Related papers: On the arithmetic of certain modular curves

For a fixed odd prime p and a representation \rho of the absolute Galois group of Q into the projective group PGL(2,p), we provide the twisted modular curves whose rational points supply the quadratic Q-curves of degree N prime to p that…

We study intersection theory on the relative Hilbert scheme of a family of nodal-or-smooth curves, over a base of arbitrary dimension. We introduce an additive group called 'discriminant module', generated by diagonal loci, node scrolls,…

We construct normalized differentials on families of curves of infinite genus. Such curves are used to investigate integrable PDE's such as the focusing Nonlinear Schr{\"o}dinger equation.

After recalling several constructions of the moduli space of curves of genus zero by different people we give our alternative construction of the moduli space. This gives a simple description of the intersection ring of this space. We give…

Let $L_1,\dots,L_s$ be line bundles on a smooth variety $X\subset \mathbb{P}^r$ and let $D_1,\dots,D_s$ be divisors on $X$ such that $D_i$ represents $L_i$. We give a probabilistic algorithm for computing the degree of intersections of…

We compute the Picard group of the moduli stack of smooth curves of genus $g$ for $3\leq g\leq 5$, using methods of equivariant intersection theory. We base our proof on the computation of some relations in the integral Chow ring of certain…

We study upper bounds on the order of automorphisms of non-singular curves $X$ satisfying at least one of the following hypothesis: 1) $X$ is an $m$-sheeted covering of exactly one non-singular curve of genus $\gamma$, where $m$ is prime;…

We compute all the top intersection numbers of divisors on the total space of the Poincare bundle restricted to the product of a curve and the abelian variety. We use these computations to find the class of the universal theta divisor and…

We compute the first, second, third, and fifth rational cohomology groups of the moduli space of stable n-pointed genus g curves, for all g and n, using (mostly) algebro-geometric techniques.

Let $E$ be a non-CM elliptic curve defined over $\mathbb {Q}$. Fix an algebraic closure $\overline{\mathbb {Q}}$ of $\mathbb {Q}$. We get a Galois representation \[\rho_E \colon Gal(\overline{\mathbb {Q}}/\mathbb {Q}) \to GL_2(\hat{\mathbb…

For a complex elliptic curve $E$ and a point $p$ of order $n$ on it, the images of the points $p_k=kp$ under the Weierstrass embedding of $E$ into $\mathbb{C}\mathbb{P}^2$ are collinear if and only if the sum of indices is divisible by $n$.…

In this paper, we study the Heegner points on more general modular curves other than $X_0(N)$, which generalizes Gross' work "Heegner points on $X_0(N)$". The explicit Gross-Zagier formula and the Euler system property are stated in this…

We study crossing numbers for systoles of congruence surfaces. Taken as a family of curves on a family of surfaces, we show that the growth rate of their intersection is optimally small among all sets of curves of the same cardinality lying…

We give a criterion for a continuous family of curves on a nodal $K$-trivial threefold $X_0$ to contribute geometrically rigid curves to a general smoothing of $X_0$. As an application, we prove the existence of geometrically rigid curves…

The goal in this preprint is to give an efficient algorithm to compute Puiseux expansions at cusps of X0(N). It is based on a relation with a hypergeometric function that holds for any N.

The moduli spaces of trigonal curves are proven to be rational when the genus is divisible by 4.

We study the automorphism groups of the reduction $X_0(N) \times \bar{\mathbb{F}}_p$ of a modular curve $X_0(N)$ over primes $ p\nmid N$.

We compute the rational Chow ring of the moduli stack of planar nodal curves of fixed degree and express it in terms of tautological classes. Along the way, we extend Vial's results on Chow groups of Brauer-Severi varieties to…

We present a Mordell-Weil sieve that can be used to compute points on certain bielliptic modular curves $X_0(N)$ over fixed quadratic fields. We study $X_0(N)(\mathbb{Q}(\sqrt{d}))$ for $N \in \{ 53,61,65,79,83,89,101,131 \}$ and $\lvert d…

We determine the splitting (isomorphism) type of the normal bundle of a generic genus-0 curve with 1 or 2 components in any projective space, as well as the (sometimes nontrivial) way the bundle deforms locally with a general deformation of…