Related papers: Bielliptic intermediate modular curves

We develop an algorithm to test whether a non-CM elliptic curve $E/\mathbb{Q}$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing…

Let $K$ be a composite field of some real quadratic fields. We give a sufficient condition on $K$ such that all elliptic curves over $K$ is modular.

We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component…

Genus 5 curves can be hyperelliptic, trigonal, or non-hyperelliptic non-trigonal, whose model is a complete intersection of three quadrics in $\mathbb{P}^4$. We present and explain algorithms we used to determine, up to isomorphism over…

We will study modular Abelian varieties with odd congruence numbers, by studying the cuspidal subgroup of $J_0(N)$. We show the conductor of such Abelian varieties must be of a special type, for example if $N$ is odd then $N=p^\alpha$ or…

Let $p$ be a prime. We study non-constant morphisms $f:X_0(p)_\mathbb \to Y$, where $Y/\mathbb Q$ is a curve of genus $\geq 2$. We prove that for $p<3000$ such an $f$ of degree $d>1$ must be isomorphic to the quotient map $X_0(p)\to…

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…

We establish asymptotic lower bounds for the number of elliptic curves over $\mathbb{Q}$ with prescribed entanglement of division fields, ordered by naive height. Such elliptic curves are obtained as $1$-parameter families arising from…

The modularity theorem implies that for every elliptic curve $E /\mathbb{Q}$ there exist rational maps from the modular curve $X_0(N)$ to $E$, where $N$ is the conductor of $E$. These maps may be expressed in terms of pairs of modular…

We prove that there exist infinitely many pairs of non-isomorphic elliptic curves over $\mathbb{Q}$ sharing the same BSD invariants -- including their Mordell--Weil groups, Tate--Shafarevich groups, Tamagawa numbers, regulators, and real…

We describe an algorithm for computing a $\Q$-rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining $q$-expansions for a basis of the…

We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these…

For positive integers $M \mid N$ and an order of discriminant $\Delta$ in an imaginary quadratic field $K$ with discriminant $\Delta_K < -4$, we determine the fiber of the morphism $X_0(M,N) \rightarrow X(1)$ over the closed point…

We study connections between self-inversive and self-reciprocal polynomials, reduction theory of binary forms, minimal models of curves, and formally self-dual codes. We prove that if $\mathcal X$ is a superelliptic curve defined over…

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…

Let $E_1, E_2 / \mathbb{C}$ be non-isomorphic elliptic curves with complex multiplication. We prove that the pair $(E_1, E_2)$ is characterised, up to isomorphism, by the difference $j(E_1) - j(E_2)$ of the respective $j$-invariants. In…

We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular…

Let E be an elliptic curve over Q, and let n=>1. The central object of study of this article is the division field Q(E[n]) that results by adjoining to Q the coordinates of all n-torsion points on E(Q). In particular, we classify all curves…

We give a sharp bound on the number of automorphisms of a stable curve of a given genus and describe all curves attaining this bound.

Ihara's proof that the reduction of the modular curve $X_0(n)$ at a prime $p$ not dividing $n$ has many points over a quadratic extension is adapted to the drinfeld modular curves $X_0(n)$. In order to do so, some properties of drinfeld…