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Ramanujan's celebrated partition congruences modulo $\ell\in \{5, 7, 11\}$ assert that $$ p(\ell n+\delta_{\ell})\equiv 0\pmod{\ell}, $$ where $0<\delta_{\ell}<\ell$ satisfies $24\delta_{\ell}\equiv 1\pmod{\ell}.$ By proving Subbarao's Conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes $\ell \geq 5,$ we give a new proof of the conclusion that there are infinitely many $m$ for which $p(\ell m+\delta_{\ell})$ is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such $m$ satisfies $m<(\ell^2-1)/24,$ representing a significant improvement to the previous bound., Comment: Corrects minor typos found by the referees

Differences in staining and imaging procedures can cause significant color variations in histopathology images, leading to poor generalization when deploying deep-learning models trained from a different data source. Various color augmentation methods have been proposed to generate synthetic images during training to make models more robust, eliminating the need for stain normalization during test time. Many color augmentation methods leverage domain labels to generate synthetic images. This approach causes three significant challenges to scaling such a model. Firstly, incorporating data from a new domain into deep-learning models trained on existing domain labels is not straightforward. Secondly, dependency on domain labels prevents the use of pathology images without domain labels to improve model performance. Finally, implementation of these methods becomes complicated when multiple domain labels (e.g., patient identification, medical center, etc) are associated with a single image. We introduce ContriMix, a novel domain label free stain color augmentation method based on DRIT++, a style-transfer method. Contrimix leverages sample stain color variation within a training minibatch and random mixing to extract content and attribute information from pathology images. This information can be used by a trained ContriMix model to create synthetic images to improve the performance of existing classifiers. ContriMix outperforms competing methods on the Camelyon17-WILDS dataset. Its performance is consistent across different slides in the test set while being robust to the color variation from rare substances in pathology images. We make our code and trained ContriMix models available for research use. The code for ContriMix can be found at https://gitlab.com/huutan86/contrimix

Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of $t$-hooks in the partitions of $n$. We prove that the limiting distribution is normal with mean $\mu_t(n)\sim \frac{\sqrt{6n}}{\pi}-\frac{t}{2}$ and variance $\sigma_t^2(n)\sim \frac{(\pi^2-6)\sqrt{6n}}{2\pi^3}.$ Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed $t\geq 4$ in partitions of $n$ converge to a shifted Gamma distribution with parameter $k=(t-1)/2$ and scale $\theta=\sqrt{2/(t-1)}.$, Comment: Minor edits and a new first paragraph on previous related work (added at the suggestion of a referee)

Hausel and Rodriguez-Villegas recently observed that work of G\"ottsche, combined with a classical result of Erd\H{o}s and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes $(\mathbb{C}^2)^{[n]}$ on $n$ points, as $n\rightarrow +\infty,$ is a \textit{Gumbel distribution}. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes $((\mathbb{C}^2)^{[n]})^{T_{\alpha,\beta}}$ that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erd\H{o}s and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer $A\geq 2.$ Furthermore, if $p_k(A;n)$ denotes the number of partitions of $n$ with exactly $k$ parts that are multiples of $A$, then we obtain the asymptotic $$ p_k(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^k}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}}, $$ a result which is of independent interest., Comment: Minor revision addressing referee comments

The classical $\mathrm{AGM}$ produces wonderful interdependent infinite sequences of arithmetic and geometric means with common limit. For finite fields $\mathbb{F}_q,$ with $q\equiv 3\pmod 4,$ we introduce a finite field analogue $\mathrm{AGM}_{\mathbb{F}_q}$ that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a $\mathit{jellyfish~swarm},$ as the 3D renderings of the connected components resemble $\mathit{jellyfish}$ (i.e. tentacles connected to a bell head). These swarms turn out to be more than the stuff of child's play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of $\mathbb{F}_q$-points, which can be used to prove that each swarm has at least $(1/2-\varepsilon)\sqrt{q}$ jellyfish. Additionally, this interpretation gives a description of the $\mathit{class~numbers}$ of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish., Comment: Corrects minor typos

We explicitly construct the Dirichlet series $$L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},$$ where $P_{\mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form with Tamagawa product $m.$ Although there are no $E/\mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{\mathrm{Tam}}(1)=0.5053\dots.$ As a corollary, we find that $L_{\mathrm{Tam}}(-1)=1.8193\dots$ is the average Tamagawa product for elliptic curves over $\mathbb{Q}.$ We give an application of these results to canonical and Weil heights., Comment: This version corrects a typographical error in a Remark on page 3

A modular grid is a pair of sequences $(f_m)_m$ and $(g_n)_n$ of weakly holomorphic modular forms such that for almost all $m$ and $n$, the coefficient of $q^n$ in $f_m$ is the negative of the coefficient of $q^m$ in $g_n$. Zagier proved this coefficient duality in weights $1/2$ and $3/2$ in the Kohnen plus space, and such grids have appeared for Poincar\'{e} series, for modular forms of integral weight, and in many other situations. We give a general proof of coefficient duality for canonical row-reduced bases of spaces of weakly holomorphic modular forms of integral or half-integral weight for every group $\Gamma \subseteq {\text{SL}}_2(\mathbb{R})$ commensurable with ${\text{SL}}_2(\mathbb{Z})$. We construct bivariate generate functions that encode these modular forms, and study linear operations on the resulting modular grids., Comment: Second revision

In this note we obtain effective lower bounds for the canonical heights of non-torsion points on $E(\mathbb{Q})$ by making use of suitable elliptic curve ideal class pairings $$\Psi_{E,-D}: E(\mathbb{Q})\times E_{-D}(\mathbb{Q})\mapsto \mathrm{CL}(-D).$$ In terms of the class number $H(-D)$ and $T_E(-D)$, a logarithmic function in $D$, we prove $$ \widehat{h}(P)> \frac{|E_{\mathrm{tor}}(\mathbb{Q})|^2}{\left( H(-D)+ |E_{\mathrm{tor}}(\mathbb{Q})|\right)^2}\cdot T_E(-D). $$, Comment: Comments. 1) This paper was published as Proceedings of the American Mathematical Society 149 (2021), 5093-5100. 2) After the paper was published, we discovered a typographical error in the example after Theorem 1.1. The number 0.043 should be 0.035