Related papers: On the arithmetic of certain modular curves

We compute the Picard group of the moduli stack of stable hyperelliptic curves of any genus, exhibiting explicit and geometrically meaningful generators and relations.

A curve X over the field Q of rational numbers is modular if it is dominated by X_1(N) for some N; if in addition the image of its jacobian in J_1(N) is contained in the new subvariety of J_1(N), then X is called a new modular curve. We…

Let k be an algebraically closed field of characteristic p. Let X(p^e;N) be the curve parameterizing elliptic curves with full level N structure (where p does not divide N) and full level p^e Igusa structure. By modular curve, we mean a…

The aim of this paper is to provide an explicit basis of the miniversal deformation of a monomial curve defined by a free semigroup -- these curves make up a notable family of complete intersection monomial curves. First, we dispense a…

We explain how to compute top-dimensional intersections of psi-classes on moduli spaces of m-stable curves. On the moduli spaces of Deligne-Mumford stable pointed curves of genus one, these intersection numbers are determined by two…

In this paper, we present a probabilistic algorithm to compute the number of $\mathbb{F}_p$-points of modular curve $X_1(n)$. Under the Generalized Riemann Hypothesis(GRH), the algorithm takes…

In this paper, we will study the arithmetic of the Eisenstein part of the modular Jacobians. In the first section, we introduce some general preliminaries of the arithmetic theory of modular curves that we will need later. In the second…

We determine all modular curves $X_0^+(N)$ that admit infinitely many cubic points over the rational field $\mathbb{Q}$.

In this paper, we first discuss the structure of the Ramond N=2 superconformal algebras. Then we also classify the modules of the intermediate series over Ramond N=2 superconformal algebra.

For every $n\geq 3$, we exhibit infinitely many extremal effective divisors on the moduli space of genus one curves with $n$ marked points.

We determine necessary conditions for ample divisors in arbitrary genus as well as for very ample divisors in genus 2 and 3. We also compute the intersection numbers $\lambda^9$ and $\lambda_{g-1}^3$ in genus 4. The latter number is…

Given an odd representation of the absolute Galois group of Q onto PGL(2,3) and a positive integer N, there exists a twisted modular curve defined over Q whose rational points classify the quadratic Q-curves of degree N realizing the…

Let $X$ be any smooth simply connected projective surface. We consider some moduli space of pure sheaves of dimension one on $X$, i.e. $\mhu$ with $u=(0,L,\chi(u)=0)$ and $L$ an effective line bundle on $X$, together with a series of…

Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with…

In this paper we build bridges between moduli theory of sheaf stable pairs on one hand and birational geometry on the other hand. We will in particular treat moduli of sheaf stable pairs on smooth projective curves in detail and present…

We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic…

We explicitly compute the moduli space pointed algebraic curves with a given numerical semigroup as Weierstrass semigroup for many cases of genus at most seven and determine the dimension for all semigroups of genus seven.

We analyze Weierstrass cycles and tautological rings in moduli space of smooth algebraic curves and in moduli spaces of integral algebraic curves with embedded disks with special attention to moduli spaces of curves having genus $\leq 6$.…

For affine algebraic plane curves we reduce a calculation of its invariants to calculation of the intersection of kernels of some derivations.

Let $N>1$ be an integer coprime to $6$ such that $N\notin\{5,7,13\}$ and let $g=g(N)$ be the genus of the modular curve $X_0(N)$. We compute the intersection matrices relative to special fibres of the minimal regular model of $X_0(N)$.…