Andrew Haddock, Paul Jenkins, Mathematics
Poincaré series are a certain type of modular form. Modular forms are complex-valued functions that satisfy certain symmetry properties. There are many different types of modular forms, and one way to classify modular forms is by their level, such as 1, 2, 3, etc. They are of much interest as a research subject because they are connected in surprising ways to many different fields in number theory—e.g., elliptic curves, quadratic forms, and partition functions, to name just a few. As we gain more insight into the various properties of modular forms, we gain more insight into how these various branches of number theory are connected.
Recently, there has been a growing interest in finding where the zeros of different classes of modular forms lie. This began with a result by Rankin and Swinnerton-Dyer that classical Eisenstein series have all of their zeros on a bottom arc of the fundamental domain (the region of the complex plane that constitutes one period of the function) for level 11.
Rankin and Swinnerton-Dyer’s results about Eisenstein series inspired similar questions about other modular forms in levels 1, 2, 3, and 4, as well as for Poincaré series in level 12, 3, 4, 5. In each of these cases, the particular classes of modular forms were shown to have most, if not all, of their zeros on the lower boundary arc of the fundamental domain. These results are significant because in general, the zeros of a given modular form tend to be scattered throughout the fundamental domain.
Poincaré series can be understood as a generalization of the Eisenstein series studied by Rankin and Swinnerton-Dyer, and they appear to be intimately related to the question of why all these different classes of modular forms have their zeros on a lower boundary arc in the complex plane5. By discovering if Poincaré series of level 2 have a similar property to those of level 1, we will better understand which types of modular forms have their zeros on a particular arc and why.
We used computations in Sage, a programming language with extended functionality specifically for modular forms, in order to explore the nature of Poincaré series in level 2. Modular forms in general can be represented in a structured way that is well understood. By doing computations on certain representative modular forms, we can use the overall structure to make wider conclusions.
We found that many aspects of Rankin’s argument in level 1 directly translate to results in level 2. Simply by constructing the correct analogs of Rankin’s reasoning, we were able to replicate his first theorem in level 2. This was, namely, that provided that under certain quantifiable properties, we can calculate a lower bound for the number of zeros of a level 2 Poincaré series that must lie on the arc in question. With an extra condition, one subclass of Poincaré series of level 2 has all its zeros on the designated arc.
The difficult portion of this project is not in proving these theorems, but in trying to identify examples of Poincaré series in level 2 that satisfy the conditions we constructed for the theorems. We hypothesize that there may not be any Poincaré series that completely satisfy all of these conditions, rendering the new theorems impotent. Instead, there are likely large classes of Poincaré series that satisfy these conditions only partially. As is often the case in higher levels, we expect that we will only be able to check for zeros on a portion of the arc associated with level 2, not the whole arc as in level 1. The process of seeking specific examples and modifying the proven theorem appropriately is an ongoing portion of this project.
While the project is not fully completed, the work done on this research project has explored other ways of proving that the zeros of a given modular function lie on a particular line or arc. Much of this method was laid out by Rankin, but the process of generalizing his results to higher levels gives us a greater understanding of the relationship between Poincaré series and other types of modular forms.
1 F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein series, Bull. London Math. Soc. 2 (1970), 169–170.
2 W. Duke and P. Jenkins, On the zeros and coefficients of certain weakly holomorphic modular forms, Pure Appl. Math. Q. B (2008), no. 4, 1327–1340.
3 S. Garthwaite and P. Jenkins, Zeros of weakly holomorphic modular forms of levels 2 and 3, Mathematical Research Letters 20, no. 4 (2013), 657-674, arXiv:1205.7050v2 [math.NT].
4 A. Haddock and P. Jenkins, Zeros of weakly holomorphic modular forms of level 4, International Journal of Number Theory 10, no. 2 (2014), 455-470, doi: 10.1142/S1793042113501030, arXiv:1305.3896v1 [math.NT].
5 R. A. Rankin, On the zeros of certain Poincaré series, Compositio Mathematica 46 (1982), 255–272.