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This paper is a continuation of the author's companion work \cite{InoueQuasi} on Haar-type measures for topological quasigroups, where the quasigroup setting was analyzed in connection with Kunen's theorem. We extend that framework to locally compact topological loops and study Haar-type (quasi-invariant) Radon measures together with modular cocycles describing the distortion of such measures under translations. Unlike the classical group case, the composition of translations in a loop is affected by the failure of associativity, which produces an additional correction term governed by an associativity deviation map. We derive the resulting cocycle relation and show that loop identities, in particular Moufang- and Kunen-type identities, impose structural restrictions on the modular data. In the associative limit, the cocycle reduces to the classical modular function of a locally compact group.

We study geometric modular flows in two-dimensional conformal field theories. We explore which states exhibit a geometric modular flow with respect to a causally complete subregion and, conversely, how to construct a state from a given geometric modular flow. Given suitable boundary conditions, we find that generic geometric modular flows in the Rindler wedge are conformally equivalent. Based on this insight, we show how conformal unitaries can be used to explicitly construct a state for each flow. We analyze these states, deriving general formulas for the energy density and entanglement entropy. We also consider geometric flows beyond the Rindler wedge setting, and in higher dimensions.

We present a practical, unconditional algorithm for determining the $S$-integral points on any elliptic moduli problem $\mathcal{Y}/\mathbb{Z}[1/S]$ -- that is, on any geometrically connected curve carrying a non-isotrivial elliptic fibration $\mathcal{E} \to \mathcal{Y}$. The associated map $Φ_M\colon \mathcal{Y} \to \mathcal{M}_{1,1}$ (the modular period map) plays the role ordinarily filled by a $p$-adic period map in Chabauty-type methods. Our Modular Chabauty method studies the image and fibres of $Φ_M$, and proceeds in two steps: an Effective Shafarevich step, in which we combine the modularity theorem with Cremona's enumeration of elliptic curves by conductor and list all rational elliptic curves with good reduction outside $S$; and a Fibre Computation step, in which we compute the $S$-integral points in the corresponding fibre of $Φ_M$. A Python/Sage implementation computes $\mathcal{Y}(\mathbb{Z}[1/S])$ for $\mathcal{Y}=\mathbb{P}^1\setminus\{0,1,\infty\}$ and for every modular curve $Y_1(N)$ with $4\le N\le 10$ or $N=12$, for all sets $S$ with $\prod_{p\in S} p^{2}\le 5\cdot 10^{5}$, within $3.5$ seconds on a standard computer.

In this paper, we attempt to explore the landscape of two-dimensional conformal field theories (2d CFTs) by efficiently searching for numerical solutions to the modular bootstrap equation using machine-learning-style optimization. The torus partition function of a 2d CFT is fixed by the spectrum of its primary operators and its chiral algebra, which we take to be the Virasoro algebra with $c>1$. We translate the requirement that this partition function is modular invariant into a loss function, which we then minimize to identify possible primary spectra. Our approach involves two technical innovations that facilitate finding reliable candidate CFTs. The first is a strategy to estimate the uncertainty associated with truncating the spectrum to the lowest dimension operators. The second is the use of a new singular-value-based optimizer (Sven) that is more effective than gradient descent at navigating the hierarchical structure of the loss landscape. We numerically construct candidate truncated CFT partition functions with central charges between 1 and $\frac{8}{7}$, a range devoid of known examples, and argue that these candidates likely come from a continuous space of modular bo

Kaneko's formula expresses the Fourier coefficients of the elliptic modular $j$-function as finite sums of singular moduli. First published as a short article in 1996, it was presented as a consequence of Zagier's work inspired by Borcherds products. Since then, the formula has developed into a broader framework that links the Fourier coefficients of modular forms to the special values of modular functions, extending in various directions. This article surveys these subsequent developments.

We study the structure of the vector space of Drinfeld quasi-modular forms for congruence subgroups. We provide representations as polynomials in the false Eisenstein series with coefficients in the space of Drinfeld modular forms (the $E$-expansion), and, whenever possible, as sums of hyperderivatives of Drinfeld modular forms. \\ Moreover, we introduce and study the double-slash operator, and use it to provide a well-posed definition for Hecke operators on Drinfeld quasi-modular forms. We characterize eigenforms and, for the special case of Hecke congruence subgroups $Γ_0(\mathfrak n)$, we give explicit formulas for the Hecke action on $E$-expansions.

Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well as natural integral structures in these realisations. We then use the method of Taylor and Wiles to verify the $λ$-part of the Tamagawa number conjecture of Bloch and Kato for $L(A,0)$ and $L(A,1)$. Here $λ$ is any prime of $K$ not dividing $Nk!$, and so that the mod $λ$ representation associated to $f$ is absolutely irreducible when restricted to the Galois group over $\mathbb{Q}(\sqrt{(-1)^{(\ell-1)/2}\ell})$ where $λ\mid \ell$. The method also establishes modularity of all lifts of the mod $λ$ representation which are crystalline of Hodge-Tate type $(0,k-1)$.

Cyber threat attribution can play an important role in increasing resilience against digital threats. Recent research focuses on automating the threat attribution process and on integrating it with other efforts, such as threat hunting. To support increasing automation of the cyber threat attribution process, this paper proposes a modular architecture as an alternative to current monolithic automated approaches. The modular architecture can utilize opinion pools to combine the output of concrete attributors. The proposed solution increases the tractability of the threat attribution problem and offers increased usability and interpretability, as opposed to monolithic alternatives. In addition, a Pairing Aggregator is proposed as an aggregation method that forms pairs of attributors based on distinct features to produce intermediary results before finally producing a single Probability Mass Function (PMF) as output. The Pairing Aggregator sequentially applies both the logarithmic opinion pool and the linear opinion pool. An experimental validation suggests that the modular approach does not result in decreased performance and can even enhance precision and recall compared to monolith

We exhibit a simple uniruledness criterion for general orthogonal modular varieties in terms of invariants of the corresponding lattice. As an application, we obtain the uniruledness of almost all Nikulin--Vinberg moduli spaces parameterizing projective K3 surfaces of Picard number at least 3 and fixed finite automorphism group.

Many complex tasks can be decomposed into simpler, independent parts. Discovering such underlying compositional structure has the potential to enable compositional generalization. Despite progress, our most powerful systems struggle to compose flexibly. It therefore seems natural to make models more modular to help capture the compositional nature of many tasks. However, it is unclear under which circumstances modular systems can discover hidden compositional structure. To shed light on this question, we study a teacher-student setting with a modular teacher where we have full control over the composition of ground truth modules. This allows us to relate the problem of compositional generalization to that of identification of the underlying modules. In particular we study modularity in hypernetworks representing a general class of multiplicative interactions. We show theoretically that identification up to linear transformation purely from demonstrations is possible without having to learn an exponential number of module combinations. We further demonstrate empirically that under the theoretically identified conditions, meta-learning from finite data can discover modular policies t

This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms, and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms), as well as the modular graph functions arising in genus one string perturbation theory. In an appendix, we use weakly holomorphic modular forms to write down modular primitives of cusp forms. Their coefficients involve the full period matrix (periods and quasi-periods) of cusp forms.

We study the modular invariance in magnetized torus models. Modular invariant flavor model is a recently proposed hypothesis for solving the flavor puzzle, where the flavor symmetry originates from modular invariance. In this framework coupling constants such as Yukawa couplings are also transformed under the flavor symmetry. We show that the low-energy effective theory of magnetized torus models is invariant under a specific subgroup of the modular group. Since Yukawa couplings as well as chiral zero modes transform under the modular group, the above modular subgroup (referred to as modular flavor symmetry) provides a new type of modular invariant flavor models with $D_4 \times \mathbb{Z}_2$, $(\mathbb{Z}_4 \times \mathbb{Z}_2) \rtimes \mathbb{Z}_2$, and $(\mathbb{Z}_8 \times \mathbb{Z}_2) \rtimes \mathbb{Z}_2$. We also find that conventional discrete flavor symmetries which arise in magnetized torus model are non-commutative with the modular flavor symmetry. Combining both two symmetries we obtain a larger flavor symmetry, where the conventional flavor symmetry is a normal subgroup of the whole group.

