Congruences for f and \omega

Congruences for Ramanujan’s f and ω functions via generalized Borcherds products

Jen Berg University of Texas at Austin, Department of Mathematics, Austin, TX 78712 Abel Castillo The University of Illinois at Chicago, Department of Mathematics, Statistics, and Computer Science, Chicago, IL 60607-70045 Robert Grizzard University of Texas at Austin, Department of Mathematics, Austin, TX 78712 Vítězslav Kala Purdue University, Department of Mathematics, West Lafayette, IN 47907 Richard Moy Northwestern University, Department of Mathematics, Evanston, IL 60208  and  Chongli Wang Columbia University, Department of Mathematics, New York, NY 10027
Abstract.

Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujan’s mock theta functions and in detail.

Key words and phrases:
mock theta functions, Borcherds products, harmonic Maass forms
11F03, 11F33

1. Introduction

The goal of this note is to prove some explicit congruences for the following two functions:

 ω(q) =∞∑n=0aω(n)qn:=∞∑n=0q2n(n+1)(1−q)2(1−q3)2⋯(1−q2n+1)2 =1+2q+3q2+4q3+6q4+8q5+10q6+14q7+18q8+22q9+29q10+36q11+⋯; f(q) =∞∑n=0af(n)qn:=∞∑n=0qn2(1+q)2(1+q2)2⋯(1+qn)2 =1+q−q2+q3−q6+q7+q9−2q10+q11−q12+2q13−2q14+2q15−q16+⋯,

where for in the complex upper half-plane . The functions and are two of Ramanujan’s famous “mock theta functions”, which he described in his famous last letter to Hardy. Thanks largely to the work of Zwegers (see [8] and [9]), we now understand the mock theta functions as the holomorphic parts of half-integral weight weak harmonic Maass forms. See for example [5] for background on these functions.

In [1] Bruinier and Ono described generalized Borcherds products for weak harmonic Maass forms and proved that they are weight 0 meromorphic modular forms. They specifically describe a family of such Borcherds products which is defined using a vector-valued form whose coefficients are defined in terms of the coefficients of and . This work indicates that one can prove a multitude of congruences for these mock theta functions by using the classical theory of congruences for modular forms; however, using this technique to obtain explicit congruences takes some effort. Bruinier and Ono showed one example of this (only for ) in [2], and in the present paper we produce additional examples.

In order to state our results, we must introduce some auxiliary objects which will be defined in Section 2. We will define a vector-valued form with coefficients , which will be given in terms of the coefficients and in Lemma 2.4. For a fixed negative fundamental discriminant and an integer such that , the aforementioned Borcherds product studied by Bruinier and Ono has a logarithmic derivative which will have a -expansion as follows:

 (1.1) ΦΔ,r(z,˜H)=∞∑n=1∑d∣nc+(|Δ|d224,rd12)(−d)∑s mod Δ(Δs)2e(sndΔ)qn

where is the quadratic residue symbol. For any prime inert or ramified in and any exponent , the normalization of this Fourier expansion will be the same modulo as that of a modular form on . Therefore it makes sense to discuss the action of Hecke operators for any prime on , modulo . Our most general result is the following.

Theorem 1.1.

Let be a negative fundamental discriminant, an integer such that , and let be the normalization of the form defined in (1.1). Fix a prime inert or ramified in and a positive integer . Let be a prime such that

 (1.2) Φ∗Δ,r | Tp,k≡b(p)Φ∗Δ,r (\textupmod ℓR),

where is the corresponding weight as in Lemma 2.8 and is the appropriate Hecke operator.

Write . Then we have

 (1.3) c(n)≡c(1)n∏p|n(b(pvp(n))−b(pvp(n)−1)(Δp)2) (\textupmod ℓR).

Moreover, for a given , , , and , the set of primes for which the above statement holds has positive arithmetic density.

For example, we have the following two corolarries.

Corollary 1.2.

Let , . Let ; , , and be as in Theorem 1.1, such that . Then for all we have

 aω(2(p4m+2−1)3) ≡−1p2m+1(−pk−1)m(−8p)2(p3)2 (\textupmod ℓR), \textupand aω(2(p4m−1)3) ≡−1p2m(−pk−1)m (\textupmod ℓR).

Explicitly, when and when ; when and when .

Moreover, for a fixed prime power , the set of primes such that the above statement holds has positive arithmetic density.

Corollary 1.3.

Let , . Let ; , , and be as in Theorem 1.1, such that . Then for all we have

 af(23p4m+2+124) ≡−1p2m+1(−pk−1)m(−23p)2(p6)2 (\textupmodℓR), \textupand af(23p4m+124) ≡−1p2m(−pk−1)m (\textupmodℓR).

