RESEARCH

My research focuses on the theory of modular and automorphic forms and their arithmetic applications. I have primarily worked on connections between modular forms and partition theory, investigating congruences and combinatorial structures, as well as applications to Galois representations in understanding the arithmetic of these objects. Building on this foundation, I am interested in expanding my research to explore the rich connections to L-functions, elliptic curves, harmonic Maass forms and mock modular forms, and emerging applications in mathematical physics.

Papers and Preprints:

1) Boylan M., Swati, Congruence properties modulo prime powers for a class of partition functions arXiv:2401.03663  (Submitted)

In this paper, we study specialized partition functions defined by certain generating functions. We show that these generating functions, when restricted to specific arithmetic progressions, lies in a Hecke-invariant subspace comprising of forms arising as multiples of a fixed eta-quotient times a holomorphic modular form, modulo prime powers l^j, j >=1 and l >=5 prime. Further, we use the Hecke-invariance of these subspaces to prove infinite family of Ramanujan-type congruences for these partition functions, modulo prime powers.

2) Boylan M., Swati, Indices of nilpotency in certain spaces of modular forms arxiv:2410.24182. (Submitted)

In this paper, we study the index of nilpotency relative to certain Hecke operators in spaces of modular forms with integer weight and level 1 with integer coefficients modulo smaller primes p. In these settings, we prove upper bounds on certain indices of nilpotency.  As an application of our bounds, we prove infinite families of congruences for p^t -core partition functions modulo p and a positive integer t, and we prove an infinite family of congruences modulo 3 for the r-th power partition function, r = 12k with k relatively prime to 6. We also include conjectures on a function which quantifies degree lowering on powers of the Delta function by the relevant Hecke operators in these settings, and on the index of nilpotency relative to a modification of this degree-lowering function.

3) Boylan M., Swati, Explicit images for the Shimura Correspondence arxiv:2505.01018 

In 2014, Y. Yang identified, for odd 1 <= r <= 23, the r-th Shimura image associated to the theta multiplier on multiples of powers of Dedekind eta-function by integral weight modular forms. Yang's isomorphism reveals that this image lies in a much smaller space than Shimura's theory predicts. His proof relies on technical trace formula computations for the Hecke operators in integral and half-integral weights.

In this paper, we present a constructive proof of Yang’s result by deriving explicit formulas for the Shimura lifts associated with the eta-multiplier. We also obtain formulas for lifts of Hecke eigenforms multiplied by theta-function eta-quotients and lifts of Rankin-Cohen brackets of Hecke eigenforms with theta-function eta-quotients.

Ongoing Projects:

• Number Theory REU, Exceptional Congruences for eta-quotient newforms (Joint with Matthew Boylan, Henry Stone, Eddie O' Sullivan, and Jin Xiaolan) 

• Rethinking Number Theory (RNT 6), Partition Numbers and Perfect Powers  (Joint with Summer Haag, Praneel Samanta, Holly Swisher, Stephanie Treener, and Robin Visser)

• Traces of Singular Moduli

Research Appointments:

• (10 Nov 2018 - 31 Mar 2019) Visiting Researcher, Department of Mathematics, The Institute of Mathematical Sciences (IMSc), Chennai, India.

 Supervisor: Prof. Srinivas Kotyada

        The project aimed at studying the growth rate of the Riemann zeta function and the gaps between the successive zeros on the critical line. Subsequently, we studied the growth of classical zeta functions, namely the Epstein zeta function and the Selberg class of L-functions, in the critical strip.

•  (20 Feb 2018 - 31 July 2018) Research Assistant- I, Department of Mathematics, The University of Hong Kong (HKU), Pokfulam, Hong Kong.

 Supervisor: Prof. Ben Kane

         My work primarily centered on a family of L-functions, which generalize the Riemann zeta function, and their connections with regularized Mellin transforms. 

• (25 Feb 2017 - 25 Jan 2018) Project Assistant, Department of Mathematics, Harish-Chandra Research Institute (HRI), Prayagraj (Allahabad), India.

Title: Development of Modules and Tools for Integer Factorization using Number Field Sieve (NFS)

Funding Agency: Defense Research and Development Organisation (DRDO)

Supervisors: Prof. Kalyan Chakraborty, Prof. R. Thangadurai

        The problem involved the factorization of a 596-bit RSA modulus using NFS. To reduce the time complexity involved in the factorization, I developed an algorithm to increase the efficiency of the sieving module of NFS.