Related papers: Hyperelliptic and trigonal modular curves in chara…

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 $X_0(N)$.…

We study a tropical analogue of the projective dual variety of a hypersurface. When $X$ is a curve in $\mathbb{P}^2$ or a surface in $\mathbb{P}^3$, we provide an explicit description of $\text{Trop}(X^*)$ in terms of $\text{Trop}(X)$, as…

Let $p$ be an odd prime number and let $X_0^+(p)$ be the quotient of the classical modular curve $X_0(p)$ by the action of the Atkin-Lehner operator $w_p$. In this paper we show how to compute explicit equations for the canonical model of…

Consider a modular curve $X_G$ defined over a number field $K$, where $G$ is a subgroup of $GL_2(\mathbb{Z}/N\mathbb{Z})$ with $N>2$. The curve $X_G$ comes with a morphism $j: X_G\to \mathbb{P}^1_K=\mathbb{A}^1_K \cup\{\infty\}$ to the…

Let $p$ be a prime number, and let $\Delta_1,\Delta_2 < 0$ be two coprime fundamental discriminants. When $p$ splits in $\mathbb{Q}(\sqrt{\Delta_1})$ and $\mathbb{Q}(\sqrt{\Delta_2})$ the height pairings of the corresponding CM divisors on…

Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of…

We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n=7,9,11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, i.e. pairs of…

There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this note we characterize the j-invariants of elliptic…

Let $\ell$ and $p \geq 3$ be distinct prime numbers. Let $E/\mathbb{Q}_{\ell}$ be an elliptic curve with $p$-torsion module $E_p$. Let $\mathbb{Q}_{\ell}(E_p)$ be the $p$-torsion field of $E$. We provide a complete description of the degree…

Given a smooth curve X of genus g we compute de dimension of the family of curves C which have an involution over X. Moreover we distinguish when the curve C is hyperelliptic.

Infinitely many elliptic curves over ${\bf Q}$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let $N_i(X)$ denote the number of elliptic curves over ${\bf Q}$…

Let $E_1$ and $E_2$ be $\overline{\mathbb{Q}}$-nonisogenous, semistable elliptic curves over $\mathbb{Q}$, having respective conductors $N_{E_1}$ and $N_{E_2}$ and both without complex multiplication. For each prime $p$, denote by…

Given an elliptic curve $E$ defined over the rational numbers and a prime $p$ at which $E$ has good reduction, we consider the Galois deformation ring parametrizing lifts of the residual representation on the $p$-torsion group $E[p]$. For a…

Let $E/\mathbb{Q}$ be an elliptic curve that has complex multiplication (CM) by an imaginary quadratic field $K$. For a prime $p$, there exists $\theta_p \in [0, \pi]$ such that $p+1-\#E(\mathbb{F}_p) = 2\sqrt{p} \cos \theta_p$. Let $x>0$…

Let $p>3$ be a fixed prime. For a supersingular elliptic curve $E$ over $\mathbb{F}_p$ with $j$-invariant $j(E)\in \mathbb{F}_p\backslash\{0, 1728\}$, it is well known that the Frobenius map $\pi=((x,y)\mapsto (x^p, y^p))\in…

We consider generalizations of Szpiro's classical discriminant conjecture to hyperelliptic curves over a number field $K$, and to smooth, projective and geometrically connected curves $X$ over $K$ of genus at least one. The main results…

For the pure $\psi$-class intersection numbers $D(\textbf{e})=\langle \tau_{e_1} \cdots \tau_{e_n} \rangle_g$ on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves, we determine for which choices of $\textbf{e}=(e_1, \ldots,…

In this study, we determine all modular curves $X_0(N)$ that admit infinitely many cubic points.

Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower…

Looking at the finite \'etale congruence covers $X(p)$ of a complex algebraic variety $X$ equipped with a variation of integral polarized Hodge structures whose period map is quasi-finite, we show that both the minimal gonality among all…