arXiv subjects

Matthew Boylan

Publications and source records attributed to Matthew Boylan.

3 recordsLinked to original sources

arXiv2025

Explicit images for the Shimura Correspondence

In 2014, Yang showed that for $F \in \mathcal{A}_{r, s, 1, 1_N}$, we have $\textup{Sh}_{r}(F \mid V_{24}) = G \otimes \chi_{12}$ where $G\in S^{new}_{r+2s - 1}(\Gamma_{0}(6), - \left( \frac{8}{r} \right), - \left( \frac{12}{r} \right))$, where $\textup{Sh}_{r}$ is the $r$-th Shimura lift associated to the theta-multiplier. He proved a similar result for $(r,6) = 3$.\:His proofs rely on trace computations in integral and half-integral weights. In this paper, we provide a constructive proof of Yang's result. We obtain explicit formulas for $\mathcal{S}_{r}(F)$, the $r$-th Shimura lift associated to the eta-multiplier defined by Ahlgren, Andersen, and Dicks, when $1\leq r\leq 23$ is odd and $N = 1$. 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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arXiv2024

Indices of nilpotency in certain spaces of modular forms

We study the index of nilpotency relative to certain Hecke operators in spaces of modular forms with integer weight and level $N$ with integer coefficients modulo primes $p$ for $(p, N) \in \{(3, 1), (5, 1), (7, 1), (3, 4)\}$. 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 $p$ for $p\in \{3, 5, 7\}$ and $t\geq 1$, and we prove an infinite family of congruences modulo $3$ for the $r$th power partition function, $p_r(n)$, when $r = 12k$ with $\gcd(k,6) = 1$. 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.

math.NT↗

arXiv2024

Congruence properties modulo prime powers for a class of partition functions

Let $p$ be prime, and let $p_{[1,p]}(n)$ denote the function whose generating function is $\prod (1-q^n)^{-1}(1 - q^{pn})^{-1}$. This function and its generalizations $p_{[c^{\ell}, d^m]}(n)$ are the subject of study in several recent papers. Let $\ell\geq 5$, let $j\geq 1$, and let $p \in \{2, 3, 5\}$. In this paper, we prove that the generating function for $p_{[1, p]}(n)$ in the progression $\beta_{p, \ell, j}$ modulo $\ell^j$ with $24\beta_{p, \ell, j} \equiv p + 1 \pmod{\ell^j}$ lies in a Hecke-invariant subspace of type $\{\eta(Dz)\eta(Dpz)F(Dz) : F(z) \in M_{s}(\Gamma_0(p), \chi)\}$ for suitable $D\geq 1$, $s\geq 0$, and character~$\chi$. When $p\in \{2, 3, 5\}$, we use the Hecke-invariance of these subspaces proved in [21] to prove, for distinct primes $\ell$ and $m\geq 5$ and $j\geq 1$, congruences of the form \[ p_{[1, p]}\left(\frac{\ell^jm^k n + 1}{D}\right)\equiv 0 \pmod{\ell^j} \] for all $n\geq 1$ with $m\nmid n$, where $k$ is explicitly computable and depends on the forms in the invariant subspace. Our proofs require adapting and extending analogous level one results on $p(n)$ in [1] and [22] to level $p$.

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