相关论文: Lifting divisors with imposed ramifications on a g…

Let $G$ be a connected graph of genus $g$. The Picard group of degree $g$, $\text{Pic}^g(G)$, is the set of equivalence classes of divisors on $G$ of degree $g$, where two divisors are equivalent if one can be reached from the other through…

Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D<0$ such that the reduction map modulo a prime above $p$ from elliptic curves with CM by $\order_{D}$ to supersingular…

We prove new results on splitting Brauer classes by genus 1 curves, settling in particular the case of degree 7 classes over global fields. Though our method is cohomological in nature, and proceeds by considering the more difficult problem…

For any family of elliptic curves over the rational numbers with fixed $j$-invariant, we prove that the existence of a long sequence of rational points whose $x$-coordinates form a non-trivial arithmetic progression implies that the…

We extend the algorithm of Darmon-Green and Darmon-Pollack for computing p-adic Darmon points on elliptic curves to the case of composite conductor. We also extend the algorithm of Darmon-Logan for computing ATR Darmon points to treat…

By means of the theory of strongly semistable sheaves and of the theory of the Greenberg transform, we generalize to higher dimensions a result on the sparsity of p-divisible unramified liftings which played a crucial role in Raynaud's…

In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for…

A realisation of a graph in the plane as a bar-joint framework is rigid if there are finitely many other realisations, up to isometries, with the same edge lengths. Each of these finitely-many realisations can be seen as a solution to a…

We introduce arrangements of rational sections over curves. They generalize line arrangements on P^2. Each arrangement of d sections defines a single curve in P^{d-2} through the Kapranov's construction of \bar{M}_{0,d+1}. We show a…

Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent…

When does topological recursion applied to a family of spectral curves commute with taking limits? This problem is subtle, especially when the ramification structure of the spectral curve changes at the limit point. We provide sufficient…

The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and…

An action of a finite group on a smooth projective curve over an algebraically closed field of positive characteristic is called restrained, if all second ramification groups are trivial (e.g., every group action on an ordinary curve is…

The canonical tree-decomposition theorem, given by Robertson and Seymour in their seminal graph minors series, turns out to be one of the most important tool in structural and algorithmic graph theory. In this paper, we provide the…

We show that on a metric graph of genus $g$, a divisor of degree $n$ generically has $g(n-g+1)$ Weierstrass points. For a sequence of generic divisors on a metric graph whose degrees grow to infinity, we show that the associated Weierstrass…

To any real rational function with generic ramification points we assign a combinatorial object, called a garden, which consists of a weighted labeled directed planar chord diagram and of a set of weighted rooted trees each corresponding to…

Following the work of Totaro and Pereira, we study sufficient conditions under which collections of pairwise-disjoint divisors on a variety over an algebraically closed field are contained in the fibers of a morphism to a curve. We prove…

The stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended Chen's multiplicative property involves the D\"urr-Connes-Kreimer…

We show that the non-Archimedean skeleton of the $d$-th symmetric power of a smooth projective algebraic curve $X$ is naturally isomorphic to the $d$-th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of…

For an elliptic curve $E$ defined over the field $\mathbb{C}$ of complex numbers, we classify all translates of elliptic curves in $E^3$ such that the $x$-coordinates satisfy a linear equation. This classification enables us to establish a…