It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of $\mathrm{GL}_2$-type over $\mathbb{Q}$ by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves.

We construct certain algebraic integers $\alpha_n$ as special values of two variable theta functions in the ray class field of a certain quartic field modulo $2^n$, and study a property of prime ideals which appear in $\alpha_n$ in connection to the relationships between cyclotomic units and exponential functions and between elliptic units and elliptic theta functions.

F.K.C. Rankin and H.P.F. Swinnerton–Dyer proved that all the zeros of the Eisenstein Series $E_k$ contained in the standard fundamental domain $\mathcal{F}$ lie on the arc $A\,{=}\, \{ e^{i\theta}| {\pi}/{3} \,{\le}\, \theta \,{\le}\, {\pi}/{2}\}$. Recently, J. Getz has generalized the method of Rankin and Swinnerton–Dyer to show that modular forms under certain conditions have similar properties. In this paper we prove similar results for certain types of cusp forms, motivated by the work of R.A. Rankin. Further, we give a closed formula for the zeros of a class of cusp forms in terms of the Fourier coefficients following the method of Kohnen.

We study the prime ideal structure of the Iwasawa algebra $\Lambda_G$ of an almost simple compact $p$-adic Lie group $G$. When the Lie algebra of $G$ contains a copy of the two-dimensional non-abelian Lie algebra, we show that the prime ideal structure of $\Lambda_G$ is somewhat restricted. We also provide a potential example of a prime c-ideal of $\Lambda_G$ in the case when the Lie algebra of $G$ is ${\frak s}{\frak l}_2(\mathbb{Q}p)$.

Analogously to extraspecial $p$-groups, we define the class of small Frattini $p$-loops and investigate some properties of small Frattini Bol 2-loops. We show that they are related to certain invariant transversals in extraspecial 2-groups. This fact enables us to describe the automorphisms and isomorphisms of small Frattini Bol 2-loops by linear algebraic tools. We use this description to obtain a lower bound for the number of non-isomorphic small Frattini Bol 2-loops of given order.

We discuss a confluence from Siegel–Whittaker functions to Whittaker functions on $Sp(2,{\mathbb{R})$ by using their explicit formulae. In our proof, we use expansion theorems of the good Whittaker functions by the secondary Whittaker functions.

We give a precise description of combed trees in terms of Kelly–Mac Lane graphs. We show that any combed tree is uniquely expressed as an allowable Kelly–Mac Lane graph of a certain shape. Conversely, we show that any such Kelly–Mac Lane graph uniquely defines a combed tree. We show how to apply this to the construction of opetopes.

We show that the Bianchi group $\mathrm{PSL}_2(\mathcal{O}_d)$, where $\mathcal{O}_d$ is the ring of integers in $\mathbb{Q}(\sqrt{d})$, $d\,{<}\,0$, has a free quotient of rank$\,{\geq}\, |d|^{\frac {1}{4}-\epsilon}$, as $|d|\,{\to}\,\infty$. To do so, we give an estimate for a sifted character sum.

In this paper we study periodic functions of one and two variables that are invariant under a subgroup of the Euclidean group. Starting with a function $f$ defined on the plane we obtain a function of one variable by two methods: (1) we project the values on a strip into its edge, by integrating $f$ along the width; and (2) we restrict $f$ to a line. If the functions had been obtained by solving a partial differential equation equivariant under the Euclidean group, how do the symmetries of the function of one variable compare to those of solutions of equations formulated directly in one dimension?

Some of the symmetries of projected and of restricted functions may be obtained knowing only the symmetries of the original functions. Extra symmetries arise at special widths of the strip and at some special positions of the line used for restriction. We obtain a general description of the two types of symmetries and discuss how they arise in the wallpaper groups (crystallographic groups of the plane). We show that the projections and restrictions of solutions of differential equations in the plane may have symmetry groups larger than those of solutions of problems formulated in one dimension.

Springer's theory of regular elements in complex reflection groups is generalized to arbitrary fields. Consequences for the cyclic sieving phenomenon in combinatorics are discussed.

We give a constructive proof of a Theorem of Izmestiev and Joswig. Namely, given $(L,\omega)$ where $L$ is a link in $S^{3}$ and $\omega$ a simple (not necessarily transitive) representation of $\pi_{1}(S^{3}\backslash L)$ onto the symmetric group $\Sigma_{4}$ of four elements $\{1,2,3,4\}$ we construct a triangulation of $S^{3}$ giving rise to $(L,\omega)$ in a natural way.

This paper shows that, given a finite subset $X$ of a finitely generated virtually free group $F$, the freeness of the subsemigroup of $F$ generated by $X$ can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup of $F$ has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.

