相关论文: Quadratic Chabauty and rational points II: General…

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We…

We develop a theory of general sheaves over weighted projective lines. We define and study a canonical decomposition, analogous to Kac's canonical decomposition for representations of quivers, study subsheaves of a general sheaf, general…

We use a global version of Heath-Brown's $p-$adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most $B$ on non-singular cubic curves defined over $\mathbb{Q}$. The bounds are…

We prove a lower bound on the Calabi functional for degenerations of polarized varieties, involving the difference of CM degrees between generically isomorphic families. This may be viewed as a discretely valued version of Donaldson's lower…

We use height arguments to prove two results about the dynamical Mordell-Lang problem. (i) For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov multiplier of any…

We prove a certain transcendence property of the unipotent Albanese map of a smooth variety, conditional on the Ax-Schanuel conjecture for variations of mixed Hodge structure. We show that this property allows the Chabauty-Kim method to be…

We prove a level raising mod $\ell=2$ theorem for elliptic curves over $\mathbb{Q}$. It generalizes theorems of Ribet and Diamond-Taylor and also explains different sign phenomena compared to odd $\ell$. We use it to study the 2-Selmer…

We generalize the work of Dem'janenko and Silverman for the Fermat quartics, effectively determining the rational points on the curves $x^{2m}+ax^m+ay^m+y^{2m}=b$ whenever the ranks of some companion hyperelliptic Jacobians are at most one.…

We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We…

Canonical heights and Arakelov geometry on semi-abelian varieties. In this paper, we propose a construction of the canonical heights on an extension of an abelian variety by the multiplicative group, in the framework of Arakelov geometry.…

Let $k$ be a field of characteristic $q$, $\cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(\cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$…

We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related…

In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties over the complex numbers of a fixed dimension…

We prove the finiteness of Selmer groups attached to lifts of certain 2-dimensional mod p representations of the absolute Galois group of Q. The mod p representation can be either even or odd. The lifts considered are the ones that were…

We study heights of motives with integral coefficients over number fields introduced by Kato. It is a generalization of the Faltings height of an abelian variety and we establish generalizations of some properties of the Faltings height in…

This paper introduces explicit Galois cohomological methods for determining the ranks of Bloch--Kato Selmer groups associated to the Tate twists of the 2-adic second \'etale cohomology of the Jacobian of a hyperelliptic curve with a…

Several recent papers construct auxiliary polynomials to bound the Weil height of certain classes of algebraic numbers from below. Following these techniques, the author gave a general method for introducing auxiliary polynomials to…

Let $E$ be an elliptic curve without complex multiplication defined over a number field $K$ which has at least one real embedding. The field $F$ generated by all torsion points of $E$ over $K$ is an infinite, non-abelian Galois extension of…

Let C : y^2=f(x) be a hyperelliptic curve defined over the rationals. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f=f_1 f_2...f_r. We shall define a "Selmer set" corresponding to this…

We develop refined methods to effectively bound the dimension of Bloch-Kato Selmer groups associated to the higher Chow group $\mathrm{CH}^2(J,1)$, where $J$ is the Jacobian of a hyperelliptic curve $X$. This extends the recent work of…