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The nature of the magnetic field structure throughout the Galactic Center (GC) has long been of interest. The recent Far-InfraREd Polarimetric Large-Area CMZ Exploration (FIREPLACE) Survey reveals preliminary connections between the seemingly distinct vertical and horizontal magnetic field distributions previously observed in the GC. We use the statistical techniques of the Histogram of Relative Orientation (HRO) and the Projected Rayleigh Statistic (PRS) to assess whether the CMZ magnetic field preferentially aligns with the column density gradient derived from CMZ molecular clouds or the morphology of the non-thermal emission of the GC NTF population. We find that there is a range of magnetic field orientations throughout the population of CMZ molecular clouds, ranging from parallel to perpendicular orientation. This range of orientations contrasts with what is observed in Galactic Disk star-forming regions. We also compare the magnetic field orientation from dust polarimetry with individual prominent NTFs, finding a preferred perpendicular relative orientation. This perpendicular orientation indicates that the vertical field component found in the FIREPLACE observations is not spatially confined to the NTFs, providing evidence for a more pervasive vertical field in the GC. We estimate an upper limit on the magnetic field strength for this vertical field of 4 mG. A field close to this upper limit would indicate that the NTFs are likely not local enhancements of a weaker background field and that the locations of the NTFs depend on proximity to sites of cosmic ray production., Comment: Submitted to ApJ

Given a correspondence $V$ between a connected Shimura variety $S$, a commutative connected algebraic group $G$, and $n \in \mathbb{N}$, we prove that the $V$-images of any $n$ special points on $S$ outside a proper Zariski closed subset are algebraically independent. Our result unifies previous unlikely intersection results on multiplicative independence and linear independence. We prove multiplicative independence of differences of singular moduli, generalizing previous results by Pila-Tsimerman, and Aslanlyan-Eterovi\'c-Fowler. We also give an application to abelian varieties by proving that the special points of $S$ whose $V$-images lie in a finite-rank subgroup of $T$ are contained in a finite union of proper special subvarieties of $S$, only dependent on the rank of the subgroup. In this way, our result is a generalization of the works of Pila-Tsimerman and Buium-Poonen., Comment: 17 pages, comments welcome. We changed the notation for the commutative connected algebraic group and corrected typos

Dust is a key component of galaxies, but its properties during the earliest eras of structure formation remain elusive. Here we present a simple semi-analytic model of the dust distribution in galaxies at $z \gtrsim 5$. We calibrate the free parameters of this model to estimates of the UV attenuation (using the IRX-$\beta$ relation between infrared emission and the UV spectral slope) and to ALMA measurements of dust emission. We find that the observed dust emission requires that most of the dust expected in these galaxies is retained (assuming a similar yield to lower-redshift sources), but if the dust is spherically distributed, the modest attenuation requires that it be significantly more extended than the stars. Interestingly, the retention fraction is larger for less massive galaxies in our model. However, the required radius is a significant fraction of the host's virial radius and is larger than the estimated extent of dust emission from stacked high-$z$ galaxies. These can be reconciled if the dust is distributed anisotropically, with typical covering fractions of $\sim 0.2-0.7$ in bright galaxies and $\lesssim 0.1$ in fainter ones., Comment: 37 pages, 15 figures, Accepted for publication in JCAP

Let $B/F$ be a quaternion algebra over a totally real number field. We give an explicit formula for heights of special points on the quaternionic Shimura variety associated with $B$ in terms of Faltings heights of CM abelian varieties. Special points correspond to CM fields $E$ and partial CM-types $\phi \subset \mathrm{Hom}(E, \mathbb{C})$. We then show that our height is compatible with the canonical height of a partial CM-type defined by Pila, Shankar, and Tsimerman. This gives another proof that the height of a partial CM-type is bounded subpolynomially in terms of the discriminant of $E$., Comment: 30 pages. Comments welcome

Let $(G, X)$ be a Shimura datum, let $\Omega$ be a connected component of $X$, let $\Gamma$ be a congruence subgroup of $G(\mathbb{Q})^{+}$, and consider the quotient map $q: \Omega \to S:=\Gamma \backslash \Omega$. Consider the Harish-Chandra embedding $\Omega\subset\mathbb{C}^{N}$, where $N=\dim X$. We prove two results that give geometric conditions which if satisfied by an algebraic variety $V \subset \mathbb{C}^{N} \times S$, ensure that there is a Zariski dense subset of $V$ of points of the form $(x,q(x))$., Comment: 14 pages. Comments welcome

Franz Lemmermeyer's previous work laid the framework for a description of the arithmetic of Pell conics, which is analogous to that of elliptic curves. He describes a group law on conics and conjectures the existence of an analogous Tate--Shafarevich group with order the squared ideals of the narrow class group. In this article, we provide a cohomological definition of the Tate--Shafarevich group and show that its order is as Lemmermeyer conjectured.