Related papers: Computing minimal Weierstrass equations

We construct unramified central simple algebras representing 2-torsion classes in the Brauer group of a hyperelliptic curve, and show that every 2-torsion class can be constructed this way when the curve has a rational Weierstrass point or…

We extend the map Exp for elliptic curves in short Weierstrass form over $ \mathbb{C} $ to Edwards curves over local fields. Subsequently, we compute the map Exp for Edwards curves over the local field $ \mathbb{Q}_{p} $ of $ p $-adic…

We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\mathbf{Q}$-curves in certain cases. This generalizes earlier…

We prove that any minimal (maximal) strongly regular surface in the three-dimensional Minkowski space locally admits canonical principal parameters. Using this result, we find a canonical representation of minimal strongly regular time-like…

We present a novel method for efficiently computing optimal transport maps and Wasserstein barycenters in high-dimensional spaces. Our approach uses conditional normalizing flows to approximate the input distributions as invertible…

In this work we use elementary methods to discuss the question of the minimal number of points with bad reduction over the projective line for elliptic curves E/k(T) which are non-constant resp. have non-constant j-invariant.

We describe a minimal set of generators of the defining ideal of the Rees Algebra associated to a proper parametrization of any monoid hypersurface. In the case of plane curves, we recover a known description for rational parametrizations…

Let $\mathcal X_g$ be a genus $g\geq 2$ superelliptic curve, $F$ its field of moduli, and $K$ the minimal field of definition. In this short note we construct an equation of the curve $\mathcal X_g$ over its minimal field of definition $K$…

We determine the probability that a random Weierstrass equation with coefficients in the $p$-adic integers defines an elliptic curve with a non-trivial $3$-torsion point, or with a degree $3$ isogeny, defined over the field of $p$-adic…

The Weierstrass curve $X$ is a smooth algebraic curve determined by the Weierstrass canonical form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\cdots + A_{r-1}(x) y + A_{r}(x)=0$, where $r$ is a positive integer, and each $A_j$ is a…

The Weierstrass curve is a pointed curve $(X,\infty)$ with a numerical semigroup $H_X$, which is a normalization of the curve given by the Weierstrass canonical form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\dots + A_{r-1}(x) y +…

We give a completely explicit upper bound for integral points on (standard) affine models of hyperelliptic curves, provided we know at least one rational point and a Mordell-Weil basis of the Jacobian. We also explain a powerful refinement…

Several problems which could be thought of as belonging to recreational mathematics are described. They are all such that solutions to the problem depend on finding rational points on elliptic curves. Many of the problems considered lead to…

We provide an efficient algorithm to compute the minimum area of a homotopy between two closed plane curves, given that they divide the plane into finite number of regions. For any positive real number $\varepsilon>0$, we construct a closed…

We study the normal map for plane projective curves, i.e., the map associating to every regular point of the curve the normal line at the point in the dual space. We first observe that the normal map is always birational and then we use…

In the 80's D. Eisenbud and J. Harris posed the following question: "What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type?" We answer their question for one-dimensional families…

We investigate minimal surfaces passing a given curve in $R^{3}$. Using the Frenet frame of a given curve and isothermal parameter, we derive the necessary and sufficient condition for minimal surface. Also we derive the parametric…

In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or…

We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we have proved the formula for 16714…

We prove upper bounds on the number of rational points on transcendental curves in arbitrary $1$-h-minimal fields, similar to the Pila--Wilkie counting theorem in the o-minimal setting. These results extend results due to…