Related papers: Defining equations of $X_0(2^{2n})$

We give a procedure to determine equations for the modular curves $X_0(N)$ which are bielliptic and equations for the 30 values of $N$ such that $X_0(N)$ is bielliptic and nonhyperelliptic are presented.

Modular curves like X_0(N) and X_1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL_2(Z), they allow for a more arithmetic description as…

We determine all modular curves $X_0(N)$ with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from $X_0(N)$ to a positive rank elliptic…

We study the real components of modular curves. Our main result is an abstract group-theoretic description of the real components of a modular curve defined by a congruence subgroup of level N in terms of the corresponding subgroup of…

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…

The 2x2 space-filling curve is a type of generalized space-filling curve characterized by a basic unit is in a "U-shape" that traverses a 2x2 grid. In this work, we propose a universal framework for constructing general 2x2 curves where…

Let $N\geq 1$ be a non-square free integer and let $W_N$ be a non-trivial subgroup of the group of the Atkin-Lehner involutions of $X_0(N)$ such that the modular curve $X_0(N)/W_N$ has genus at least two. We determine all pairs $(N,W_N)$…

This note gives explicit equations for the elliptic curves (in characteristic not 2 or 3) with mod 2 representation isomorphic to that of a given one.

In this work, we estimate the genus of the intermediate modular curves between $X_1(N)$ and $X_0(N),$ and determine all the trigonal ones.

For each open subgroup $H\leq \operatorname{GL}_2(\widehat{\mathbb{Z}})$, there is a modular curve $X_H$, defined as a quotient of the full modular curve $X(N)$, where $N$ is the level of $H$. The genus formula of a modular curve is well…

In the classical setting, the modular equation of level $N$ for the modular curve $X_0(1)$ is the polynomial relation satisfied by $j(\tau)$ and $j(N\tau)$, where $j(\tau)$ is the standard elliptic $j$-function. In this paper, we will…

We determine conditions that guarantee that a hyperelliptic or plane curve over a field of characteristic not equal to 2 can be defined over its field of moduli. We also give new examples of curves not definable over their fields of moduli.

We give recursive formulas for the generating elements in the Milnor basis of the mod 2 motivic Steenrod algebra.

We construct plane models of the modular curve $X_H(\ell)$, and use their explicit equations to compute Galois representations associated to modular forms for values of $\ell$ that are significantly higher than in prior works.

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…

Quadratic functions have applications in cryptography. In this paper, we investigate the modular quadratic equation $$ ax^2+bx+c=0 \quad (mod \,\, 2^n), $$ and provide a complete analysis of it. More precisely, we determine when this…

In this paper, we will construct the projective resolution of any $\cR$-2-module, define the derived 2-functor and give some related properties of the derived 2-functor.

Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of…

We study the special value at 2 of L-functions of modular forms of weight 2 on congruence subgroups of the modular group. We prove an explicit version of Beilinson's theorem for the modular curve X_1(N). When N is prime, we deduce that the…

We provide an explicit description of two torsion points on the classical Bianchi elliptic quintic curve in terms of Ramanujan's functions. As a byproduct, we describe generators and defining equations of several modular function fields of…