The Griffiths group $\text{G}{{\text{r}}^{r}}\left( X \right)$ of a smooth projective variety $X$ over an algebraically closed field is defined to be the group of homologically trivial algebraic cycles of codimension $r$ on $X$ modulo the subgroup of algebraically trivial algebraic cycles. The main result of this paper is that the Griffiths group $\text{G}{{\text{r}}^{2}}\left( {{A}_{{\bar{k}}}} \right)$ of a supersingular abelian variety ${{A}_{{\bar{k}}}}$ over the algebraic closure of a finite field of characteristic $p$ is at most a $p$-primary torsion group. As a corollary the same conclusion holds for supersingular Fermat threefolds. In contrast, using methods of $\text{C}$. Schoen it is also shown that if the Tate conjecture is valid for all smooth projective surfaces and all finite extensions of the finite ground field $k$ of characteristic $p\,>\,2$, then the Griffiths group of any ordinary abelian threefold ${{A}_{{\bar{k}}}}$ over the algebraic closure of $k$ is non-trivial; in fact, for all but a finite number of primes $\ell \,\ne \,p$ it is the case that $\text{G}{{\text{r}}^{2}}\left( {{A}_{{\bar{k}}}} \right)\,\otimes \,{{\mathbb{Z}}_{\ell }}\,\ne \,0$.

Gromov introduced the concept of uniform embedding into Hilbert space and asked if every separable metric space admits a uniform embedding into Hilbert space. In this paper, we study uniform embedding into Hilbert space and answer Gromov’s question negatively.

We introduce the notion of Lie algebras with plus-minus pairs as well as regular plus-minus pairs. These notions deal with certain factorizations in universal enveloping algebras. We show that many important Lie algebras have such pairs and we classify, and give a full treatment of, the three dimensional Lie algebras with plus-minus pairs.

A partial differential equation, the local Möbius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local Möbius equation and an additional partial differential equation.

An irreducible supercuspidal representation $\pi$ of $G\,=\,\text{GL}\left( n,\,F \right)$, where $F$ is a nonarchimedean local field of characteristic zero, is said to be “distinguished” by a subgroup $H$ of $G$ and a quasicharacter $\text{ }\!\!\chi\!\!\text{ }$ of $H$ if $\text{Ho}{{\text{m}}_{H}}\left( \pi ,\,\text{ }\!\!\chi\!\!\text{ } \right)\,\ne \,0$. There is a suitable global analogue of this notion for an irreducible, automorphic, cuspidal representation associated to $\text{GL}\left( n \right)$. Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.

As a consequence of a theorem of Rost-Springer, we establish that the cyclicity problem for central simple algebra of degree 5 on fields containg a fifth root of unity is equivalent to the study of anisotropic elements of order 5 in the split group of type ${{E}_{8}}$.

We show that the set of values of an additive polynomial in several variables with arguments in a formal Laurent series field over a finite field has the optimal approximation property: every element in the field has a (not necessarily unique) closest approximation in this set of values. The approximation is with respect to the canonical valuation on the field. This property is elementary in the language of valued rings.

We prove polytopality of the generalized associahedra introduced in [5].

On a locally rectifiable arc going to infinity, each continuous function can be approximated by entire functions.

We classify the subalgebras of the general Lie conformal algebra $\text{g}{{\text{c}}_{N}}$ that act irreducibly on $\mathbb{C}\,{{\left[ \partial \right]}^{N}}$ and that are normalized by the $\text{s}{{\text{l}}_{2}}$-part of a Virasoro element. The problem turns out to be closely related to classical Jacobi polynomials $P_{n}^{\left( -\sigma ,\sigma \right)}$, $\sigma \,\in \,C$. The connection goes both ways—we use in our classification some classical properties of Jacobi polynomials, and we derive from the theory of conformal algebras some apparently new properties of Jacobi polynomials.

