imposing the additional constraint a(0) = 0.
Our examples of forms from M !,+ k+1/2 with k = 2 involved in the proof of Theorem 1 are the following three: where θ(τ ) = n∈Z q n 2 and The modular form g 0 (τ ) is known by the name of normalised Cohen-Eisenstein series of weight 5/2.
Proof. Indeed, we only need to check that f 4a , f 4b have vanishing constant term and that the first three coefficients in the q-expansions of Ψ(f 4a ), Ψ(f 4b ) agree with those of the predicted meromorphic modular forms; we choose to check the first seven coefficients.
As we will see further, for certain forms n≫−∞ a(n)q n ∈ S !,+ 5/2 with integral qexpansions (in particular, for the forms 64f 4a and 108f 4b ) one can make use of Hecke operators to conclude with the divisibility n | a(n 2 ) for n > 0. This readily implies that 64F 4a and 108F 4b in Theorem 1 are strongly magnetic modular forms, since the relation in (5) translates the divisibility into A detailed analysis below reveals that the factors 64 and 108 can be also removed.

The square part and Hecke operators
We refer the reader to [8] and [4] for the definition of Hecke operators T p and T p 2 on integral weight 2k and half-integral weight k + 1/2 modular forms (including weakly holomorphic or meromorphic), respectively. As in the case of the Shimura-Borcherds lift Ψ = Ψ k in (6), these definitions make perfect sense for any Laurent series f = n≫−∞ a(n)q n , not necessarily of modular origin but with the weight 2k or k + 1/2 additionally supplied. We refer to the finite sum n<0 a(n)q n as to the principal part of f . We take for a character χ : Z → C, and define is the Kronecker-Jacobi symbol. A simple calculation shows that Ψ k (f ) | (T p , 2k) = Ψ k f | (T p 2 , k + 1/2) , which we can reproduce in a simplified form when k is fixed. Lemma 2. Given a positive integer k, assume that there are no cusp forms of weight 2k. For a prime p, let f ∈ M !,+ k+1/2 have p-integral coefficients and satisfy p 2 > − ord q (f ). Then f | T n p 2 ≡ 0 mod p (k−1)n . Proof. Following the argument in [4, proof of Lemma 3.1], we can write where α a,b,c,r are some integers. This writing can be easily deduced from V p 2 χ p = χ p U p 2 = 0 and the fact that V p 2 U p 2 is the identity. We only need to analyse the principal part of f | T n p 2 which, by the hypothesis dim S 2k = 0, determines it uniquely. If r < a, then f | U a−r has no principal part, because the latter is killed by a single action of U p 2 (since a −p 2 m = 0 for any m ≥ 0). Therefore, we may assume that a = r ≤ c. This implies that (2k − 1)c + k+1/2 whose elements have all coefficients integral (see [8,Proposition 2]).
The definitions immediately lead to the following conclusions. .
In addition to this, we list some other easily verifiable properties about the interaction of Hecke operators and square parts. (a) Ψ(f ) | T n p = Ψ(f | T n p 2 ) for n = 1, 2, . . . .
If the coefficients of f are integral and k ≥ 2, then f | T p 2 ≡ f | U 2 p mod p. Proof of Theorem 1. Consider f ∈ {f 4a , f 4b }. For a prime p ≥ 5, the form f is pintegral and we have ord q (f ) ≥ −4; therefore Lemma 2 with k = 2 applies to result in f | T n p 2 ≡ 0 mod p n . Applying Shimura-Borcherds map (6) we deduce that, for F = Ψ(f ) ∈ {F 4a , F 4b }, we have F | T n p ≡ 0 mod p n for all n ≥ 1, hence F | U n p ≡ 0 mod p n ; in other words, F = m>0 A(m)q m has the strong p-magnetic property: for any prime p ≥ 5. This argument also works for f = f 4a in the case of p = 3, because f 4a is 3-integral. Consider now p = 3 and f = f 4b , in which case we only know that 27f is 3-integral. Take the (unique!) element g r ∈ M !,+ 5/2 with q-expansion g r = q −4·9 r + O(q); by [8, Proposition 2] it has integral coefficients. We first show that g 0 | T n 9 ≡ 0 mod 3 n+3 . For n = 0 this is true, because g 0 = −108 · f 4b and f 4b is in qZ [[q]]. For n = 1 we observe that Ψ(− 1 meaning that g 0 | T n 9 ≡ 0 mod 3 n+3 is true when n = 1. Since g 0 | T 9 = 27g 1 − 3g 0 we also deduce from this that g 1 ≡ 0 mod 3.
