相关论文: On uniformization of Burnside's curve $y^2=x^5-x$

We establish the following uniformization result for metric spaces $X$ of finite Hausdorff 2-measure. If $X$ is homeomorphic to a smooth 2-manifold $M$ with non-empty boundary, then we show that $X$ admits a quasiconformal almost…

In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a…

We prove two types of results. First we develop the decoupling theory for hypersurfaces with nonzero Gaussian curvature, which extends our earlier work from \cite{BD3}. As a consequence of this we obtain sharp (up to $\epsilon$ losses)…

Let $X$ be a curve of genus $g\geq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic…

We compute the fundamental groups of non-singular analytic Deligne-Mumford curves, classify the simply connected ones, and classify analytic Deligne-Mumford curves by their uniformization type. As a result, we find an explicit presentation…

We study genus $g$ hyperelliptic curves with reduced automorphism group $A_5$ and give equations $y^2=f(x)$ for such curves in both cases where $f(x)$ is a decomposable polynomial in $x^2$ or $x^5$. For any fixed genus the locus of such…

We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric K\"ahler-Einstein and…

We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point, and additional analyticity properties. This situation occurs…

We show that a Fuchsian differential equation having five regular singular points admits solutions in terms of a single generalized hypergeometric function for infinitely many particular choices of equation parameters. Each solution assumes…

We find a new class of the Fuchsian equations, which have an algebraic geometric solutions with the parameter belonging to a hyperelliptic curve. Methods of calculating the algebraic genus of the curve, and its branching points, are…

In this article, we functorially associate definable sets to $k$-analytic curves, and definable maps to analytic morphisms between them, for a large class of $k$-analytic curves. Given a $k$-analytic curve $X$, our association allows us to…

We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation…

This paper develops analityc methods for investigating uniform hypergraphs. Its starting point is the spectral theory of 2-graphs, in particular, the largest and the smallest eigenvalues of 2-graphs. On the one hand, this simple setup is…

We study generalisations to totally real fields of methods originating with Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction.…

We show that four exceptional Fuchsian equations, each determined by the four parabolic singularities, known as the Chudnovsky equations, are transformed into each other by algebraic transformations. We describe equivalence of these…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…

We show that certain statements related to the Fourier-Walsh expansion of functions with respect to a biased measure on the discrete cube can be deduced from the respective results for the uniform measure by a simple reduction. In…

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…

We describe a strategy for computing Yukawa couplings and the mirror map, based on the Picard-Fuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes in the case of quintic hypersurfaces.) We…

We describe a strategy for computing Yukawa couplings and the mirror map, based on the Picard-Fuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes in the case of quintic hypersurfaces.) We…