We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds $M$ with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of $M$, for there to exist a family of $\mathbb{Z}_n$ covering spaces, $M_n$, in which $\beta_1(M_n)$ increases linearly with $n$. The latter generalizes work of M. Katz and C. Lescop, by showing that the non-vanishing of any one of these invariants of $M$ is sufficient to guarantee certain optimal systolic inequalities for $M$ (by work of Ivanov and Katz).

The Euler graph has vertices labelled $(n,k)$ for $n=0,1,2,\ldots$ and $k=0,1,\ldots,n$, with $k+1$ edges from $(n,k)$ to $(n+1,k)$ and $n-k+1$ edges from $(n,k)$ to $(n+1,k+1)$. The number of paths from (0,0) to $(n,k)$ is the Eulerian number $A(n,k)$, the number of permutations of $1,2,\ldots,n+1$ with exactly $n-k$ falls and k rises. We prove that the adic (Bratteli–Vershik) transformation on the space of infinite paths in this graph is ergodic with respect to the symmetric measure.

We study the properties of shifted vertex operator algebras, which are vertex algebras derived from a given theory by shifting the conformal vector. In this way, we are able to exhibit large numbers of vertex operator algebras which are regular (rational and $C_2$-cofinite) and yet are pathological in one way or another.

We introduce a category of basic combinatorial objects, encompassing PCAs and locales. Such a basic combinatorial object is to be thought of as a pre-realizability notion. To each such object we can associate an indexed preorder, generalizing the construction of triposes for various notions of realizability. There are two main results: first, the characterization of triposes which arise in this way, in terms of ordered PCAs equipped with a filter. This will include “Effective Topos-like” triposes, but also the triposes for relative, modified and extensional realizability and the dialectica tripos. Localic triposes can be identified as those arising from ordered PCAs with a trivial filter. Second, we give a classification of geometric morphisms between such triposes in terms of maps of the underlying combinatorial objects. Altogether, this shows that the category of ordered PCAs with non-trivial filters serves as a framework for studying a wide variety of realizability notions.

A basis, denoted $\{Q_{\lambda,\mu\}$, for the full Homfly skein of the annulus $\mathcal{C}$ has been introduced by the authors, where $\lambda$ and $\mu$ are partitions of integers $n$ and $p$ into $k$ and $k^*$ parts respectively. The basis consists of eigenvectors of the two meridian maps on $\mathcal{C}$; these maps are the linear endomorphisms of $\mathcal{C}$ induced by the insertion of a meridian loop with either orientation around a diagram in the annulus.

Here we present an explicit expression for each $Q_{\lambda,\mu}$ as the determinant of a $(k^*+k)\times(k^*+k)$ matrix whose entries are simple elements $h_n, h_n^*$ in the skein $\mathcal{C}$. In the case $p=0$ ($\mu=\phi$) the determinant gives the Jacobi–Trudy formula for the Schur function $s_{\lambda}$ of $N$ variables as a polynomial in the complete symmetric functions $h_n$ of the variables. The Jacobi–Trudy determinants have previously been used by Kawagoe and Lukac in discussing the elements in the skein of the annulus represented by closed braids in which all strings are oriented in the same direction. Our results and techniques form a natural extension of the work of Lukac.

We show that the Nielsen–Thurston classification of mapping classes of the sphere with four marked points is determined by the quantum $SU(n)$ representations, for any fixed $n\geq 2$. In the Pseudo–Anosov case we also show that the stretching factor is a limit of eigenvalues of (non-unitary) $SU(2)$-TQFT representation matrices. It follows that at big enough levels, Pseudo–Anosov mapping classes are represented by matrices of infinite order.

Using the concept of $s$-formality we are able to extend the bounds of a Theorem of Miller and show that a compact $k$-connected $(4k+3)$- or $(4k+4)$-manifold with $b_{k+1}=1$ is formal. We study $k$-connected $n$-manifolds, $n=4k+3, 4k+4$, with a hard Lefschetz-like property and prove that in this case if $b_{k+1}=2$, then the manifold is formal, while, in $4k+3$-dimensions, if $b_{k+1}=3$ all Massey products vanish. We finish with examples inspired by symplectic geometry and manifolds with special holonomy.

On a smooth Banach manifold $M$, the equivalence classes of curves that agree up to acceleration form the second order tangent bundle $T^{2}M$ of $M$. This is a vector bundle in the presence of a linear connection $\nabla$ on $M$ and the corresponding local structure is heavily dependent on the choice of $\nabla$. In this paper we study the extent of this dependence and we prove that it is closely related to the notions of conjugate connections and second order differentials. In particular, the vector bundle structure on $T^{2}M$ remains invariant under conjugate connections with respect to diffeomorphisms of $M$.