Related papers: Points on a curve with a power on a curve

Let $X$ be the product of two projective spaces and consider the general CICY threefold $Y$ in $X$ with configuration matrix $A$. We prove the finiteness part of the analogue of the Clemens' conjecture for such a CICY in low bidegrees. More…

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…

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,…

In this paper, the conjecture on the Zalcmanness of $\mathbb C^n \ (n\geq 2)$ and $(\mathbb C^*)^2$, which is posed in \cite{Do}, is proved in the case where the derivatives of limit holomorphic curves are bounded. Moreover, several…

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…

Given a prime power q, for every pair of positive integers m and n with m dividing the GCD of n and q-1, we construct a modular curve over F_q that parametrizes elliptic curves over F_q along with F_q-defined points P and Q of order m and…

We introduce deformation theoretic methods for determining when a curve $X$ in a non-hyperelliptic jacobian $JC$ will deform with $JC$ to a non-jacobian. We apply these methods to a particular class of curves in the second symmetric power…

Let C be a 2-connected Gorenstein curve either reduced or contained in a smooth algebraic surface and let S be a subcanonical cluster (i.e. a 0-dim scheme such that the space H^0(C, I_S K_C) contains a generically invertible section). Under…

Let $C\subset \mathbb {P}^n$ be a smooth curve and $N_C$ its normal bundle. $N_C$ satisfies strong interpolation if for all integers $s>0$ and $\lambda _i\in \{0,1,\dots ,n-1\}$, $1\le i \le s$, there are distinct points $P_1,\dots ,P_s\in…

Fix a hyperelliptic curve $C/\mathbb{Q}$ of genus $g$, and consider the number fields $K/\mathbb{Q}$ generated by the algebraic points of $C$. In this paper, we study the number of such extensions with fixed degree $n$ and discriminant…

There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free…

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 ${\bf Q}$…

Let $C_1$ be an irreducible component of a reduced projective curve $C\subset \mathbb P^2$ defined over the field $\mathbb C$, $\mathrm{deg} C_1\geq 2$, and let $T$ be the set of lines $l\subset \mathbb P^2$ meeting $C$ transversally. In…

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…

Let $E_1$ and $E_2$ be elliptic curves in Legendre form with integer parameters. We show there exists a constant $C$ such that for almost all primes, for all but at most $C$ pairs of points on the reduction of $E_1 \times E_2$ modulo $p$…

We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible…

We prove that every immersed $C^2$-curve $\gamma$ in $\mathbb R^n$, $n\geqslant 3$ with curvature $k_{\gamma}$ can be $C^1$-approximated by immersed $C^2$-curves having prescribed curvature $k>k_{\gamma}$. The approximating curves satisfy a…

Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on $\mathbb P^2$ and $V_{\alpha}(T)$ be the upper level sets of Lelong numbers $\nu(T,x)$ of $T$. For any $\alpha\geq \frac{1}{3}$, we show that…

In [DJL07] it was shown that if A is an affine hyperplane arrangement in C^n, then at most one of the L^2-Betti numbers of its complement is non--zero. We will prove an analogous statement for complements of any algebraic curve in C^2.…

Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one…