For n general, we want to write g 0 | T n 9 as a Z-linear combination of g r with r = 0, 1, . . . , n. Looking at the principal part of g 0 | T n 9 , one finds out that only terms of the form q −4·3 2m appear, so that subtracting the related linear combination of f r leads to a holomorphic cusp form, which then must vanish. To examine this linear combination in more details we proceed as in the proof of Lemma 2: (see equation (8)). As already noticed in that proof, only the terms with r = a ≤ c contribute to the principal part, thus to the linear combination; the terms with r = a contribute by the subsum Now notice that if 2c ≥ a + 3, then the coefficient is divisible by 3 n+3 . In the remaining situations we have 2a ≤ 2c < a + 3, in particular a ∈ {0, 1, 2}, and we use the following analysis: (a) If a = 2, then the inequalities imply that c = 2, hence b = n − 4; the corresponding term is then a multiple of 3 3·2+n−4 g 0 . (b) If a = 1, then c = 1, hence b = n − 2; the corresponding term happens to be a multiple of 3 3·1+n−2 g 0 .
(c) If a = 0, then c ∈ {0, 1}. The term corresponding to c = 0 is a multiple of 3 n g 0 , while the term corresponding to c = 1 is a multiple of 3 n+2 · g 1 . Gathering all the terms, we end up with an expression g 0 | T n 9 = 3 n+3 g + 3 n+2 α · g 1 + 3 n β · g 0 , where g is integral and both α and β are integers. Taking the square parts on both sides and using the results for n = 0, 1 we deduce that g 0 | T n 9 ≡ 0 mod 3 n+3 for any n = 0, 1, . . . . Finally, we apply the Shimura-Borcherds map to this congruence to deduce that F 4b | T n 3 ≡ 0 mod 3 n for all n ≥ 0. In other words, this implies the congruences (9) for p = 3.
Turning now our attention to the prime p = 2, notice that the Hecke operator T 4 does not respect the Kohnen plus space. However, if we define the projection then the operator T ′ 4 = K + • T 4 maps the space M !,+ k+1/2 onto itself and inherits all the properties used above for T p 2 when p > 2. We use this operator T ′ 4 in place of T 4 to complete the proof of our Theorem 1. Notice that in both cases f = f 4a and f = f 4b has powers of 2 in the denominator of its main term. For an ease of the argument, we treat the two cases separately, though the same strategy is used for both, along the line with the proof above of relation (9) for p = 3.
When f = f 4b , we need to prove that F 4b | T n 2 ≡ 0 mod 2 n , which is in turn implied by the congruence f 4b | T ′ 4 n ≡ 0 mod 2 n . Introduce g r = q −4·4 r + O(q) ∈ M !,+
For f = f 4a , we introduce the family g r = q −3·4 r +O(q) ∈ M !,+ 5/2 , where r = 0, 1, . . . , which is invariant under the action of the operator T ′ 4 , and proceed similarly to get exactly the same recursion g r | T ′ 4 = 8g r+1 + g r−1 for r ≥ 0 with g −1 = 0. On using g 0 = 1 64 f 4a , g 0 | T ′ 4 n = 2 n+6 g + 2 n+5 α · g 2 + 2 n+4 β · g 1 + 2 n+3 γ · g 0 for n ≥ 3, and F 4a ≡ ∆ mod 8, we conclude with g 0 | T ′ 4 n ≡ 0 mod 2 n+6 implying F 4a | T n 2 ≡ 0 mod 2 n as required. Proof of Theorem 2. We now work with k = 3. Consider where f * b+1/2 is the weight b + 1/2 modular form from the table in [8,Appendix]. One can easily check (through the first few coefficients) that Ψ(f ) = F 6 and from the expression above we also know that f has p-integral coefficients for any p ≥ 5. It follows from Lemma 2 (applied this time with k = 3) that f | T n p 2 ≡ 0 mod p 2n . Therefore, F 6 | T n p ≡ 0 mod p 2n for all n ≥ 0 implying that F 6 | U n p ≡ 0 mod p 2n and that for F 6 = m>0 A(m)q m we have for any prime p ≥ 5. Since 384 = 3 · 2 7 , for p = 3 we see that 3f is 3-integral. Repeating the argument from Lemma 2 and using the fact that f is a multiple of the unique element in M !,+ 7/2 with the integral q-expansion q −1 + O(q), we deduce