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Abstract We prove an asymptotic formula for the second moment of central values of DirichletL-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulodinside the full group of characters moduloq. Suppose that$$\nu _p(d) \geq \nu _p(q)/2$$for all primespdividingq. In this range, we obtain an asymptotic formula with a power-saving error term; curiously, there is a secondary main term of rough size$$q^{1/2}$$here which is not predicted by the integral moments conjecture of Conrey, Farmer, Keating, Rubinstein, and Snaith. The lower-order main term does not appear in the second moment of the Riemann zeta function, so this feature is not anticipated from the analogous archimedean moment problem. We also obtain an asymptotic result for smallerd, with$$\nu _p(q)/3 \leq \nu _p(d) \leq \nu _p(q)/2$$, with a power-saving error term fordlarger than$$q^{2/5}$$. In this more difficult range, the secondary main term somewhat changes its form and may have size roughlyd, which is only slightly smaller than the diagonal main term. 

We develop representation theoretic techniques to construct three dimensional non-semisimple topological quantum field theories which model homologically truncated topological B-twists of abelian Gaiotto--Witten theory with linear matter. Our constructions are based on relative modular structures on the category of weight modules over an unrolled quantization of a Lie superalgebra. The Lie superalgebra, originally defined by Gaiotto and Witten, is associated to a complex symplectic representation of a metric abelian Lie algebra. The physical theories we model admit alternative realizations as Chern--Simons-Rozansky--Witten theories and supergroup Chern--Simons theories and include as particular examples global forms of gl(1,1)-Chern--Simons theory and toral Chern--Simons theory. Fundamental to our approach is the systematic incorporation of non-genuine line operators which source flat connections for the topological flavour symmetry of the theory. 

We present FARR (Finite-difference time-domain ARRay), an open source, high-performance, finite-difference time-domain (FDTD) code. FARR is specifically designed for modeling radio wave propagation in collisional, magnetized plasmas like those found in the Earth’s ionosphere. The FDTD method directly solves Maxwell’s equations and captures all features of electromagnetic propagation, including the effects of polarization and finite-bandwidth wave packets. By solving for all vector field quantities, the code can work in regimes where geometric optics is not applicable. FARR is able to model the complex interaction of electromagnetic waves with multi-scale ionospheric irregularities, capturing the effects of scintillation caused by both refractive and diffractive processes. In this paper, we provide a thorough description of the design and features of FARR. We also highlight specific use cases for future work, including coupling to external models for ionospheric densities, quantifying HF/VHF scintillation, and simulating radar backscatter. The code is validated by comparing the simulated wave amplitudes in a slowly changing, magnetized plasma to the predicted amplitudes using the WKB approximation. This test shows good agreement between FARR and the cold plasma dispersion relations for O, X, R, and L modes, while also highlighting key differences from working in the time-domain. Finally, we conclude by comparing the propagation path of an HF pulse reflecting from the bottomside ionosphere. This path compares well to ray tracing simulations, and demonstrates the code’s ability to address realistic ionospheric propagation problems. 

We introduce an unrolled quantization U_q^E(gl(1,1)) of the complex Lie superalgebra gl(1,1) and use its categories of weight modules to construct and study new three dimensional non-semisimple topological quantum field theories. These theories are defined on categories of cobordisms which are decorated by ribbon graphs and cohomology classes and take values in categories of graded super vector spaces. Computations in these theories are enabled by a detailed study of the representation theory of U_q^E(gl(1,1)). We argue that by restricting to subcategories of integral weight modules we obtain topological quantum field theories which are mathematical models of Chern--Simons theories with gauge supergroups psl(1,1,) and U(1,1) coupled to background flat \mathbb{C}^{\times}-connections, as studied in the physics literature by Rozansky--Saleur and Mikhaylov. In particular, we match Verlinde formulae and mapping class group actions on state spaces of non-generic tori with results in the physics literature. We also obtain explicit descriptions of state spaces of generic surfaces, including their graded dimensions, which go beyond results in the physics literature. 

ABSTRACT Avian irruptions are facultative, often periodic, migrations of thousands of birds outside of their resident range. Irruptive movements produce regional anomalies of abundance that oscillate over time, forming ecological dipoles (geographically disjunct regions of low and high abundance) at continental scales. Potential drivers of irruptions include climate and food variability, but these relationships are rarely tested over broad geographic scales. We used community science data on winter bird abundance (1989–2021) to identify spatiotemporal patterns of irruption for nine boreal birds across the United States and Canada and compared them to time series of winter climate and annual tree seed production. We hypothesized that, during irruption, bird abundance would decrease in regions experiencing colder winter climates (climate variability hypothesis) or low seed production resulting from the boom‐and‐bust of widespread mast‐seeding patterns (resource variability hypothesis). Across all species, we detected latitudinal or longitudinal irruption modes, or both, demonstrating north–south and east–west migration dynamics across the northern United States and southern Canada. Seven of nine species displayed associations consistent with the climate variability hypothesis and six with the resource variability hypothesis. While irruption dynamics are likely entrained by multiple environmental drivers, future climate change could alter the spatial and temporal characteristics of avian irruption. 

We construct a two dimensional unoriented open/closed topological field theory from a finite graded group $$\pi:\Gh \twoheadrightarrow \{1,-1\}$$, a $$\pi$$-twisted $$2$$-cocycle $$\hat{\theta}$$ on $$B \hat{G}$$ and a character $$\lambda: \hat{G} \rightarrow U(1)$$. The underlying oriented theory is a twisted Dijkgraaf--Witten theory. The construction is based on a detailed study of the $$(\hat{G}, \hat{\theta},\lambda)$$-twisted Real representation theory of $$\textnormal{ker} \pi$$. In particular, twisted Real representations are boundary conditions of the unoriented theory and the generalized Frobenius--Schur element is its crosscap state. 

Abstract LetEbe an elliptic curve over$${{\mathbb {Q}}}$$ $Q$. We conjecture asymptotic estimates for the number of vanishings of$$L(E,1,\chi )$$ $L(E,1,\chi )$as$$\chi $$ $\chi$varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis onE. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order$$\ell $$ $\ell$characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the sub-family of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order$$\ell $$ $\ell$twisted elliptic curveL-functions, with$$\ell $$ $\ell$even and composite, is a mixed family with both unitary and orthogonal aspects.