Related papers: Hyperelliptic and trigonal modular curves in chara…

Hyperbolism of a given curve with respect to a point and a line is an interesting construct, a special kind of geometric locus, not frequent in the literature. While networking between two different kinds of mathematical software, we…

This paper gives a criterion for a moduli point to be a point of non-transversal intersection of the hyperelliptic locus and the supersingular locus in the Siegel moduli stack $\mathfrak{A}_3 \times \mathbb{F}_p$. It is shown that for…

We study families of n-gonal curves with maximal variation of moduli, which have a rational section. Certain numerical results on the degree of the modular map are obtained for such families of hyperelliptic and trigonal curves. In the last…

The full automorphism group of a certain elementary abelian $p$-cover of the Hermitian curve in characteristic $p>0$ is determined. It is remarkable that the order of Sylow $p$-groups of the automorphism group is close to Nakajima's bound…

The modular curve X_0(N) parametrizes elliptic curves together with a cyclic subgroup of order N, and hence cyclic N-isogenies. While explicit moduli descriptions of X_1(N) are well developed, a comparable construction for X_0(N) has…

We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular…

Let $N$ be a positive integer. For every $d\mid N$ such that $(d,N/d)=1$ there exists an Atkin-Lehner involution $w_d$ of the modular curve $X_0(N)$. The curve $X_0^*(N)$ is a quotient curve of $X_0(N)$ by $B(N)$, the group of all…

In this paper, we explicitly classify the minimal discriminants of all elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion subgroup. This is done by considering various parameterized families of elliptic curves with the property that…

We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $\nu(D)\neq 1$,…

In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by…

Let $K_i$ be a number field for all $i \in \mathbb{Z}_{> 0}$ and let $\mathcal{E}$ be a family of elliptic curves containing infinitely many members defined over $K_i$ for all $i$. Fix a rational prime $p$. We give sufficient conditions for…

We give regular models for modular curves associated with (normalizer of) split and non-split Cartan subgroups of ${\mathrm{GL}}_2 ({\mathbb F}_p )$ (for $p$ any prime, $p\ge 5$). We then compute the group of connected components of the…

In this paper I prove that for any prime $p$ there is a constant $C_p>0$ such that for any $n>0$ and for any $p$-power $q$ there is a smooth, projective, absolutely irreducible curve over $\mathbb{F}_q$ of genus $g\leq C_p q^n$ without…

We produce an integral model for the modular curve $X(Np^m)$ over the ring of integers of a sufficiently ramified extension of $\mathbf{Z}_p$ whose special fiber is a {\em semistable curve} in the sense that its only singularities are…

We develop a strategy for bounding from above the height of rational points of modular curves with values in number fields, by functions which are polynomial in the curve's level. Our main technical tools come from effective Arakelov…

Let $C$ be a non-hyperelliptic algebraic curve. It is known that its canonical image is the intersection of the quadrics that contain it, except when $C$ is trigonal (that is, it has a linear system of degree 3 and dimension 1) or…

Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of $A_s$ should contain an element of order $p$ for a positive proportion of…

By means of an {\it ad hoc} modification of the so-called ``Castelnuovo-Harris analysis" we derive an upper bound for the genus of integral curves on the three dimensional nonsingular quadric which lie on an integral surface of degree $2k$,…

We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p > 2. Using this, we prove that the Z/\ell-monodromy of every irreducible component of the stratum…

Modular Galois representations into GL_2(F_p) with cyclotomic determinant arise from elliptic curves for p = 2,3,5. We show (by constructing explicit examples) that such elliptic curves cannot be chosen to have conductor as small as…