Related papers: Period Constraints on Hyperelliptic Branch Points

In the study of the curve shortening flow on general closed curves, Abresch and Langer posed a conjecture that the homothetic curves can be regarded as saddle points between multi-folded circles and some singular curves. In other words,…

We describe our recent work on deformations of hyperelliptic curves by means of integrable hierarchy of hydrodynamic type (nlin.SI/0205012). We also discuss a further extension to the case of non-hyperelliptic curves.

Several problems which could be thought of as belonging to recreational mathematics are described. They are all such that solutions to the problem depend on finding rational points on elliptic curves. Many of the problems considered lead to…

For certain complex projective manifolds (such as K3 surfaces and their higher dimensional analogues, the complex symplectic projective manifolds) the period map takes values in a locally symmetric variety of type IV. It is often an open…

We study hyperelliptic curves y^2=f(x) over local fields of odd residue characteristic. We introduce the notion of a "cluster picture" associated to the curve, that describes the p-adic distances between the roots of f(x), and show that…

For a given elliptic curve, its associated $L$-function evaluated at $1$ is closely related to its real period. In this article, we generalize this principle to a rational curve. We count the rational points over all finite fields and use…

The periods of arbitrary abelian forms on hyperelliptic Riemann surfaces, in particular the periods of the meromorphic Seiberg-Witten differential, are shown to be in one-to-one correspondence with the conformal blocks of correlation…

We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show…

We establish GIT semistability of the 2nd Hilbert point of every Gieseker-Petri general canonical curve by a simple geometric argument. As a consequence, we obtain an upper bound on slopes of general families of Gorenstein curves. We also…

This paper defines a symplectic form on the infinite dimensional Fr\'echet manifold of framed curves of fixed length over a simply connected Riemannian manifold of constant curvature. The paper then considers Hamiltonian systems generated…

In \cite{MP} we have shown that if a compact Riemann surface admits a Strebel differential with rational periods, then the Riemann surface is the complex model of an algebraic curve defined over the field of algebraic numbers. We will show…

Effects of geometric constraints on a steady flow potential are described by an elliptic-hyperbolic generalization of the harmonic map equations. Sufficient conditions are given for global triviality.

The identifiability problem arises naturally in a number of contexts in mathematics and computer science. Specific instances include local or global rigidity of graphs and unique completability of partially-filled tensors subject to rank…

Temporal graphs arise when modeling interactions that evolve over time. They usually come in several flavors, depending on the number of parameters used to describe the temporal aspects of the interactions: time of appearance, duration,…

In this paper, we will consider the period index problems of elliptic curves and introduce a value called generic index which is closed related to the essential dimension of Picard stacks. In particular, we will use examples to see that…

The Information-Theoretic Schottky Problem treats the period matrix of a compact Riemann Surface as a compressible signal. In this case, the period matrix of a smooth plane curve is characterized by only 4 of its columns, a significant…

We define the period as a multiplicative characteristic of stably symmetric monoidal $\infty$-categories, develop its basic properties, and study many examples, with a focus on `ordinary' equivariant and motivic homotopy theory. We apply…

We consider a special class of periodic continued fractions (called alpha-fractions) and discuss the related algebraic and geometric problems. A classical description of the Jacobi variety of a hyperelliptic curve due to Jacobi naturally…

We investigate planar piecewise-smooth vector fields with a discontinuity line, focusing on the bifurcation of crossing limit cycles that arise when one of the vector fields is translated along the discontinuity set. We establish…

I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number…