Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication, and for each prime $p$ of good reduction, let ${{n}_{E}}(p)\,=\,\left| E\left( {{\mathbb{F}}_{p}} \right) \right|$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let ${{Q}_{E,B}}(x)$ be the number of primes $p\le x$ such that ${{b}^{{{n}_{E}}(p)}}\,\equiv \,b\left( \bmod \,{{n}_{E}}\left( p \right) \right)$, and let $\pi _{E,b}^{\text{pseu}}(x)$ be the number of compositive${{n}_{E}}(p)$ such that ${{b}^{{{n}_{E}}(p)}}\,\equiv \,b\left( \bmod \,{{n}_{E}}\left( p \right) \right)$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for ${{Q}_{E,B}}(x)$ and $\pi _{E,b}^{\text{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting.

We discuss a clean level lowering theorem modulo prime powers for weight 2 cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations $a{{x}^{n}}\,+\,b{{y}^{n\,}}\,+\,c{{z}^{n}}\,=\,0$.

In this paper we study the extension problem, the averaging problem, and the generalized Erdős–Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

We introduce the fundamental group $\mathcal{F}\left( A \right)$ of a simple $\sigma $-unital ${{C}^{*}}$–algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple${{C}^{*}}$-algebras with Unique Trace I and II by Nawata and Watatani. Our definition in this paper makes sense for stably projectionless ${{C}^{*}}$-algebras. We show that there exist separable stably projectionless ${{C}^{*}}$-algebras such that their fundamental groups are equal to $\mathbb{R}_{+}^{\times }$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

In this article we refine the method of Bertolini and Darmon $\left[ \text{BD}1 \right],\,\left[ \text{BD2} \right]$ and prove several finiteness results for anticyclotomic Selmer groups of Hilbert modular forms of parallel weight two.

We introduce a $q,\,t$-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory $\nabla $ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for $\nabla {{e}_{n}}\left[ X \right]$. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of $q,\,t$-Catalan sequences, and we prove a number of identities involving these functions.

Let $C$ and $D$ be digraphs. A mapping $f:V\left( D \right)\to V\left( C \right)$ is a $C$-colouring if for every arc $uv$ of $D$, either $f\left( u \right)f\left( v \right)$ is an arc of $C$ or $f\left( u \right)=f\left( v \right)$, and the preimage of every vertex of $C$ induces an acyclic subdigraph in $D$. We say that $D$ is $C$-colourable if it admits a $C$-colouring and that $D$ is uniquely $C$-colourable if it is surjectively $C$-colourable and any two $C$-colourings of $D$ differ by an automorphism of $C$. We prove that if a digraph $D$ is not $C$-colourable, then there exist digraphs of arbitrarily large girth that are $D$-colourable but not $C$-colourable. Moreover, for every digraph $D$ that is uniquely $D$-colourable, there exists a uniquely $D$-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number $r\ge 1$, there are uniquely circularly $r$-colourable digraphs with arbitrarily large girth.

We formulate an analog of Hermite's constant for function fields over a finite field and state a conjectural value for this analog. We prove our conjecture in many cases, and prove slightly weaker results in all other cases.

A body moves in a rarefied medium composed of point particles at rest. The particles make elastic reflections when colliding with the body surface and do not interact with each other. We consider a generalization of Newton’s minimal resistance problem: given two bounded convex bodies ${{C}_{1}}$ and ${{C}_{2}}$ such that ${{C}_{1}}\,\subset \,{{C}_{2}}\,\subset \,{{\mathbb{R}}^{3}}$ and $\partial {{C}_{1}}\,\cap \,\partial {{C}_{2}}\,=\,\varnothing $, minimize the resistance in the class of connected bodies $B$ such that ${{C}_{1}}\,\subset \,B\,\subset \,{{C}_{2}}$. We prove that the infimum of resistance is zero; that is, there exist “almost perfectly streamlined” bodies.

We introduce a flow of a spatial graph and see how invariants for spatial graphs and handlebody-links are derived from those for flowed spatial graphs. We define a new quandle (co)homology by introducing a subcomplex of the rack chain complex. Then we define quandle colorings and quandle cocycle invariants for spatial graphs and handlebody-links.

