In this paper, we classify the possible torsion subgroup structures of elliptic curves defined over the compositum of all quadratic extensions of the rational number field, whose j-invariant is a rational number not equal to 0 or 1728.
Let $n$ be a cubefree natural number and $p\geq 5$ be a prime number. Assume that $n$ is not expressible as a sum of the form $x^3+y^3$, where $x,y\in \mathbb{Q}$. In this note, we study the solutions (or lack thereof) to the equation $n=x^3+y^3$, where $x$ and $y$ belong to the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, consider the case when $n$ is not a sum of rational cubes. Then, we prove that $n$ cannot be a sum of two cubes in certain large families of prime cyclic extensions of $\mathbb{Q}$.
As proven in a celebrated theorem due to Vaisman, pure locally conformally Kähler metrics do not exist on compact Kähler manifolds. In a previous paper, we extended this result to the singular setting, more precisely to Kähler spaces which are locally irreducible. Without the additional assumption of local irreducibility, there are counterexamples for which Vaisman's theorem does not hold. In this article, we give a much broader sufficient condition under which Vaisman's theorem still holds for compact Kähler spaces which are locally reducible.
Let S be a compact surface with boundary and F be the set of the orbits of a traversing flow on S. If the flow is generic, its orbit space is a spine G of S, namely G is a graph embedded in S and S is a regular neighbourhood of G. Moreover an extra structure on G turns it into a flow-spine, from which one can reconstruct S and F. In this paper we study properly immersed curves C in S. We do this by considering generic C's and their apparent contour relative to F, namely the set of points of G corresponding to orbits that either are tangent to C, or go through a self-intersection of C, or meet the boundary of C. We translate this apparent contour into a decoration of G that allows one to reconstruct C, and then we allow C to vary up to homotopy within a fixed generic F, and next also F to vary up to homotopy, and we identify a finite set of local moves on decorated graphs that translate these homotopies.
We give a non-left-orderability criterion for involutory quandles of non-split links. We use this criterion to show that the involutory quandle of any non-trivial alternating link is not left-orderable, thus improving Theorem 8.1. proven by Raundal et al. (Proceedings of the Edinburgh Mathematical Society (2021 64), page 646). We also use the criterion to show that the involutory quandles of augmented alternating links are not left-orderable. We introduce a new family of links containing all non-alternating and quasi-alternating 3-braid closures and show that their involutory quandles are not left-orderable. This leads us to conjecture that the involutory quandle of any quasi-alternating link is not left-orderable.
We consider a generalized Cantor set () for an infinite sequence = ()infinity=1 E (0, 1)N, and consider the moduli space () for which are the set of ' for which (') is conformally equivalent to (). In this paper, we may give a necessary and sufficient condition for ( ) := C \ () to be a uniform domain. As a byproduct, we give a condition for () to belong to (0), the moduli space of the standard middle one-third Cantor set. We also show that the volume of the moduli space ( ) with respect to the standard product measure on (0, 1)N vanishes under a certain condition for .
A number of questions related to the length spectrum of surfaces are discussed and in particular the existence of pairs of surfaces which though not isometric are isospectral. Here by isospectral we mean that a pair of bodies have the same distribution of chord lengths. In the Euclidean setting, we study isospectral convex dodecagons found by Mallows and Clark in the 1970's. Starting from their idea, we give constructions for isospectral pairs of hyperbolic surfaces that have no common cover. Since the work of Mallows and Clark is probably unfamiliar to readers with a background in topology/hyperbolic geometry we include expository material on other related topics about the distribution of chord lengths.
We study a modified version of the classical Ulrich modules, which we call $c$-Ulrich. Unlike the traditional setting, $c$-Ulrich modules always exist. We prove that these modules retain many of the essential properties and applications observed in the literature. Additionally, we reveal their significance as obstructions to Cohen-Macaulay properties of tensor products. Leveraging this insight, we show the utility of these modules in testing the finiteness of homological dimensions across various scenarios.
Congruences, or $2$-parameter families of lines in $3$-space are of interest in many situations, in particular in geometric optics. In this paper we consider elements of their geometry which are invariant under affine changes of co-ordinates, for example that associated with their focal sets, and less well studied focal planes. We use tools from singularity theory to describe some generic phenomena. In particular we determine the generic singularities of various surfaces in affine 3-space associated to these congruences. We identify a projective quadric in the projectivised tangent space to the manifold of lines which plays a key role in understanding the affine geometry of ruled surfaces, congruences and $3$-parameter families or complexes. Many of the results generalise to lines in $\mathbb R^n, n>3$.
We consider the Segre embedding of the product CP & times; Cq into CP+q+Pq and study the biharmonicity of P & times; Cq and 1P & times; 2qas submanifolds of CP+q+Pq, where and 1 are Lagrangian submanifolds of CP and 2 is a Lagrangian submanifold of Cq. We find two new large classes of biharmonic submanifolds in complex projective space forms.
We study the behaviour of principal bundles under pullback along proper surjective morphisms of either schemes over an algebraically closed field of characteristic 0 or complex analytic spaces.
In this paper, we establish a logarithmic vanishing theorem on weakly pseudoconvex Kähler manifolds, where the divisor may have infinitely many irreducible components. This result serves as a generalization of Norimatsu's findings on compact Kähler manifolds. We derive vanishing theorems for certain direct image sheaves as a direct corollary.
It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. In this paper, certain cases of specific (non-real analytic) smooth functions are precisely investigated. In particular, we give asymptotic limits of local zeta functions at some singularities along one direction. It follows from these behaviors that the respective local zeta functions have singularities different from poles. Then we show the optimality of the lower estimates of certain quantity concerning with meromorphic continuation of local zeta functions in some smooth model cases.
We first investigate geometric properties of geodesic spheres with sufficiently big radii in a complex projective space. Motivated by this result, we redefine Berger spheres. Our aim is to study such Berger spheres from the viewpoints of contact geometry, submanifold geometry and length spectral geometry.
Let C = A(r, r ') be a closed annulus of radii r and r ' (r < r ' is an element of Q >= 0) over a complete discrete valuation field with algebraically closed residue field of characteristic p > 0. To an & eacute;tale sheaf of F-& ell;-modules F on C, ramified at most at a finite set of rigid points of C, we associate an Abbes-Saito Swan conductor function sw(F,) : [r, r '] -> Q which, for the variable t, measures the ramification of the restriction of F to the sub-annulus of C of radius t with 0-thickness, along the special fiber of the normalized integral model of said sub-annulus. We show that this function is continuous, convex and piecewise linear outside the radii of the ramification points of F, with finitely many slopes which are all integers. For two distinct radii t and t ' lying between consecutive radii of ramification points of F, we compute the difference of the slopes of sw(F,) at t and t ' as the difference of the orders of the characteristic cycles of Fat t and t '.
