Related papers: Abel pairs and modular curves

The 4IM+1CM problem is determining all pairs (f,g) of meromorphic functions in the complex plane that are not Moebius transformations of each other and share five pairs of complex values, one of them counting multiplicities. It is shown…

We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in\Bbb F_q(X)\setminus\{0\}$ be such that $q$ is…

In this note we discuss a class of hyperelliptic curves introduced by Abel in a 1826 paper. After some indications of the context in which he introduced them and a description of his main result we give some results on the moduli space of…

In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field $k$ and the intersection of the moduli space $\M_3^b$ of such curves with the hyperelliptic moduli $\H_3$. Such intersection $\S$ is an…

In the Hurwitz space of rational functions on CP^1 with poles of given orders, we study the loci of multisingularities, that is, the loci of functions with a given ramification profile over 0. We prove a recursion relation on the Poincare…

Given a properly normalized parametrization of a genus-0 modular curve, the complex multiplication points map to algebraic numbers called singular moduli. In the classical case, the maps can be given analytically. However, in the Shimura…

We pose some questions about spaces parametrizing rational curves on rationally connected varieties. We give a partial answer for cubic threefolds. Many of our results were previously proved by Iliev, Markushevich and Tikhimirov by…

In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves.…

Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $\mathcal{MS}_C(n,L)$ and $\mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We…

The strata of the moduli spaces of Abelian differentials are non-homogenous spaces carrying natural bi-algebraic structures. Partly inspired by the case of homogenous spaces carrying bi-algebraic structures (such as torii, Abelian varieties…

Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties,…

We provide a generalization of an algebraic linear combination for the trace of certain elliptic modular forms, and through specializing the expression at a suitable pair consisting of an elliptic curve over algebraic number fields and its…

We study singular curves from analytic point of view. We give completely analytic proofs for the Serre duality and a generalized Abel's theorem. We also reconsider Picard varieties, Albanese varieties and generalized Jacobi varieties of…

We study the moduli space of stable sheaves of Euler characteristic 2, supported on curves of arithmetic genus 2 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers and we give a…

We exhibit an algorithm that, given input a curve $X$ over a number field, computes as output the minimal degree of a Belyi map $X \to \mathbb{P}^1$.

In this survey article we give an overview of how noncongruence modular curves can be viewed as Hurwitz moduli spaces of covers of elliptic curves at most branched above the origin. We describe some natural questions that arise, and…

We quantize the coordinate ring of the moduli space of B-bundles on the elliptic curve. Here B is a Borel subgroup of some semisimple Lie group. We construct some representations of these algebras and study intertwining operators for these…

We study the compactification of the moduli space of a certain class of rank-two irregular connections on the Riemann sphere, presenting one double pole and two simple poles. To construct the compactification explicitly, we identify a class…

We describe the moduli space of Higgs pairs on an irreducible nodal curve of arithmetic genus one and its geometric structures in terms of the Hitchin map and a flat degeneration of the moduli space of Higgs bundles on an elliptic curve.

Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form $E_{A,B}:\, A(x)-B(y)=0$, where $A, B\in\mathbb C(z)$. We also investigate "series" of…