Michael Griffin and Dr. Paul Jenkins, Department of Mathematics
Modular forms are constructs of complex analysis that possess many intricate connections to widely-separated branches of mathematics. In the most well-known application of these functions, Andrew Wiles established a connection between the Fourier coefficients of modular forms and elliptic curves–objects of analytic geometry–in order to prove Fermat’s Last Theorem, a classical number theory problem that had resisted proof for more than 350 years. Modular forms have also found applications ranging from cryptography to the million-dollar Birch and Swinerton-Dyer conjecture.
More than simply being important as functions, modular forms encode remarkable information within their structures. Many of the most mathematically-interesting sequences of integers, including the values of many important number-theoretic functions, appear as the Fourier coefficients of certain modular forms. By studying theses coefficients, Mathematicians gather valuable information about the behavior of integers.
A weakly holomorphic modular form (WHMF) of weight is a periodic analytic function defined on the upper half of the complex plane that is meromorphic at infinity, and satisfies the transformation
for any matrix
with integer entries and determinant . A modular function is a weight WHMF. The -function is a modular function important in the study of elliptic curves, and whose Fourier coefficients are integers that appear as the degrees of a representation of the Monster group. In fact, modular functions are exactly the set of polynomials in the j-function (including the constant functions). In 1949, Lehner demonstrated that high powers of the primes and divide many of the coefficients of the j-function (Lehner, 1949). In Fact, Lehner’s findings are specific examples of
Modular functions have Fourier expansions of the form
With this notation, the operator is defined by
Lehner’s results show that
modulo powers of small primes. Ahlgren and Ono (Ahlgren & Ono, 2005) define a congruence (mod p) for a modular function as the existence of another modular functions so that .
Ahlgren and Ono (Ahlgren & Ono, 2005), and Guerzhoy (Guerzhoy, 2007) have given a necessary and sufficient condition for the existence of congruences (mod p) for modular functions. Letbe Ramanujan’s delta function (a weight 12 modular form), and letbe the Eisenstein series of weight k. We note that given any even integer k, there are unique integers, d, e, f, so that k=12f + 4e + 6d, with e=0,1 or 2 and d=0 or 1. Given these d, e, f, let . Then if where P(x) is a polynomial with integer coefficients, then has a congruence if and only if where is a modular function with integer coefficients.
The purpose of this research project was to find similar results for congruences modulo powers of primes. I was able to conjecture the conditions that would lead to such congruences, be computationally experimenting on many different modular functions, using SAGE mathematics software. I then tackled these theoretically in order to prove (and adjust) my conjectures. I have been successful in demonstrating a sufficient condition for the existence of a congruence modulo any power of any prime. I have also accrued a great deal of evidence (both computational and theoretic) that this condition is necessary as well as sufficient. However, to date I have been unsuccessful in proving this, with the single exception of finding an alternate proof that Ahlgren, Ono, and Guerzhoys condition is necessary and sufficient.
My condition can be stated as follows:
Over the course of this project, my conception of the original problem has changed dramatically. Originally, I conceived the project as concerning the divisibility properties of modular functions, hoping to find some results similar to Lehner’s for larger primes. The result above answers this question, but also answers a slightly more interesting and more general question. As my concept of the problem evolved, so did my approaches to solving it. Although it has taken me longer to find the solution than I originally thought it would take, my final solutions is simpler than I expected, and simpler than several of the attacks I waged unsuccessfully. Through the process I have I have become aware of several related and interesting problems, which I think I may be able to solve using similar methods. I intend to publish my work in a mathematics journal, but first I would like to tackle some of these related problems, and hopefully give a more complete answer to a slightly more general question. I also hope to provide a proof that my condition given above is necessary as well as sufficient.
- Ahlgren, S., & Ono, K. (2005). Arithmetic of singular moduli and class polynomials. Compos. Math., 293-312.
- Guerzhoy, P. (2007). On Up-congruences. Proc. Amer. Math. Soc, 2743–2746.
- Lehner, J. (1949). Further congruence properties of the Fourier coefficients of the modular invariant j(t). American Journal of Mathematics, 373-386.