arXiv subjects
Publications and source records attributed to Catherine Cossaboom.
In this paper, we generalize Andrews' partitions separated by parity to overpartitions in two ways. We investigate the generating functions for 16 overpartition families whose parts are separated by parity, and we prove various $q$-series identities for these functions. These identities include relations to modular forms, $q$-hypergeometric series, and mock modular forms.
We establish a hook length bias between self-conjugate partitions and partitions of distinct odd parts, demonstrating that there are more hooks of fixed length $t \geq 2$ among self-conjugate partitions of $n$ than among partitions of distinct odd parts of $n$ for sufficiently large $n$. More precisely, we derive asymptotic formulas for the total number of hooks of fixed length $t$ in both classes. This resolves a conjecture of Ballantine, Burson, Craig, Folsom, and Wen.
For $j=1,2$, let $f_j(z) = \sum_{n=1}^{\infty} a_{j}(n) e^{2πi nz}$ be a holomorphic, non-CM cuspidal newform of even weight $k_j \ge 2$ with trivial nebentypus. For each prime $p$, let $θ_{j}(p)\in[0,π]$ be the angle such that $a_j(p) = 2p^{(k-1)/2} \cos θ_{j}(p)$. The now-proven Sato--Tate conjecture states that the angles $(θ_j(p))$ equidistribute with respect to the measure $dμ_{\mathrm ST} = \frac{2}π\sin^2θ\,dθ$. We show that, if $f_1$ is not a character twist of $f_2$, then for subintervals $I_1,I_2 \subset [0,π]$, there exist infinitely many bounded gaps between the primes $p$ such that $θ_1(p) \in I_1$ and $θ_2(p) \in I_2$. We also prove a common generalization of the bounded gaps with the Green--Tao theorem.
A well-known observation of Serre and Tate is that the Hecke algebra acts locally nilpotently on modular forms mod 2 on $\mathrm{SL}_2(\mathbb{Z})$. We give an algorithm for calculating the degree of Hecke nilpotency for cusp forms, and we obtain a formula for the total number of cusp forms mod 2 of any given degree of nilpotency. Using these results, we find that the degrees of Hecke nilpotency in spaces $M_k$ have no limiting distribution as $k \rightarrow \infty$. As an application, we study the parity of the partition function using Hecke nilpotency.