Related papers: Tetragonal intermediate modular curves

In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves $X_0(N)$ of genus up to $8$, and genus up to $10$ with $N$ prime,…

In this paper, we determine all tetraelliptic modular curves $X_1(N)$ over $\mathbb Q$, and find some tetraelliptic maps $\phi_N$ from $X_1(N)$ to elliptic curves for those tetraelliptic $X_1(N)$. Also we will construct $\phi_N$ explicitly…

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.

In this paper we compute the degree of a curve which is the image of a mapping $z\longmapsto (f(z): g(z): h(z))$ constructed out of three linearly independent modular forms of the same even weight $\ge 4$ into $\mathbb P^2$. We prove that…

Associated to an open subgroup $G$ of $\GL_2(\Zhat)$ satisfying conditions $-I \in G$ and $\det(G) \subsetneq (\Zhat)^{\times}$ there is a modular curve $X_G$ which is a smooth compact curve defined over an extension of $\Q.$ In this…

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 $(N,W_N)$…

Let $\mathcal N$ be the moduli space of sextics with 3 (3,4)-cusps. The quotient moduli space ${\mathcal N}/G$ is one-dimensional and consists of two components, ${\mathcal N}_{torus}/G$ and ${\mathcal N}_{gen}/G$. By quadratic…

We determine the quadratic points on the modular curves $X_0(N)$ for $N\leq 100$ for which this has not been previously done, namely the cases $$N\in\{66,70,78,82,84,86,87,88,90,96,99\}.$$ We accomplish this by improving on the ``going down…

We determine all modular curves $X_0^+(N)$ that admit infinitely many cubic points over the rational field $\mathbb{Q}$.

Mumford and Newstead generalized the classical Torelli theorem to higher rank i.e., a smooth, projective curve $X$ is uniquely determined by the second intermediate Jacobian of the moduli space of stable rank $2$ bundles on $X$, with fixed…

We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined…

We determine all integers $N$ for which the modular curve $X_0(N)$ admits a sporadic CM point (of any degree), as well as all $N$ for which $X_0(N)$ admits a sporadic point, whether CM or non-CM. In a sense, our results generalize the…

Let $X$ be a smooth irreducible projective curve of genus $g$ and gonality 4. We show that the canonical model of $X$ is contained in a uniquely defined surface, ruled by conics, whose geometry is deeply related to that of $X$. This surface…

Answering a question of Zureick-Brown, we determine the cubic points on the modular curves $X_0(N)$ for $N \in \{53,57,61,65,67,73\}$ as well as the quartic points on $X_0(65)$. To do so, we develop a "partially relative" symmetric Chabauty…

Infinitely many elliptic curves over ${\bf Q}$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let $N_i(X)$ denote the number of elliptic curves over ${\bf Q}$…

Bruin and Najman, Ozman and Siksek, and Box described all the quadratic points on the modular curves of genus $2\leq g(X_0(n)) \leq 5$. Since all the hyperelliptic curves $X_0(n)$ are of genus $\leq 5$ and as a curve can have infinitely…

A curve $C$ defined over $\mathbb Q$ is modular of level $N$ if there exists a non-constant morphism from $X_1(N)$ onto $C$ defined over $\mathbb Q$ for some positive integer $N$. We provide a sufficient and necessary condition for the…

We obtain the modular automorphism group of any quotient modular curve of level $N$, with $4,9\nmid N$. In particular, we obtain some non-expected automorphisms of order 3 that appear for the quotient modular curves when the Atkin-Lehner…

We completely determine the $1085$ open subgroups $H$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ of prime-power level that satisfy $-I \in H$ and $\operatorname{det}(H)=\widehat{\mathbb{Z}}^{\times}$ for which the corresponding modular…

From the Modularity Theorem proven by Wiles, Taylor, et al, we know that all elliptic curves are modular. It has been shown by Martin and Ono exactly which are represented by eta-quotients, and some examples of elliptic curves represented…