相关论文: Explicit arithmetic of Jacobians of generalized Le…

Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/\mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian…

We study abelian varieties defined over function fields of curves in positive characteristic $p$, focusing on their arithmetic within the system of Artin-Schreier extensions. First, we prove that the $L$-function of such an abelian variety…

Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ over the field $\Fp(t)$ where $p$ is an odd prime. We study the arithmetic of $E$ over extensions $\Fq(t^{1/d})$ where $q$ is a power of $p$ and $d$ is an integer prime to $p$. The rank of $E$…

Consider a genus 2 curve defined over $\mathbb{Q}$ given by an affine equation of the form $y^2 = f(x)$ for some polynomial $f$ of degree 5, and let $p$ be an odd prime. Extending work of Perrin-Riou for elliptic curves, we construct a…

We study an explicit $(2g-1)$-dimensional family of Jacobian varieties of dimension $\frac{d-1}2(g-1)$, arising from quotient curves of unramified cyclic coverings of prime degree $d$ of hyperelliptic curves of genus $g\ge 2$. By using a…

For a prime $\mathfrak{p} \subseteq \mathbb{F}_{q}[T]$ and a positive integer $r$, we consider the generalised Jacobian $J_{0}(\mathfrak{n})_{\mathbf{m}}$ of the Drinfeld modular curve $X_{0}(\mathfrak{n})$ of level…

We study the family of algebraic curves of genus $\geq 1$ defined by the affine equations $y^s=ax^r+b$ over a number field $k$, where $r \geq 2$ and $s\geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture on varieties…

We study the elliptic curve E given by y^2=x(x+1)(x+t) over the rational function field k(t) and its extensions K_d=k(\mu_d,t^{1/d}). When k is finite of characteristic p and d=p^f+1, we write down explicit points on E and show by…

The Birch and Swinnerton-Dyer conjecture predicts that the parity of the algebraic rank of an abelian variety over a global field should be controlled by the expected sign of the functional equation of its $L$-function, known as the global…

Given a smooth projective curve $C$ of genus $g$ over the complex numbers, Torelli's thoerem asserts that the pair $(J(C),W^{g-1})$ determines $C$, where $W^{g-1}$ is an image of the $g-1$st symmetric power of $C$ inside the Jacobian under…

We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield…

We consider the question: which elliptic curves appear as the Jacobian of a smooth curve of genus one splitting a Severi--Brauer variety? We provide three new examples. First, we show that if $E$ is any elliptic curve over an algebraically…

Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. In this paper, we compute the Sato-Tate group of $J_m$. Currently,…

For a finite field $\mathbb{F}_q$ of characteristic $p\geq 5$ and $K=\mathbb{F}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$ given by $y^2+xy - t^dy=x^3$ for all integers $d$ coprime to $q$. We provide an explicit…

Let $J$ be the Jacobian of a superelliptic curve defined by the equation $y^{\ell} = f(x)$, where $f$ is a separable polynomial of degree non-divisible by $\ell$. In this article we study the "exponential" (i.e. $\ell$-power) torsion of…

Let k be a field and f be a Siegel modular form of weight h \geq 0 and genus g>1 over k. Using f, we define an invariant of the k-isomorphism class of a principally polarized abelian variety (A,a)/k of dimension g. Moreover when (A,a) is…

We continue our study of the Legendre elliptic curve $y^2=x(x+1)(x+t)$ over function fields $K_d=\mathbf{F}_p(\mu_d,t^{1/d})$. When $d=p^f+1$, we have previously exhibited explicit points generating a subgroup $V_d$ of $E(K_d)$ of rank…

We develop a cohomological description of various explicit descents in terms of generalized Jacobians, generalizing the known description for hyperelliptic curves. Specifically, given an integer $n$ dividing the degree of some reduced…

A simply laced Dynkin diagram gives rise to a family of curves over $\mathbb{Q}$ and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a…

Let $K$ be a number field, let $g \geq 1$ be an integer and let $f(x) = (x - a_1) \cdots (x - a_{2g + 1}) \in O_K[x]$ be a polynomial that splits into $2g + 1$ distinct linear factors. Write $C$ for the hyperelliptic curve given by $C: y^2…