We study properties of composition operators induced by symbols acting from the unit disk to the polydisk. This result will be involved in the investigation of weighted composition operators on the Hardy space on the unit disk and, moreover, be concerned with composition operators acting from the Bergman space to the Hardy space on the unit disk.

We consider the stochastic difference equation ${{\eta }_{k}}\,=\,{{\xi }_{k}}\phi \left( {{\eta }_{k-1}} \right),\,\,k\,\in \,\mathbb{Z}$ on a locally compact group $G$, where $\phi $ is an automorphism of $G$, ${{\xi }_{K}}$ are given $G$-valued random variables, and ${{\eta }_{k}}$ are unknown $G$-valued random variables. This equation was considered by Tsirelson and Yor on a one-dimensional torus. We consider the case when ${{\xi }_{K}}$ have a common law $\mu $ and prove that if $G$ is a distal group and $\phi $ is a distal automorphism of $G$ and if the equation has a solution, then extremal solutions of the equation are in one-to-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu $ is supported on $Kz\,=\,z\phi \left( K \right)$ for any $z$ in the support of $\mu $. We also provide a necessary and sufficient condition for the existence of solutions to the equation.

We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group.

In this article, we study the correspondence between the geometry of del Pezzo surfaces ${{s}_{r}}$ and the geometry of the $r$-dimensional Gosset polytopes (${{(r-4)}_{21}}$. We construct Gosset polytopes ${{(r-4)}_{21}}$ in Pic ${{S}_{r}}\,\otimes \,\mathbb{Q}$ whose vertices are lines, and we identify divisor classes in Pic ${{s}_{r}}$ corresponding to $(a-1)$-simplexes $(a\le r)$, $(r-1)$-simplexes and $(r-1)$-crosspolytopes of the polytope ${{(r-4)}_{21}}$. Then we explain how these classes correspond to skew $a$-lines$(a\le r)$, exceptional systems, and rulings, respectively.

As an application, we work on the monoidal transform for lines to study the local geometry of the polytope ${{(r-4)}_{21}}$. And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes ${{3}_{21}}$ and ${{4}_{21}}$, respectively.

We consider cubic polynomials $f\left( z \right)\,=\,{{z}^{3}}\,+\,az\,+\,b$ defined over $\mathbb{C}\left( \lambda \right)$, with a marked point of period $N$ and multiplier $\lambda$. In the case $N\,=\,1$, there are infinitely many such objects, and in the case $N\,\ge \,3$, only finitely many (subject to a mild assumption). The case $N\,=\,2$ has particularly rich structure, and we are able to describe all such cubic polynomials defined over the field ${{\cup }_{n\ge 1}}\,\mathbb{C}\left( {{\lambda }^{1/n}} \right)$.

We study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropical degenerations to construct a natural, “multiplicity-free” parameterization of Trop$\left( X \right)$ by a topological space ${{\Gamma }_{X}}$ and give a geometric interpretation of the cohomology of ${{\Gamma }_{X}}$ in terms of the action of a monodromy operator on the cohomology of $X$. This gives bounds on the Betti numbers of ${{\Gamma }_{X}}$ in terms of the Betti numbers of $X$ which constrain the topology of Trop$\left( X \right)$. We also obtain a description of the top power of the monodromy operator acting on middle cohomology of $X$ in terms of the volume pairing on ${{\Gamma }_{X}}$.

It is shown that simplicity of the crossed product of a unital $\text{AH}$-algebra with slow dimension growth by an endomorphism implies that the algebra is also purely infinite, provided only that the endomorphism leaves no trace state invariant and takes the unit to a full projection.

We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another, i.e., heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces.

Fix an elliptic curve $E/\mathbf{Q}$and assume the Riemann Hypothesis for the $L$-function $L({{E}_{D}},\,s)$ for every quadratic twist ${{E}_{D}}$ of $E$ by $D\,\in \,\mathbf{Z}$. We combine Weil's explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of ${{E}_{D}}$. We derive from this an upper bound for the density of low-lying zeros of $L({{E}_{D}},\,s)$ that is compatible with the randommatrixmodels of Katz and Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbf{R}$, the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer conjecture) the number of integral points of ${{E}_{D}}$ are less than $f(D)$ for almost all $D$.

In this paper we study the balayage of semi-Dirichlet forms. We present new results on balayaged functions and balayaged measures of semi-Dirichlet forms. Some of the results are new even in the Dirichlet forms setting.