General SectionRomik's conjecture for the Jacobi theta function
Introduction
Dan Romik published a 2019 paper [7] considering the Taylor expansion of the classical Jacobi theta function , defined as It satisfies an important modular transformation given by It is natural to consider the Taylor expansion of around , the fixed point of this transformation. In fact, the natural function to study is The reason for considering such a rescaled function is that the Möbius transformation conformally maps the right half-plane to the unit disc and sends the inversion map to the reflection , so that the modular transformation satisfied by is then equivalent to the statement , and Taylor expanding around is equivalent to Taylor expanding the simpler around .
Although Romik found a recurrence for the Taylor coefficients (Definition 4), there does not seem to exist a closed form expression for these coefficients. They depend on a sequence , which in turn depends on a sequence , 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 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, exhibits the following behavior. For any prime and , for . For any prime or , is periodic.
We prove a stronger result in the case . Theorem 2 Consider . Then for , we have
This encodes more arithmetic information, and on iterating this times we see that is periodic with period , which is not necessarily minimal. The proof structure is tripartite:
- (1)
show that for in Lemma 6, a result conjectured in Scherer's paper [8];
- (2)
use this to simplify the expression for in Lemma 10;
- (3)
use the expression for to show the desired periodicity of in Theorem 16.
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
We begin with the definition of the and coefficients, in terms of hypergeometric functions and a recurrence. We require the Gauss hypergeometric series with denoting a Pochhammer symbol. Definition 3 [7] We define our coefficients in terms of the following generating functions: and Equivalently, we set and calculate them using the recurrences
Reduction of
Now that we have good control over the behavior of the u and v coefficients, we can reduce the coefficients. Recall the definition where denotes the coefficient of . We can then do the obvious thing and expand the product, collecting coefficients of each unique multinomial . 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
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 and a fixed prime p, let denote the elements formed of k p-cycles and one-cycles. Then and consists of all the elements in of order p. We then appeal to a classical theorem of Frobenius [5].
Theorem 12 Let G be a finite group with m dividing . Then m divides the number of solutions in G 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 instead. Conjecture 18 We formulate the following conjectures: Let p be any prime. Then for any positive k, there exists an such that for all , we have . Consider a prime . Then there exists a constant such that Some first examples are
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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