Elsevier

Journal of Number Theory

Volume 215, October 2020, Pages 275-296
Journal of Number Theory

General Section
Romik's conjecture for the Jacobi theta function

https://doi.org/10.1016/j.jnt.2020.01.009Get rights and content

Abstract

Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point i. He then conjectured that the Taylor coefficients d(n) either vanish or are periodic modulo any prime p; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, with the latter trio considering arbitrary half integral weight modular forms. We refine previous work for p1(mod4) by displaying a concise algebraic relation between d(n+p12) and d(n) related to the p-adic factorial, from which we can deduce periodicity with an effective period.

Introduction

Dan Romik published a 2019 paper [7] considering the Taylor expansion of the classical Jacobi theta function θ3, defined asθ3(x):=n=eπn2x,x>0. It satisfies an important modular transformation given byθ3(1x)=xθ3(x). It is natural to consider the Taylor expansion of θ3 around x=1, the fixed point of this transformation. In fact, the natural function to study isσ3(z):=11+zθ3(1z1+z)=θ3(1)n=0d(n)(2n)!(Γ8(14)27π4)nz2n. The reason for considering such a rescaled function is that the Möbius transformation xz=1x1+x conformally maps the right half-plane to the unit disc and sends the inversion map x1x to the reflection zz, so that the modular transformation satisfied by θ3 is then equivalent to the statement σ3(z)=σ3(z), and Taylor expanding θ3(x) around x=1 is equivalent to Taylor expanding the simpler σ3(z) around z=0.

Although Romik found a recurrence for the Taylor coefficients d(n) (Definition 4), there does not seem to exist a closed form expression for these coefficients. They depend on a sequence s(n,k), which in turn depends on a sequence u(n), which is given by a recurrence relation. This triply nested definition makes it rather unwieldy to work with the Taylor coefficients directly, though Romik conjectured several nice properties of the d(n) coefficients modulo any prime. This paper is dedicated towards refining the second half of Romik's conjecture, which was proved by the combined efforts of Scherer [8] and Guerzhoy-Mertens-Rolen [3]. Guerzhoy, Mertens, and Rolen in fact prove a stronger statement in the context of an arbitrary half integer weight modular form.

Theorem 1

[8], [3] Modulo any prime p, d(n) exhibits the following behavior.

  • For any prime p3(mod4) and n0=p212, d(n)0(modp) for n>n0.

  • For any prime p1(mod4) or p=2, d(n)(modp) is periodic.

We prove a stronger result in the case p1(mod4).

Theorem 2

Consider p1(mod4). Then for np+12, we haved(n+p12)2p123272(2p3)2d(n)(modp).

This encodes more arithmetic information, and on iterating this p1 times we see that d(n) is periodic with period (p1)22, which is not necessarily minimal. The proof structure is tripartite:

  • (1)

    show that u(n),v(n)0(modp) for np+12 in Lemma 6, a result conjectured in Scherer's paper [8];

  • (2)

    use this to simplify the expression for s(n,k)(modp) in Lemma 10;

  • (3)

    use the expression for s(n,k) to show the desired periodicity of d(n)(modp) in Theorem 16.

Our methods are elementary, and consist of a tour through classical number theory and group theory. We essentially use the method of Scherer [8], who studied the p=5 case; however, the proof for arbitrary p becomes rather technically complex. The modular forms approach of Guerzhoy-Mertens-Rolen [3] proves eventual periodicity for any half integral weight modular form at any CM point. However, with our method we can show not just periodicity but also a finer algebraic relation between d(n+p12) and d(n).

The study of the Fourier coefficients of modular forms is a prominent thread of twentieth century mathematics; this gives us a first hint at a similarly deep theory, where Taylor coefficients of various modular forms expanded around complex multiplication points have p-adic properties analogous to the Fourier coefficients. As a first step, it would be nice to see the Taylor coefficients of other fundamental modular forms studied explicitly, such as the elliptic j-invariant or Dedekind η-function.

Section snippets

Vanishing of u(n)

We begin with the definition of the u(n) and v(n) coefficients, in terms of hypergeometric functions and a recurrence. We require the Gauss hypergeometric seriesF12(a,bc;z).:=n=0(a)n(b)nn!(c)nzn, with (a)n:=i=0n1(a+i) denoting a Pochhammer symbol.

Definition 3

[7] We define our coefficients in terms of the following generating functions:n=0u(n)(2n+1)!tn:=F12(34,3432;4t)F12(14,1412;4t), andn=0v(n)2n(2n)!tn:=F12(14,1412;4t). Equivalently, we set u(0)=v(0)=1 and calculate them using the recurrencesu(n)=

Reduction of s(n,k)

Now that we have good control over the behavior of the u and v coefficients, we can reduce the s(n,k) coefficients. Recall the definitions(n,k)=(2n)!(2k)![z2n](j=0u(j)(2j+1)!z2j+1)2k, where [z2n] denotes the coefficient of z2n. We can then do the obvious thing and expand the product, collecting coefficients of each unique multinomial u(j1)c1u(j2)c2. This was done by Scherer, but before presenting his result we need to introduce some notation.

Given an integer n, a partition of n is a tuple λ=(

Final steps

We now take a detour through the theory of the symmetric group, which forms the last link in our proof. Given the symmetric group on n letters Sn and a fixed prime p, let XnkSn denote the elements formed of k p-cycles and npk one-cycles. Then|Xnk|=n!k!(npk)!pk andXn:=k=0npXnk consists of all the elements in Sn of order p. We then appeal to a classical theorem of Frobenius [5].

Theorem 12

Let G be a finite group with m dividing |G|. Then m divides the number of solutions in G to xm=1.

Applying this to

Extensions

After extensive numerical investigation, it appears that we should be able to lift our result to arbitrary powers of a prime. Additionally, there appears to be a similar result to our main theorem, with period p14 instead.

Conjecture 18

We formulate the following conjectures:

  • (1)

    Let p be any prime. Then for any positive k, there exists an nk such that for all n>nk, we have u(n)v(n)0(modpk).

  • (2)

    Consider a prime p1(mod4). Then there exists a constant Cp such thatd(n+p14)Cpd(n)(modp). Some first examples are d(n+3

Declaration of Competing Interest

This is a single author paper, with all relevant work completed by Tanay Wakhare.

Acknowledgments

Many thanks to Larry Washington, Christophe Vignat, Lin Jiu, and Karl Dilcher, for chatting over coffee, emailing me back at 2 am, and contributing endless blackboard space as we discussed this. I'd also like to thank the anonymous reviewer, whose comments were very insightful and helpful. The bulk of this work was completed during an idyllic summer in Halifax, and I'd like to again thank Karl Dilcher for the invitation. I'd also like to thank Chris, Brodie, and Pascal for easing the creative

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