Related papers: Quadratic points on $X_0(163)$
We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a…
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…
For Atkin-Lehner quotients $X_0^+(N)$, of prime level and of genus at least 2, we provide an algorithm for computing one of the main objects in the quadratic Chabauty algorithm in terms of weakly holomorphic modular forms associated to the…
We study the splitting of primes in number fields generated by points on modular curves. Momose was the first to notice that quadratic points on $X_1(n)$ generate quadratic fields over which certain primes split in a particular way and his…
In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian…
We describe an algorithm to compute the number of points over finite fields on a broad class of modular curves: we consider quotients $X_H/W$ for $H$ a subgroup of $\GL_2(\mathbb Z/n\mathbb Z)$ such that for each prime $p$ dividing $n$, the…
In this article, we present a method for computing rational points on hyperelliptic curves of genus~3 and isolated quadratic points on hyperelliptic curves of genus~2 and~3 whose Jacobians have rank~0. Our approach begins by computing the…
We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces $A_f$ with quaternionic multiplication attached to a normalized newform $f$ without complex multiplication. We include…
We determine the Shimura modular curve X_0(3) and the Jacobian of the Shimura modular curve X_1(3) associated with the congruence subgroups Gamma_0(3), Gamma_1(3) of the (2,3,7) triangle group. This group is known to be arithmetic and…
We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the…
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 $n$.…
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…
A central problem in arithmetic geometry is to construct non-torsion rational points on elliptic curves. We study a canonical quadratic point $\xi_C \in {\rm Jac}(C)$ attached to a smooth non-hyperelliptic curve of genus 4 and use it to…
An isolated point on an algebraic curve is a closed point not belonging to a collection of points of the same degree parametrized by $\mathbf{P}^1$ or a positive rank abelian subvariety of the curve's Jacobian. We study the sets of…
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…
We present the geometry lying behind counting twin prime polynomials in $\mathbb{F}_q[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties…
We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of F-rational points. In the case of the Jacobian, we show that any…
It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…
For small odd primes $p$, we prove that most of the rational points on the modular curve $X_0(p)/w_p$ parametrize pairs of elliptic curves having infinitely many supersingular primes. This result extends the class of elliptic curves for…
Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give…