Moreover, for a fixed prime power , the set of primes such that the above statement holds has positive arithmetic density.

We will prove these and related results in Section 3. We will describe the statements on the density of primes in Section 2.2. In Section 4, we will describe how to prove explicit congruences that follow from our theorem, and give the following example in detail.

Corollary 1.4.

We have

 aω(2(52M−1)3)≡(−5)−m−ε\textup(mod 23)

for each positive integer , .

acknowledgments

This paper arose from a project led by Ken Ono at the Arizona Winter School 2013. The authors wish to thank Professor Ono and the organizers of the AWS for providing this opportunity and for the huge amount of support and help with the project. The authors also thank William Stein and SAGE for help with our computations.

2. Nuts and bolts

2.1. Modular forms modulo primes

Let be the Hasse invariant. We recall that is a modified Eisenstein series that vanishes precisely at the supersingular locus, and that for , can be lifted to the Eisenstein series (see [3, Sections 2.0 and 2.1]). Moreover, we know that [4, p. 6]. We combine these facts in the following lemma.

Lemma 2.1.

For any rational prime and any positive integer , there exists a holomorphic modular form , of weight such that and vanishes precisely at for all elliptic curves supersingular over .

2.2. “Almost all” statements for congruences of modular forms

Following Ono [4, p. 43], we make the following observations which result from the existence of Galois representations of modular forms and the Chebotarev Density Theorem.

Lemma 2.2.

Let be an integral weight modular form in whose Fourier coefficients are in , the ring of algebraic integers of a number field . Let be an ideal of . Then a positive proportion of primes have the property that

 g(z)∣Tp,k≡0(modl).
Lemma 2.3.

Assuming the same notation as above, a positive proportion of primes have the property that

 g(z)∣Tp,k≡2g(z)(modl).

In particular, for sufficient rational primes (see Lemma 2.8 below), we will have that the logarithmic derivative is congruent modulo to a modular form in , and is therefore an eigenfunction for the Hecke operator modulo with eigenvalue 0 or 2 for a positive proportion of primes . Additionally, [4, Theorem 2.65] tells us that “almost all” of the coefficients of will be zero modulo . These results justify the statements on the density of primes satisfying the hypotheses of Theorem 1.1 and Corollaries 1.2 and 1.3.

2.3. Generalized Borcherds products

We follow [1, Section 8.2], and write a weight 1/2 harmonic Maass for whose coefficients are related to those of and . Define the following cuspidal weight 3/2 theta functions

 g1(τ) :=∑n∈Z(−1)n(n+13)e3πi(n+13)2τ, g2(τ) :=∑n∈Z−(n+16)e3πi(n+16)2τ, and g3(τ) :=∑n∈Z(n+13)e3πi(n+13)2τ.

Let

 F(τ):=(q−124f(q),2q13ω(q12),2q13ω(−q12))T

and

 G(τ):=∫i∞−¯¯τ(g1(z),g2(z),g3(z))T√−i(τ−z)dz;

Zwegers showed that is a vector-valued weight 1/2 harmonic Maass form (see [9, Theorem 3.6]; the term “harmonic Maass form” was defined later, cf. [1]). Bruinier and Ono show in [1, Lemma 8.1] that gives rise to ; the lemma below gives an explicit relation between the coefficients of and those of and .

Lemma 2.4.

Let denote the coefficients of the holomorphic part of .

1.  c+(m,112+Z) =−c+(m,512+Z)=c+(m,712+Z)=−c+(m,1112+Z) =af(m+124)
2.  c+(m,212+Z)=−c+(m,1212+Z)=2((−1)2(m−13)−1)aω(2(m−13))
3.  c+(m,412+Z)=−c+(m,812+Z)=−2((−1)2(m−13)+1)aω(2(m−13))
Proof.

This is a simple computation following from the definition of in [1, Section 8.2]. ∎

One also observes that satisfies the conditions for the generalized twisted Borcherds lift described in [1, (8.10)] to be a meromorphic modular function of weight for , whose divisor is supported on CM points. We then use the following well-known lemma stated below without proof.

Lemma 2.5.

Let be a meromorphic modular function of weight for a congruence subgroup of . Then, the logarithmic derivative of , i.e.

 qddqgg,

is a weight 2 meromorphic modular form for of weight , with simple poles supported on the divisor of , and no other poles.

We take the logarithmic derivative of , that is,

 ΦΔ,r(z,˜H):=qddq[ΨΔ,r(z,˜H)]ΨΔ,r(z,˜H).

By the lemma above, is a weight 2 meromorphic modular function for , with at most simple poles at CM points and no other poles.

The -expansion of is given by

 (2.1) ΦΔ,r(z,˜H)=∞∑n=1∑d∣nc+(|Δ|d224,rd12)(−d)∑s mod Δ(Δs)2e(sndΔ)qn,

giving us an explicit relationship between the Fourier coefficients of and the coefficients of the mock theta functions and .

