Related papers: A unified finiteness theorem for curves

We study isolated points on the modular curves $X_{H}$, for $H$ a subgroup of $\operatorname{GL}_{2}(\mathbb{Z}/n \mathbb{Z})$ for some $n \geq 1$. In particular, we prove a single-sink theorem for such isolated points, which traces the…

We introduce the new notion of the "quasi-Galois point" in Algebraic geometry, which is a generalization of the Galois point. A point $P$ in projective plane is said to be quasi-Galois for a plane curve if the curve admits a non-trivial…

We prove an equidistribution statement for the reduction of Galois orbits of CM points on the special fiber of a Shimura curve over a totally real field, considering both the split and the ramified case. The main novelty of the ramified…

We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in ${\mathbb R}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is…

We obtain a general lower bound for the number of fixed points of a circle action on a compact almost complex manifold $M$ of dimension $2n$ with nonempty fixed point set, provided the Chern number $c_1c_{n-1}[M]$ vanishes. The proof…

Let K be a number field, let f: P_1 --> P_1 be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K, and suppose that u, w in P_1(K) are not preperiodic under f. We prove that the set of (m,n) in N^2…

We give a classification of the degrees of the points with rational $j$-invariant on the modular curves $X_{0}(n)$ and $X_{1}(n)$. The degrees which occur infinitely often are computed unconditionally, while those which occur finitely often…

We consider N=2 supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to…

Let $K$ be a number field, and $S$ a finite set of non-archimedean places of $K$, and write $\mathcal{O}_S^\times$ for the group of $S$-units of $K$. A famous theorem of Siegel asserts that the $S$-unit equation $\varepsilon+\delta=1$, with…

We compute and explicitly describe the Bieri-Neumann-Strebel invariants $\Sigma^1$ for the full and pure braid groups of the sphere $\mathbb{S}^2$, the real projective plane $\mathbb{R}P^2$ and specially the torus $\mathbb{T}$ and the Klein…

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which…

A common practice in arithmetic geometry is that of generalizing rational points on projective varieties to integral points on quasi-projective varieties. Following this practice, we demonstrate an analogue of a result of L. Caporaso, J.…

Let $G$ be a group. The orbits of the natural action of $\mbox{Aut}(G)$ on $G$ are called "automorphism orbits" of $G$, and the number of automorphism orbits of $G$ is denoted by $\omega(G)$. In this paper the finite nonsolvable groups $G$…

We define new higher-order Alexander modules $\mathcal{A}_n(C)$ and higher-order degrees $\delta_n(C)$ which are invariants of the algebraic planar curve $C$. These come from analyzing the module structure of the homology of certain…

This article presents a new proof of a theorem concerning bounds of the spectrum of the product of unitary operators and a generalization for differentiable curves of this theorem. The proofs involve metric geometric arguments in the group…

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which…

We classify the finite primitive permutation groups which have a cyclic subgroup with two orbits. This extends classical topics in permutation group theory, and has arithmetic consequences. By a theorem of C. L. Siegel, affine algebraic…

We present a theory for splitting algebras of monic polynomials over rings, and apply the results to symmetric functions, and Galois theory. Our main result is that the ring of invariants of a splitting algebra under the symmetric group…

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More…

For a non-CM elliptic curve $E$ defined over $\mathbb{Q}$, the Galois action on its torsion points gives rise to a Galois representation $\rho_E: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to…