Creating neural text encoders for written Swiss German is challenging due to a dearth of training data combined with dialectal variation. In this paper, we build on several existing multilingual encoders and adapt them to Swiss German using continued pre-training. Evaluation on three diverse downstream tasks shows that simply adding a Swiss German adapter to a modular encoder achieves 97.5% of fully monolithic adaptation performance. We further find that for the task of retrieving Swiss German sentences given Standard German queries, adapting a character-level model is more effective than the other adaptation strategies. We release our code and the models trained for our experiments at https://github.com/ZurichNLP/swiss-german-text-encoders

Circuit algebras are a symmetric analogue of Jones's planar algebras introduced to study finite-type invariants of virtual knotted objects. Circuit algebra structures appear, in different forms, across mathematics. This paper provides a dictionary for translating between their diverse incarnations and describing their wider context. A formal definition of a broad class of circuit algebras is established and three equivalent descriptions of circuit algebras are provided: in terms of operads of wiring diagrams, modular operads and categories of Brauer diagrams. As an application, circuit algebra characterisations of algebras over the orthogonal and symplectic groups are given.

Circuit algebras are a symmetric version of Jones's planar algebras. They originated in quantum topology as a framework for encoding virtual crossings. This paper extends existing results for modular operads to construct a graphical calculus and monad for general circuit algebras and prove an abstract nerve theorem. The proof relies on a subtle interplay between distributive laws and abstract nerve theory, and provides extra insights into the underlying structures. Oriented circuit algebras are equivalent to wheeled props and specialisations of the results to wheeled props follow as straightforward corollaries.

In this work, we study the local zeta functions of Calabi-Yau fourfolds. This is done by developing arithmetic deformation techniques to compute the factor of the zeta function that is attributed to the horizontal four-form cohomology. This, in turn, is sensitive to the complex structure of the fourfold. Focusing mainly on examples of fourfolds with a single complex structure parameter, it is demonstrated that the proposed arithmetic techniques are both applicable and consistent. We present a Calabi-Yau fourfold for which a factor of the horizontal four-form cohomology further splits into two pieces of Hodge type $(4,0)+(2,2)+(0,4)$ and $(3,1)+(1,3)$. The latter factor corresponds to a weight-3 modular form, which allows expressing the value of the periods in terms of critical values of the L-function of this modular form, in accordance with Deligne's conjecture. The arithmetic considerations are related to M-theory Calabi-Yau fourfold compactifications with background four-form fluxes. We classify such background fluxes according to their Hodge type. For those fluxes associated to modular forms, we express their couplings in the low-energy effective action in terms of L-function v

Modular inflation is the restriction to two fields of automorphic inflation, a general group based framework for multifield scalar field theories with curved target spaces, which can be parametrized by the comoving curvature perturbation ${\cal R}$ and the isocurvature perturbation tensor $S^{IJ}$. This paper describes the dynamics and observables of these perturbations and considers in some detail the special case of modular inflation as an extensive class of two-field inflation theories with a conformally flat target space. It is shown that the nonmodular nature of derivatives of modular forms leads to CMB observables in modular invariant inflation theories that are in general constructed from almost holomorphic modular forms. The phenomenology of the model of $j$-inflation is compared to the recent observational constraints from the Planck satellite and the BICEP2/Keck Array data.

Superconformal indices of four-dimensional $\mathcal{N}=1$ gauge theories factorize into holomorphic blocks. We interpret this as a modular property resulting from the combined action of an $SL(3,\mathbb{Z})$ and $SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2$ transformation. The former corresponds to a gluing transformation and the latter to an overall large diffeomorphism, both associated with a Heegaard splitting of the underlying geometry. The extension to more general transformations leads us to argue that a given index can be factorized in terms of a family of holomorphic blocks parametrized by modular (congruence sub)groups. We find precise agreement between this proposal and new modular properties of the elliptic $Γ$ function. This leads to our conjecture for the ``modular factorization'' of superconformal lens indices of general $\mathcal{N}=1$ gauge theories. We provide evidence for the conjecture in the context of the free chiral multiplet and SQED, and sketch the extension of our arguments to more general gauge theories. Based on this result, we systematically prove that a normalized part of superconformal lens indices defines a non-trivial first cohomology class associated with

The range of robot activities is expanding from industries with fixed environments to diverse and changing environments, such as nursing care support and daily life support. In particular, autonomous construction of robots that are personalized for each user and task is required. Therefore, we develop an actuator module that can be reconfigured to various link configurations, can carry heavy objects using a locking mechanism, and can be easily operated by human teaching using a releasing mechanism. Given multiple target coordinates, a modular robot configuration that satisfies these coordinates and minimizes the required torque is automatically generated by Tree-structured Parzen Estimator (TPE), a type of black-box optimization. Based on the obtained results, we show that the robot can be reconfigured to perform various functions such as moving monitors and lights, serving food, and so on.