Related papers: An explicit solution to the weak Schottky problem

Jacobi-Forms can be decomposed as a linear combination of Thetafunctions with modular forms as coefficients. It is shown that the space of these coefficient modular forms of Fourier-Jacobi-Forms, which come from Siegel cusp forms, has full…

Within the Schottky problem, the study of special subvarieties of the Torelli locus has long been of great interest. We describe a representation-theoretic criterion for a Jacobian variety arising from a $G$-Galois cover of $\mathbb{P}^1$…

In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. These polynomials represent the weighted Schubert classes of the cohomology of the weighted Grassmannian introduced by…

In this paper we prove a conjecture of Hershel Farkas that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the…

It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the…

Let $\X$ be an irreducible, smooth, projective curve of genus $g \geq 2$ defined over the complex field $\C.$ Then there is a covering $\pi: \X \longrightarrow \P^1,$ where $\P^1$ denotes the projective line. The problem of expressing…

We associate to any holomorphic vertex algebra a collection of Teichm\"{u}ller modular forms, one in each genus. In genus one we obtain the character of the vertex algebra, and we thus reprove Zhu's modularity result. In higher genus, we…

We show that the degree of Gauss maps on abelian varieties is semicontinuous in families, and we study its jump loci. As an application we obtain that in the case of theta divisors this degree answers the Schottky problem. Our proof…

Let $C$ be a smooth projective curve of genus $g\geq 2$. Fix an integer $r\geq 0$, and let $\underline{k}=(k_1,\ldots,k_n)$ be a sequence of positive integers with $k_1+\ldots+k_n=g-1$. We study $n$-pointed curves $(C,p_1,\ldots,p_n)$ such…

Let K be a complete, non-archimedean valued field with a residue field of characteristic different from 2. A Whittaker group G is a discontinuous subgroup of PGL(2,K), freely generated by elements s_0,...,s_g of order two, each defined by a…

We study a fractional $p$-Laplace equation involving a variable exponent singular nonlinearity in the framework of the Heisenberg group. We first establish the existence and regularity of weak solutions. In the case of a constant singular…

A comprehensive algebro-geometric integration of the two component Nonlinear Vector Schr\"odinger equation (Manakov system) is developed. The allied spectral variety is a trigonal Riemann surface, which is described explicitly and the…

Existing algorithms to compute genus 2 theta constants in quasi-linear time use Borchardt sequences, an analogue of the arithmetic-geometric mean for four complex numbers. In this paper, we show that these Borchardt sequences are given by…

In this paper, we discuss the problem of derivation of kinetic equations from the theory of weak turbulence for the quintic Schr\"odinger equation. We study the quintic Schr\"odinger equation on $L\mathbb T$, with $L\gg 1$ and with a…

We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R^n. We prove well-posedness in…

We discuss the conjecture of Buchstaber and Krichever that their multi-dimensional vector addition formula for Baker-Akhiezer functions characterizes Jacobians among principally polarized abelian varieties, and prove that it is indeed a…

We give an affirmative answer to the rectangular peg problem for a large class of continuous Jordan curves that contains all rectifiable curves and Stromquist's locally monotone curves. Our proof is based on microlocal sheaf theory and…

Let $C$ be a smooth projective absolutely irreducible curve of genus $g \geq 2$ over a number field $K$ of degree $d$, and denote its Jacobian by $J$. Denote the Mordell--Weil rank of $J(K)$ by $r$. We give an explicit and practical…

Let $M$ be a compact Riemannian homogeneous space (e.g. a Euclidean sphere). We prove existence of a global weak solution of the stochastic wave equation \mathbf D_t\partial_tu=\sum_{k=1}^d\mathbf…

This article considers the spatially inhomogeneous, non-cutoff Boltzmann equation. We construct a large-data classical solution given bounded, measurable initial data with uniform polynomial decay of mild order in the velocity variable. Our…