相关论文: Bounding the $j$-invariant of integral points on c…

In this paper, we give some effective bounds for the $j$-invariant of integral points on arbitrary modular curves over arbitrary number fields assuming that the number of cusps is not less than 3.

We bound the j -invariant of integral points on a modular curve in terms of the congruence group defining the curve. We apply this to prove that the modular curve Xsplit (p3) has no non-trivial rational point if p is a sufficiently large…

For prime $p\ge 7$, by using Baker's method we obtain two explicit bounds in terms of $p$ for the $j$-invariant of an integral point on $X_{\ns}^{+}(p)$ which is the modular curve of level $p$ corresponding to the normalizer of a non-split…

We bound the j-invariant of S-integral points on arbitrary modular curves over arbitrary fields, in terms of the congruence group defining the curve, assuming a certain Runge condition is satisfied by our objects. We then apply our bounds…

In this paper, we give a constant $C$ in \cite[Theorem 1.2]{sha2014bounding} by using an explicit Baker's inequality, hence we have an explicit bound of the integral points on modular curves.

For a fixed $j$-invariant $j_0$ of an elliptic curve without complex multiplication we bound the number of $j$-invariants $j$ that are algebraic units and such that elliptic curves corresponding to $j$ and $j_0$ are isogenous. Our bounds…

For every $n\geq 3$, we exhibit infinitely many extremal effective divisors on the moduli space of genus one curves with $n$ marked points.

In this paper we determine the quadratic points on the modular curves X_0(N), where the curve is non-hyperelliptic, the genus is 3, 4 or 5, and the Mordell--Weil group of J_0(N) is finite. The values of N are 34, 38, 42, 44, 45, 51, 52, 54,…

We prove that integral points can be effectively determined on all but finitely many modular curves, and on all but one modular curve of prime power level.

We define a computable topological invariant $\mu(\gamma)$ for generic closed planar regular curves $\gamma$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we…

In this article, we study how to compute the number of $K$-rational points with a given $j$-invariant on an arbitrary modular curve. As an application, for each positive integer $n$, we determine the list of possible numbers of cyclic…

We give effective upper bounds for the number of purely inseparable points on non isotrivial curves over function fields of positive characteristic and of transcendence degree one. These bounds depend on the genus of the curve, the genus of…

We give a classification of the degrees of the points with rational $j$-invariant on the modular curves $X_{0}(n)$ and $X_{1}(n)$. The degrees which occur infinitely often are computed unconditionally, while those which occur finitely often…

We consider families of smooth projective curves of genus 2 with a single point removed and study their integral points. We show that in many such families there is a dense set of fibres for which the integral points can be effectively…

In this paper we describe a method for computing a basis for the space of weight $2$ cusp forms invariant under a non-split Cartan subgroup of prime level $p$. As an application we compute, for certain small values of $p$, explicit…

We study a 3-dimensional stratum $\mathcal{M}_{3,V}$ of the moduli space $\mathcal{M}_3$ of curves of genus $3$ parameterizing curves $Y$ that admit a certain action of $V= C_2\times C_2$. We determine the possible types of the stable…

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 transform the positive-genus real Gromov-Witten invariants of many real-orientable symplectic threefolds into signed counts of curves. These integer invariants provide lower bounds for counts of real curves of a given genus that pass…

The slope of the moduli space of genus g curves is bounded from below by 60/(g+4) via a descendent calculation.

In this article, we construct the reduced genus-two Gromov-Witten invariants for certain almost K\"{a}hler manifold $(X, \omega, J)$ such that $J$ is integrable and satisfies some regularity conditions. In particular, the standard…