For simplicity of notation, let us denote and , suppressing the dependence on and wherever it will not cause confusion. Also, let

 Φ∗=Φ∗Δ,r(z,˜H):=ΦΔ,r(z,˜H)/⎛⎝−c∑s mod Δ(Δs)2e(sndΔ)⎞⎠

be the normalized logarithmic derivative (i.e., so that its -coefficient is 1).

To obtain a simpler expression for the coefficients of we evaluate the “Gauss sum”:

Lemma 2.6.

Let . Let .

1. If , then .

2. In particular, for , we have

 G(a,−8)=⎧⎪ ⎪⎨⎪ ⎪⎩−√−8 if a≡1,3(mod8)√−8 if a≡5,7(mod8)0 if a≡0(mod2).
Proof.

a) Let be the inverse of . We have

 G(a,Δ) =∑s mod Δ(Δs)2e(asΔ) =(Δa−1)2∑s mod Δ(Δs)2e(sΔ)

by a simple substitution in the summation and the multiplicativity of the Kronecker symbol. Therfore, a) follows since .

The proof of b) is an explicit calculation of the values. ∎

Proposition 2.7.

Take coprime to . Then the coefficient of the -expansion of is

 b(n):=1c∑d|nc(d)d(Δn/d)2.

Moreover,

 c(n)=cn∑d|nb(d)μ(n/d)(Δn/d)2,

where is the Möbius function.

Proof.

The -coefficient of is . Thus the formula for follows immediately from Lemma 2.6.

Then note that we have the convolution formula , where and . Therefore, is multiplicative, and so its Dirichlet (convolution) inverse is . Therefore , which gives exactly the formula for . ∎

In order to prove congruences in the next section, we need that is congruent to modular forms modulo powers of primes:

Lemma 2.8.

Suppose (and thus also ) has simple poles. Let be a rational prime that remains inert or ramifies in . Then there exists a holomorphic modular form of weight such that

Proof.

Let be a pole of . Then, corresponds to a zero of the holomorphic modular form defined in Lemma 2.1. We know that will be a CM point defined over , and that does not split in this field. Suppose is a zero of with ; letting and be elliptic curves with -invariant and respectively, we can multiply by

 (F⋅j(z)−j(α)j(z)−j(β))R

to obtain a meromorphic modular form congruent to modulo with one less simple pole. Observe that this increases the weight of the form by . ∎

3. Proofs of congruences

Throughout this section, will denote a fundamental discriminant and an integer such that .

Lemma 3.1.

Let be a normalized (i.e., ) modular form on of weight .

1. If is a prime such that , then and .

2. Let be a set of primes such that for all we have . Then for all coprime divisible only by primes lying in . Also, for we have .

Proof.

This is very standard, so we just provide a sketch:

The -coefficient of is . Part (a) then follows by easy induction using this formula. Similarly, part (b) follows by using the formulas for coefficients of . ∎

Proposition 2.7 allows us to obtain congruences for the coefficients assuming we know the values of modulo for a suitable prime power . For example, such information can come from our logarithmic derivative’s being an eigenfunction for a Hecke operator modulo , or even an eigenfunction for all the Hecke operators (i.e. congruent modulo to a Hecke eigenform).

Theorem 3.2.

Fix a prime which is inert or ramified in and . Let be the corresponding weight as in Lemma 2.8 and the appropriate Hecke operator. Let be a prime such that .

If also does not divide , then:

 c(p2m)≡−cp2m(−pk−1)m(modℓR)

Moreover, (fixing , , , ), the set of primes such that has positive density.

Proof.

By Proposition 2.7 we have

 c(pM)=cpM∑N≤Mb(pN)μ(pM−N)(ΔpM−N)2=cpM(b(pM)−b(pM−1)(Δp1)2),

because when , is not square-free, and so . The statement now immadiately follows from Lemma 3.1. ∎

Proof of Corollaries 1.2 and 1.3.

Note that in this case . By Lemma 2.4 we have , where the sign is exactly . The statement now follows immediately from 3.2.

The second part and Corollary 1.3 follow in the same way. ∎

Similarly we can obtain congruences for coefficients at , where is divisible only by primes as in Theorem 3.2.

Proof of Theorem 1.1.

All the congruences in this proof are , and so we omit writing this. Throughout the proof, let be a positive integer divisible only by primes in . Note that by Lemma 3.1 we have . Also the quadratic symbol is multiplicative and is 0 if is not square-free.

