This paper concerns the Weil representation of the semidirect product of the metaplectic and Heisenberg groups. First we present a canonical construction of the metaplectic group as a central extension of the symplectic group by a subquotient of the Witt group. This leads to simple formulas for the character, for the inverse Weyl transform, and for the transfer factor appearing in J. Adams’s work on character lifting. Along the way, we give formulas for outer automorphisms of the metaplectic group induced by symplectic similitudes. The approach works uniformly for finite and local fields.

We develop a theory of Selmer groups for analytic families of Galois representations, which are only assumed “ordinary” on the level of their underlying $\left(\phi ,\Gamma \right)$-modules. Our approach brings the finite-slope nonordinary case of Iwasawa theory onto an equal footing with ordinary cases in which $p$ is inverted.

We describe explicit multiplicative excellent families of rational elliptic surfaces with Galois group isomorphic to the Weyl group of the root lattices $E_7$ or $E_8$. The Weierstrass coefficients of each family are related by an invertible polynomial transformation to the generators of the multiplicative invariant ring of the associated Weyl group, given by the fundamental characters of the corresponding Lie group. As an application, we give examples of elliptic surfaces with multiplicative reduction and all sections defined over $\mathbb{Q}$ for most of the entries of fiber configurations and Mordell–Weil lattices described by Oguiso and Shioda, as well as examples of explicit polynomials with Galois group $W\left(E_7\right)$ or $W\left(E_8\right)$.

Let $G$ be an algebraic group over a field $F$. As defined by Serre, a cohomological invariant of $G$ of degree $n$ with values in $\mathbb{Q}∕\mathbb{Z}\left(j\right)$ is a functorial-in-$K$ collection of maps of sets ${Tors}_G\left(K\right)\to H^n\left(K,\mathbb{Q}∕\mathbb{Z}\left(j\right)\right)$ for all field extensions $K∕F$, where ${Tors}_G\left(K\right)$ is the set of isomorphism classes of $G$-torsors over Spec $K$. We study the group of degree $3$ invariants of an algebraic torus with values in $\mathbb{Q}∕\mathbb{Z}\left(2\right)$. In particular, we compute the group $H_{nr}^3\left(F\left(S\right),\mathbb{Q}∕\mathbb{Z}\left(2\right)\right)$ of unramified cohomology of an algebraic torus $S$.

In this paper, we continue our study, begun in an earlier paper, of abstract representations of elementary subgroups of Chevalley groups of rank $\ge 2$. First, we extend the methods to analyze representations of elementary groups over arbitrary associative rings and, as a consequence, prove the conjecture of Borel and Tits on abstract homomorphisms of the groups of rational points of algebraic groups for groups of the form ${SL}_{n,D}$, where $D$ is a finite-dimensional central division algebra over a field of characteristic $0$. Second, we apply the previous results to study deformations of representations of elementary subgroups of universal Chevalley groups of rank $\ge 2$ over finitely generated commutative rings.

The emergence of Boij–Söderberg theory has given rise to new connections between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro recently showed that every Betti diagram of an ideal with a $k$-linear minimal resolution arises from that of the Stanley–Reisner ideal of a simplicial complex. In this paper, we extend their result for the special case of $2$-linear resolutions using purely combinatorial methods. Specifically, we show bijective correspondences between Betti diagrams of ideals with $2$-linear resolutions, threshold graphs, and anti-lecture-hall compositions. Moreover, we prove that any Betti diagram of a module with a $2$-linear resolution is realized by a direct sum of Stanley–Reisner rings associated to threshold graphs. Our key observation is that these objects are the lattice points in a normal reflexive lattice polytope.

We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal’s category of species. Such structure is carried by the collection of class function spaces on those groups and also by the collection of superclass function spaces in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices.