RESEARCH

An equation for me has no meaning unless it expresses a thought of God.

Srinivasa Ramanujan

My research interests span the theory of modular forms, partition theory, L-functions, Harmonic Maass forms, and Galois representations.

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. 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.

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