Related papers: Bielliptic intermediate modular curves

We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic…

We prove that there exist infinitely many elliptic curves over \Q with given modular invariant, and rank >=2. Furthermore, there exist infinitely many elliptic curves over $\Q$ with invariant equal at 0 (resp. 1728) and rank >=6 (resp.…

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…

In this paper we compute the gonality over Q of the modular curve X1(N) for all N <= 40 and give upper bounds for each N <= 250. This allows us to determine all N for which X1(N) has infinitely points of degree <= 8. We conjecture that the…

We say a closed point $x$ on a curve $C$ is sporadic if there are only finitely many points on $C$ of degree at most deg$(x)$. In the case where $C$ is the modular curve $X_1(N)$, most known examples of sporadic points come from elliptic…

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…

Bruin--Najman and Ozman--Siksek have recently determined the quadratic points on all modular curves $X_0(N)$ of genus 2, 3, 4, and 5 whose Mordell--Weil group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3, 4, and 5…

We prove some results on algebraic curves $X$ of genus $g\geq 2$ in characteristic $0$. For example: Assume that $X$ has an automorphism $\sigma$ of prime order $p\geq 5$. If $\sigma$ has no fixed points, then $X$ cannot be trigonal. On the…

We determine all modular curves $X_0(N)$ with density degree $5$, i.e. all curves $X_0(N)$ with infinitely many points of degree $5$ and only finitely many points of degree $d\leq4$. As a consequence, the problem of determining all curves…

We determine all complex hyperelliptic curves with many automorphisms and decide which of their jacobians have complex multiplication.

We show that if p is a prime, then all elliptic curves defined over the cyclotomic Z_p extension of Q are modular.

Among all the dynamical modular curves associated to quadratic polynomial maps, we determine which curves have infinitely many quadratic points. This yields a classification statement on preperiodic points for quadratic polynomials over…

For a square-free integer $N$, we present a procedure to compute $\mathbb{Q}$-curves parametrized by rational points of the modular curve $X_0^*(N)$ when this is hyperelliptic.

In this long survey article we show that the theory of elliptic and hyperelliptic curves can be extended naturally to all superelliptic curves. We focus on automorphism groups, stratification of the moduli space $\mathcal{M}_g$, binary…

For each open subgroup $G$ of ${\rm GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the…

We study the automorphism groups of the reduction $X_0(N) \times \bar{\mathbb{F}}_p$ of a modular curve $X_0(N)$ over primes $ p\nmid N$.

The modular degree m_E of an elliptic curve E/Q is the minimal degree of any surjective morphism X_0(N) -> E, where N is the conductor of E. We give a necessarily set of criteria for m_E to be odd. Specializing to N prime our results imply…

In this paper we construct parameterizations of elliptic curves over the rationals which have many consecutive integral multiples. Using these parameterizations, we perform searches in GMP and Magma to find curves with points of small…

We work towards completely classifying all bielliptic Shimura curves $X_0^D(N)$ with nontrivial level $N$ coprime to $D$, extending a result of Rotger that provided such a classification for level one. Combined with prior work, this allows…

Let X be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two. If X admits a nontrivial automorphism \sigma that fixes pointwise all the order two…