In this paper, we define and study a ring associated to a graph that we call the cographic toric face ring or simply the cographic ring. The cographic ring is the toric face ring defined by the following equivalent combinatorial structures of a graph: the cographic arrangement of hyperplanes, the Voronoi polytope, and the poset of totally cyclic orientations. We describe the properties of the cographic ring and, in particular, relate the invariants of the ring to the invariants of the corresponding graph.

Our study of the cographic ring fits into a body of work on describing rings constructed from graphs. Among the rings that can be constructed from a graph, cographic rings are particularly interesting because they appear in the study of compactified Jacobians of nodal curves.

Following up on our earlier work and the work of N. Karpenko and A. Merkurjev, we study the essential $p$-dimension of linear algebraic groups $G$ whose connected component $G^0$ is a torus.

We consider an analog, when $\mathbb{Z}$ is replaced by ${\mathbb{F}}_q\left[t\right]$, of Wilson primes, namely the primes satisfying Wilson’s congruence $\left(p-1\right)!\equiv -1$ to modulus $p^2$ rather than the usual prime modulus $p$. We fully characterize these primes by connecting these or higher power congruences to other fundamental quantities such as higher derivatives and higher difference quotients as well as higher Fermat quotients. For example, in characteristic $p>2$, we show that a prime $\wp$ of ${\mathbb{F}}_q\left[t\right]$ is a Wilson prime if and only if its second derivative with respect to $t$ is $0$ and in this case, further, that the congruence holds automatically modulo ${\wp }^{p-1}$. For $p=2$, the power $p-1$ is replaced by $4-1=3$. For every $q$, we show that there are infinitely many such primes.

We consider the (finite) $W$-algebra $W_{m|n}$ attached to the principal nilpotent orbit in the general linear Lie superalgebra ${\mathfrak{g}\mathfrak{l}}_{m|n}\left(ℂ\right)$. Our main result gives an explicit description of $W_{m|n}$ as a certain truncation of a shifted version of the Yangian $Y\left({\mathfrak{g}\mathfrak{l}}_{1|1}\right)$. We also show that $W_{m|n}$ admits a triangular decomposition and construct its irreducible representations.

The Rankin–Cohen bracket of two Eisenstein series provides a kernel yielding products of the periods of Hecke eigenforms at critical values. Extending this idea leads to a new type of Eisenstein series built with a double sum. We develop the properties of these series and their nonholomorphic analogs and show their connection to values of $L$-functions outside the critical strip.

Let $K$ be the fraction field of a two-dimensional, henselian, excellent local domain with finite residue field $k$. When the characteristic of $k$ is not $2$, we prove that every quadratic form of rank $\ge 9$ is isotropic over $K$ using methods of Parimala and Suresh, and we obtain the local-global principle for isotropy of quadratic forms of rank $5$ with respect to discrete valuations of $K$. The latter result is proved by making a careful study of ramification and cyclicity of division algebras over the field $K$, following Saltman’s methods. A key step is the proof of the following result, which answers a question of Colliot-Thélène, Ojanguren and Parimala: for a Brauer class over $K$ of prime order $q$ different from the characteristic of $k$, if it is cyclic of degree $q$ over the completed field $K_v$ for every discrete valuation $v$ of $K$, then the same holds over $K$. This local-global principle for cyclicity is also established over function fields of $p$-adic curves with the same method.

We describe the modular operad structure on the moduli spaces of pointed stable curves equipped with an admissible $G$-cover. To do this we are forced to introduce the notion of an operad colored not by a set but by the objects of a category. This construction interpolates in a sense between “framed” and “colored” versions of operads; we hope that it will be of independent interest. An algebra over the cohomology of this operad is the same thing as a $G$-equivariant CohFT, as defined by Jarvis, Kaufmann and Kimura. We prove that the (orbifold) Gromov–Witten invariants of global quotients $\left[X∕G\right]$ give examples of $G$-CohFTs.

Let $K$ be a complete discrete valuation field of mixed characteristic $\left(0,p\right)$ and $G_K$ the absolute Galois group of $K$. In this paper, we will prove the $p$-adic monodromy theorem for $p$-adic representations of $G_K$ without any assumption on the residue field of $K$, for example the finiteness of a $p$-basis of the residue field of $K$. The main point of the proof is a construction of $\left(\phi ,G_K\right)$-module ${\tilde{\mathbb{N}}}_{rig}^{\nabla +}\left(V\right)$ for a de Rham representation $V$, which is a generalization of Pierre Colmez’s ${\tilde{\mathbb{N}}}_{rig}^{+}\left(V\right)$. In particular, our proof is essentially different from Kazuma Morita’s proof in the case when the residue field admits a finite $p$-basis.

We also give a few applications of the $p$-adic monodromy theorem, which are not mentioned in the literature. First, we prove a horizontal analogue of the $p$-adic monodromy theorem. Secondly, we prove an equivalence of categories between the category of horizontal de Rham representations of $G_K$ and the category of de Rham representations of an absolute Galois group of the canonical subfield of $K$. Finally, we compute $H^1$ of some $p$-adic representations of $G_K$, which is a generalization of Osamu Hyodo’s results.

We prove that in positive characteristic, the Manin–Mumford conjecture implies the Mordell–Lang conjecture in the situation where the ambient variety is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is a finitely generated group. In particular, in the setting of the last sentence, we provide a proof of the Mordell–Lang conjecture that does not depend on tools coming from model theory.