Exact analytical expressions for the inverse Laplace transforms of the functions $\frac{{{I}_{n}}\left( s \right)}{sI_{n}^{\prime }\left( s \right)}$ are obtained in the form of trigonometric series. The convergence of the series is analyzed theoretically, and it is proven that those diverge on an infinite denumerable set of points. Therefore it is shown that the inverse transforms have an infinite number of singular points. This result, to the best of the author’s knowledge, is new, as the inverse transforms of $\frac{{{I}_{n}}\left( s \right)}{sI_{n}^{\prime }\left( s \right)}$ have previously been considered to be piecewise smooth and continuous. It is also found that the inverse transforms have an infinite number of points of finite discontinuity with different left- and right-side limits. The points of singularity and points of finite discontinuity alternate, and the sign of the infinity at the singular points also alternates depending on the order $n$. The behavior of the inverse transforms in the proximity of the singular points and the points of finite discontinuity is addressed as well.

For an integer $n\,\ge \,3$, let ${{M}_{n}}$ be the moduli space of spatial polygons with $n$ edges. We consider the case of odd $n$. Then ${{M}_{n}}$ is a Fano manifold of complex dimension $n\,-\,3$. Let ${{\Theta }_{{{M}_{n}}}}$ be the sheaf of germs of holomorphic sections of the tangent bundle $T{{M}_{n}}$. In this paper, we prove ${{H}^{q}}\left( {{M}_{n}},\,{{\Theta }_{{{M}_{n}}}} \right)\,=\,0$ for all $q\,\ge \,0$ and all odd $n$. In particular, we see that the moduli space of deformations of the complex structure on ${{M}_{n}}$ consists of a point. Thus the complex structure on ${{M}_{n}}$ is locally rigid.

We begin by reviewing Monstrous Moonshine. The impact of Moonshine on algebra has been profound, but so far it has had little to teach number theory. We introduce (using ‘postcards’) a much larger context in which Monstrous Moonshine naturally sits. This context suggests Moonshine should indeed have consequences for number theory. We provide some humble examples of this: new generalisations of Gauss sums and quadratic reciprocity.

Let $k$ be a cyclic extension of odd prime degree $p$ of the field of rational numbers. If $t$ denotes the number of primes that ramify in $k$, it is known that the Hilbert $p$-class field tower of $k$ is infinite if $t\,>\,3\,+\,2\sqrt{p}$. For each $t\,>\,2\,+\,\sqrt{p}$, this paper shows that a positive proportion of such fields $k$ have infinite Hilbert $p$-class field towers.

The geometry of indicatrices is the foundation of Minkowski geometry. A strongly convex indicatrix in a vector space is a strongly convex hypersurface. It admits a Riemannian metric and has a distinguished invariant—(Cartan) torsion. We prove the existence of non-trivial strongly convex indicatrices with vanishing mean torsion and discuss the relationship between the mean torsion and the Riemannian curvature tensor for indicatrices of Randers type.

We construct a class of fermions (or bosons) by using a Clifford (or Weyl) algebra to get two families of irreducible representations for the extended affine Lie algebra $\widetilde{\mathfrak{g}{{\mathfrak{l}}_{N}}\left( {{\mathbb{C}}_{q}} \right)}$ of level (1, 0) (or (−1, 0)).

We give an upper bound on the essential dimension of the group $\mathbb{Z}/q\,\rtimes \,{{\left( \mathbb{Z}/q \right)}^{*}}$ over the rational numbers, when $q$ is a prime power.

Using a classical generalization of Jacobi’s derivative formula, we give an explicit expression for Gunning’s prime form in genus 2 in terms of theta functions and their derivatives.

We combine the deep methods of Frey, Ribet, Serre and Wiles with some results of Darmon, Merel and Poonen to solve certain explicit diophantine equations. In particular, we prove that the area of a primitive Pythagorean triangle is never a perfect power, and that each of the equations ${{X}^{4}}\,-\,4{{Y}^{4}}\,=\,{{Z}^{p}},\,{{X}^{4}}\,+\,4{{Y}^{p}}\,=\,{{Z}^{2}}$ has no non-trivial solution. Proofs are short and rest heavily on results whose proofs required Wiles’ deep machinery.