Noncommutative near-group fusion categories were completely classified in the previous work of the first author by using an operator algebraic method (and hence under the assumption of unitarity), and they were shown to be group theoretical though the corresponding pointed categories were not identified. In this note we give a purely algebraic construction of the noncommutative near-group fusion categories starting from pointed categories categorically Morita equivalent to them.

We show that the shape of a complex cubic field lies on the geodesic of the modular surface defined by the field’s trace-zero form. We also prove a general such statement for all orders in étale $\mathbf{Q}$-algebras. Applying a method of Manjul Bhargava and Piper H to results of Bhargava and Ariel Shnidman, we prove that the shapes lying on a fixed geodesic become equidistributed with respect to the hyperbolic measure as the discriminant of the complex cubic field goes to infinity. We also show that the shape of a complex cubic field is a complete invariant (within the family of all cubic fields).

We completely classify all subbundles of a given vector bundle on the Fargues–Fontaine curve, where by a subbundle we mean a locally free subsheaf. Our classification is given in terms of a simple and explicit condition on Harder–Narasimhan polygons. Our proof is inspired by the proof of the main theorem in our previous work (Hong 2019), but also involves a number of nontrivial adjustments.

We investigate cohomological support varieties for finite-dimensional Lie superalgebras defined over fields of odd characteristic. Verifying a conjecture from our previous work, we show the support variety of a finite-dimensional supermodule can be realized as an explicit subset of the odd nullcone of the underlying Lie superalgebra. We also show the support variety of a finite-dimensional supermodule is zero if and only if the supermodule is of finite projective dimension. As a consequence, we obtain a positive characteristic version of a theorem of Bøgvad, showing that if a finite-dimensional Lie superalgebra over a field of odd characteristic is absolutely torsion free, then its enveloping algebra is of finite global dimension.

Let $F$ be a finite extension of ${\mathbb{Q}}_p$. Let $W\left(k\right)$ denote the Witt vectors of an algebraically closed field $k$ of characteristic $\ell$ different from $p$, and let $\mathcal{𝒵}$ be the spherical Hecke algebra for ${GL}_n\left(F\right)$over $W\left(k\right)$. Given a Hecke character $\lambda :\mathcal{𝒵}\to R$, where $R$ is an arbitrary $W\left(k\right)$-algebra, we introduce the universal unramified module ${\mathcal{ℳ}}_{\lambda ,R}$. We show ${\mathcal{ℳ}}_{\lambda ,R}$ embeds in its Whittaker space and is flat over $R$, resolving a conjecture of Lazarus. It follows that ${\mathcal{ℳ}}_{\lambda ,k}$ has the same semisimplification as any unramified principal series with Hecke character $\lambda$.

In the setting of mod-$\ell$ automorphic forms Clozel, Harris, and Taylor (2008) formulated a conjectural analogue of Ihara’s lemma. It predicts that every irreducible submodule of a certain cyclic module $V$ of mod-$\ell$ automorphic forms is generic. Our result on the Whittaker model of ${\mathcal{ℳ}}_{\lambda ,k}$ reduces the Ihara conjecture to the statement that $V$ is generic.

We consider a special theta lift $𝜃\left(f\right)$ from cuspidal Siegel modular forms $f$ on ${Sp}_4$ to “modular forms” $𝜃\left(f\right)$ on $SO\left(4,4\right)$ in the sense of our prior work (Pollack 2020a). This lift can be considered an analogue of the Saito–Kurokawa lift, where now the image of the lift is representations of $SO\left(4,4\right)$ that are quaternionic at infinity. We relate the Fourier coefficients of $𝜃\left(f\right)$ to those of $f$, and in particular prove that $𝜃\left(f\right)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $𝜃\left(f\right)$ to $G_2\subseteq SO\left(4,4\right)$, we obtain cuspidal modular forms on $G_2$ of arbitrarily large weight with all algebraic Fourier coefficients. In the case of level one, we obtain precise formulas for the Fourier coefficients of $𝜃\left(f\right)$ in terms of those of $f$. In particular, we construct nonzero cuspidal modular forms on $G_2$ of level one with all integer Fourier coefficients.

The aim of this paper is to study some modular contractions of the moduli space of stable pointed curves ${\overline{M}}_{g,n}$. These new moduli spaces, which are modular compactifications of $M_{g,n}$, are related to the minimal model program for ${\overline{M}}_{g,n}$ and have been introduced by Codogni et al. (2018). We interpret them as log canonical models of adjoint divisors and we then describe the Shokurov decomposition of a region of boundary divisors on ${\overline{M}}_{g,n}$.

We consider some additive properties of integers with restricted digit expansions. Let $b\ge 3$ be an integer and $B_b$ be the set of integers whose base $b$ expansions have only digits $\left\{0,1\right\}$. Let $a,b,c$ be three integers greater than $2$. We give some estimates on the size of $\left(B_a+B_b\right)\cap B_c$. In particular, under mild conditions, $\left(B_a+B_b\right)\cap B_c$ is a very thin set in the sense that for each , as $N\to \infty$,

$$\#\left(\left(B_a+B_b\right)\cap B_c\cap \left[1,N\right]\right)=O\left(N^{𝜖}\right).$$

Let $p$ and $q$ be multiplicatively independent natural numbers, and $K$ the field $ℂ\left(x^{1∕s}\mid s=1,2,3\dots \right)$. Let $p$ and $q$ act on $K$ as the Mahler operators $x\mapsto x^p$ and $x\mapsto x^q$. Schäfke and Singer (2019) showed that a finite-dimensional vector space over $K$, carrying commuting structures of a $p$-Mahler module and a $q$-Mahler module, is obtained via base change from a similar object over $ℂ$. As a corollary, they gave a new proof of a conjecture of Loxton and van der Poorten, which had been proved before by Adamczewski and Bell (2017). When $K=ℂ\left(x\right)$, and $p$ and $q$ are complex numbers of absolute value greater than 1, acting on $K$ via dilations $x\mapsto px$ and $x\mapsto qx$, a similar theorem has been obtained by Bézivin and Boutabaa (1992). Underlying these two examples are the algebraic groups ${\mathbb{𝔾}}_m$ and ${\mathbb{𝔾}}_a$, respectively, with $K$ the function field of their universal covering, and $p$, $q$ acting as endomorphisms.

Replacing the multiplicative or additive group by the elliptic curve $ℂ∕\mathrm{\Lambda }$, and $K$ by the maximal unramified extension of the field of $\mathrm{\Lambda }$-elliptic functions, we study similar objects, which we call elliptic $\left(p,q\right)$-difference modules. Here $p$ and $q$ act on $K$ via isogenies. When $p$ and $q$ are relatively prime, we give a structure theorem for elliptic $\left(p,q\right)$-difference modules. The proof is based on a periodicity theorem, which we prove in somewhat greater generality. A new feature of the elliptic modules is that their classification turns out to be fibered over Atiyah’s classification of vector bundles on elliptic curves (1957).

Only the modules whose associated vector bundle is trivial admit a $ℂ$-structure as in thc case of ${\mathbb{𝔾}}_m$ or ${\mathbb{𝔾}}_a$, but all of them can be described explicitly with the aid of (logarithmic derivatives of) theta functions. We conclude with a proof of an elliptic analogue of the conjecture of Loxton and van der Poorten.