that f | T n 9 = 3 2n · (g + αf ) with α an integer and g a 3-integral modular form. Indeed, the principal part of f | T n 9 is an Z-linear combination of the principal parts of is 3-integral; when c = a the principal part of f | χ b 3 will be an integral multiple of the principal part of f . Thus, f | T n 9 = 3 2n · (g + α · f ) implies (applying the Shimura-Borcherds lift to both sides) that F 6 | T n 3 ≡ 0 mod 3 2n , hence we deduce that (10) is true also for p = 3. To prove the relation (10) for p = 2, we proceed as in the proof of Theorem 1. We introduce the T ′ 4 -invariant family of weight 7/2 weakly holomorphic modular forms g r = q −4 r +O(q) with integral q-expansions with the help of [8, Proposition 2]. Again, we write the expression of g 0 | T ′ 4 n as Z-linear combination of g r with r = 0, 1, . . . , n and analyse the powers of 2 appearing in the coefficients; similarly, we can prove that g 0 | T ′ 4 n ≡ 0 mod 2 2n+7 for any n ≥ 0. For n = 0 this comes from the integrality of f , while for n = 1 we get it, again, by noticing that F 6 ≡ E 6 ∆ mod 2 4 while E 6 ∆ being an eigenform of weight 18 with slope 4 at the prime 2. The induction argument follows mutatis mutandis as in the proof of Theorem 1.

Miscellania on half-integral weight modular forms
In this part, not well related to the proofs of Theorems 1 and 2, we indicate a different strategy of constructing half-integral weight weakly holomorphic modular forms using a traditional rising operator.
Standard examples of weight 1/2 modular forms (see [6, Sect. 14, Example 2]) include the theta function θ(τ ) = n∈Z q n 2 and where E 2,4 (τ ) is given in (7). The images of 12θ and 4θ + h 0 under the multiplicative Borcherds lift Ψ mult : are the modular forms ∆(τ ) and E 4 (τ ), respectively (see [6,Theorem 14.1] for the definition of h). Although it is not useful for our results in this note, we remark that the two weakly holomorphic modular forms can serve as constructors of some weight 5/2 modular forms from Section 4.
Lemma 5. The raising operator Observe that E 2 (τ )−4E 2 (4τ ) is a modular form of weight 2 for Γ 0 (4), so that the difference between the usual raising operator and D is the multiplication by a weight 2 modular form, thus indeed D : M ! k+1/2 → M ! k+5/2 . On the other hand, both δ and multiplication by any modular form f (4τ ) preserve the Kohnen plus space condition, and the lemma follows.
For the functions g 0 , f 4a and f 4b in Section 4 we find out that

Concluding remarks
Though we expect that our discussion above exhausts all elements with the magnetic property in W 0 4 , many such exist for W 0 2k with k > 2; for example, the q-series E m 2 · (δE j )/E j for j = 4, 6 and m = 1, 2, 3, 4, 6 (but not for m = 5). Constructing magnetic modular forms -meromorphic ones with poles at quadratic irrationalities from the upper half-plane -is a routine on the basis of Shimura-Borcherds (SB) lift (6); Table 1 lists a few instances of this production explicitly in terms of the j-ivariant j(τ ) = E 3 4 /∆. Generating the forms with multiple magnetic property in higher weights is a tougher task; one such example E 2 4 (j − 3 · 2 10 )/j 2 can be found in the more recent work [18] of Löbrich and Schwagenscheidt; another example of a triply magnetic form of weight 8 is We have observed that in all such instances the related numerators, viewed as polynomials in j, have real zeroes only. Furthermore, there are weaker divisibility conditions (resembling the Honda-Kaneko congruences [13]) for individual summands of magnetic forms; for example, the anti-derivatives of E 4 j (j − 2 · 30 3 ) 2 and E 4 (j − 2 · 30 3 ) 2 are already p-integral for primes p ≡ 5 (mod 6). We have not tried to investigate this arithmetic subphenomenon.