Thus by Proposition 2.7 we have

Writing with , and using the multiplicativity of , , and , we get

 ncc(n)≡∑(εp)p∏p|nb(pvp(n)−εp)(−1)εp(Δp)2,

the sum being over all possible tuples

But the last sum is just the product , as we wanted to show. ∎

4. Explicit computation

Our theorems, combined with the “almost all” theorems mentioned in Section 2.2, tell us that we can obtain infinitely many explicit congruences for the coefficients and . Writing any explicit congruence down, however, is a different matter. If we want to prove that for certain primes and , then we must compute enough coefficients of the expansion of to be able to apply Sturm’s Theorem (see [7] and [4, section 2.9]); this useful theorem gives an upper bound on the number of coefficients of a modular form that can be zero modulo without all of them being zero.

There are two methods for computing these coefficients. The first method uses (1.1) directly. These coefficients are given in terms of the coefficients of and . In practice, this is quite difficult, as we now describe.

Suppose we wanted to show that (mod ), by verifying this congruence for the first coefficients, where is determined using Sturm’s Theorem. This means we need to have at least coefficients computed for . Using Lemma 2.4, we can verify that this will require computing coefficients of and coefficients of . For example, if , , and , this would mean computing over 8 million coefficients for and over 16 million for .

The second method is to find an explicit expression for , as in [1, Section 8.2], in terms of well-known series. The authors of [1] wrote an expression for as a rational function in Eisenstein series and Dedekind eta functions (see formulas on page 2175 of [1]). Using this, one can compute as many coefficients for as one can compute for the Eisenstein and eta products. For the above example, this is faster by several orders of magnitude.

Using this second method, we were able to compute enough coefficients to prove some explicit examples of congruences for using . We proved the following computationally.

Example 4.1.

Let be the normalized -series defined in Section 2.3. We have

1. Let be the normalized newform in . We have that

 Φ∗−8,4(q)≡g(mod5).

We will only explicitly discuss part (a). Part (b) follows in the same manner. We know has at most two poles from Bruinier and Ono’s explicit formula [1, p. 2174–2175]. We multiply by two twisted Hasse invariants to obtain a holomorphic modular form of weight on by Lemma 2.8. Using SAGE (see[6]), we calculated that to precision . Sturm’s Theorem gives us that if , is a prime, and the coefficients of in the -expansion of vanish modulo for all , then . In this case, the Sturm bound is (a coincidence), and were able to verify that

 (4.1) Φ∗−8,4(q) | T5≡Φo(q) | T5≡0(mod23)

computationally, as described above.

Equation (4.1), along with Corollary 1.2 gives us the explicit congruence

 aω(2(52M−1)3)≡(−5)−m−ε\textup(mod 23)

for each positive integer , . The first few cases of this congruence were verified by computer:

congruence class mod 23
16 101 9
416 147019574355949 9
10416 61055287817898262553386166573674890 71828343984275529702652915791815662 46045801 12
260416 47135043177557696996583864908485299 02877317903609618663661969973372815 28667228657386190726273269144570224 19421615118007256010585973847924915 81461923182898155305890742543478218 22773223303943277462010996573525810 14479578234053700509911978165482992 25987091881940041827267379817078651 91857035767135962999286799635887103 00979543276318001547509333358824720 65861382823046394902072007486105992 62211650315449 12

We can also produce an infinite family of such congruences modulo 5, because we verified that is congruent modulo 5 to a Hecke eigenform.

References

• [1] Jan H. Bruinier and Ken Ono. Heegner divisors, -functions and harmonic weak Maass forms. Ann. of Math. (2), 172(3):2135–2181, 2010.
• [2] Jan H. Bruinier and Ken Ono. Identities and congruences for Ramanujan’s . Ramanujan J., 23(1-3):151–157, 2010.
• [3] Nicholas M. Katz. -adic properties of modular schemes and modular forms. In Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 69–190. Lecture Notes in Mathematics, Vol. 350. Springer, Berlin, 1973.
• [4] Ken Ono. The web of modularity: arithmetic of the coefficients of modular forms and -series, volume 102 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2004.
• [5] Ken Ono. Unearthing the visions of a master: harmonic Maass forms and number theory. In Current developments in mathematics, 2008, pages 347–454. Int. Press, Somerville, MA, 2009.
• [6] W. A. Stein et al. Sage Mathematics Software (Version 5.6). The SAGE Development Team, 2013.
• [7] Jacob Sturm. On the congruence of modular forms. In Number theory (New York, 1984–1985), volume 1240 of Lecture Notes in Math., pages 275–280. Springer, Berlin, 1987.
• [8] Sander P. Zwegers. Mock Theta Functions. PhD thesis, Universiteit Utrecht.
• [9] Sander P. Zwegers. Mock -functions and real analytic modular forms. In -series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000), volume 291 of Contemp. Math., pages 269–277. Amer. Math. Soc., Providence, RI, 2001.
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