Daeyeol Jeon, Chang Heon Kim, Andreas Schweizer
We determine which of the modular curves $X_Δ(N)$, that is, curves lying between $X_0(N)$ and $X_1(N)$, are bielliptic. Somewhat surprisingly, we find that one of these curves has exceptional automorphisms. Finally we find all $X_Δ(N)$ that have infinitely many quadratic points over $\mathbb{Q}$.
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We determine all modular curves X(N) (with $N\geq 7$) which are hyperelliptic or bielliptic. We make available a proof that the automorphism group of X(N) coincides with the normalizer of $Γ(N)$ in $\operatorname{PSL}_2(\mathbb{R})$.
Let $N\geq 1$ be a square-free integer such that the modular curve $X_0^*(N)$ has genus $\geq 2$. We prove that $X_0^*(N)$ is bielliptic exactly for $19$ values of $N$, and we determine the automorphism group of these bielliptic curves. In particular, we obtain the first examples of nontrivial $Aut(X_0^*(N))$ when the genus of $X_0^*(N)$ is $\geq 3$. Moreover, we prove that the set of all quadratic points over $\mathbb{Q}$ for the modular curve $X_0^*(N)$ with genus $\geq 2$ and $N$ square-free is not finite exactly for $51$ values of $N$.
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 $(N,W_N)$ such that $X_0(N)/W_N$ is a bielliptic curve and the pairs $(N,W_N)$ such that $X_0(N)/W_N$ has an infinite number of quadratic points over $\mathbb{Q}$.
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.
Computing points on bielliptic modular curves over fixed quadratic fields
本文提出 Mordell-Weil 筛，用于计算固定二次域上若干双椭圆模曲线 $X_0(N)$ 的点，并考察八个水平及 $|d|<100$ 的二次域。
Quadratic Chabauty Experiments on Genus 2 Bielliptic Modular Curves in the LMFDB
本文对LMFDB中亏格2双椭圆模曲线进行二次Chabauty实验，在有理数和二次虚域上应用方法，发现代数无理点，并提出模曲线水平与潜在数域关系的猜想。
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 many quadratic points only if it is either of genus $\leq 1$, hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves $X_0(n)$ naturally arises; this question has recently also been posed by Mazur. We answer Mazur's question completely and describe the quadratic points on all the bielliptic modular curves $X…
Classical modular curves are of deep interest in arithmetic geometry. In this survey we show how the work of Fumiyuki Momose is involved in order to list the classical modular curves which satisfy that the set of quadratic points over $\mathbb{Q}$ is infinite. In particular we recall results of Momose on hyperelliptic modular curves and on automorphisms groups of modular curves. Moreover, we fix some inaccuracies of the existing literature in few statements concerning automorphism groups of modular curves and we make available different results that are difficult to find a precise reference, f…
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 classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this …
Modular curves of prime-power level with infinitely many quadratic points
该研究确定了1085个素数幂水平的模曲线子群，这些子群对应的模曲线具有无限多个二次点。
Intermediate modular curves with infinitely many cubic points over $\mathbb{Q}$
本文确定了所有在有理数域$\mathbb{Q}$上具有无穷多个三次点的中间模曲线$X_Δ(N)$。
Intermediate modular curves with infinitely many quartic points
本文确定了所有在有理数域上具有无穷多个四次点的中间模曲线 $X_Δ(N)$，并开发了计算其到椭圆曲线的有理态射次数的方法。
Tetragonal intermediate modular curves
本文确定了中间模曲线 $X_Δ(N)$ 在 $\mathbb{Q}$-亏格为4、$\mathbb{C}$-亏格为4和 $\mathbb{Q}$-亏格为5时的所有情形，并给出了 $N\leq 40$ 时所有非平凡 $Δ$ 的 $\mathbb{Q}$-亏格。
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.
Consider three normalised cuspidal eigenforms of weight $2$ and prime level $p$. Under the assumption that the global root number of the associated triple product $L$-function is $+1$, we prove that the complex Abel-Jacobi image of the modified diagonal cycle of Gross-Kudla-Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel-Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow-Heegner points associat…
In this article, we examine the arithmetic aspect of the Kummer-surface-type CY 3-folds $\hat{T/G}$, characterized by the crepant resolution of 3-torus-orbifold $T/G$ with only isolated singularities. Up to isomorphisms, there are only two such space $\hat{T/G}$ with $|G|=3, 7$, and both $T$ carrying the structure of triple-product structure of a CM elliptic curve. The (Griffiths) intermediate Jacobians of these $\hat{T/G}$ are identified explicitly as the corresponding elliptic curve appeared in the structure of $T$. We further provide the $\QZ$-structure of $\hat{T/G}$ and verify their modul…
We consider certain elliptic modular graph functions that arise in the asymptotic expansion around the non--separating node of genus two string invariants that appear in the integrand of the $D^8 R^4$ interaction in the low momentum expansion of the four graviton amplitude in type II superstring theory. These elliptic modular graphs have links given by the Green function, as well its holomorphic and anti--holomorphic derivatives. Using appropriate auxiliary graphs at various intermediate stages of the analysis, we show that each graph can be expressed solely in terms of graphs with links given…
Hyperelliptic and trigonal modular curves in characteristic $p$
针对N级中间模曲线X_Δ(N)，研究其在特征p下为超椭圆或三角曲线的素数p的确定问题，通过Castelnuovo-Severi不等式建立N的界并研究典范理想明确满足条件的p，还证明其不会是光滑平面五次曲线。
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