Related papers: Chai's conjecture for semiabelian Jacobians

We construct natural relative compactifications for the relative Jacobian over a family $X/S$ of reduced curves. In contrast with all the available compactifications so far, ours admit a universal sheaf, after an etale base change. Our…

Suppose $C$ is a smooth projective curve of genus 1 over a perfect field $F$, and $E$ is its Jacobian. In the case that $C$ has no $F$-rational points, so that $C$ and $E$ are not isomorphic, $C$ is an $E$-torsor with a class $\delta(C)\in…

Let $X$ be a proper, smooth, and geometrically connected curve over a non-archimedean local field $K$. In this paper, we relate the component group of the N\'eron model of the Jacobian of $X$ to the Brauer group of $X$.

In this paper we explore the link between the theory of sheaves on graphs and noncommutative geometry showing that many concepts and constructions in the latter can be generalized and enhanced using methods coming from the former. They…

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…

In this work we consider constructions of genus three curves $X$ such that $\mathrm{End}(\mathrm{Jac} (X))\otimes Q$ contains the totally real cubic number field $Q(\zeta _7 +\bar{\zeta}_7 )$. We construct explicit three-dimensional…

We develop the theory of algebraic groups over real closed fields and apply the results to construct a geometric object $\mathcal{B}$ and to prove that $\mathcal{B}$ is an affine $\Lambda$-building. We use a model theoretic transfer…

Let $X$ be a complex smooth projective variety of dimension $d$. Under some assumption on the cohomology of $X$, we construct mutually orthogonal idempotents in $CH_d(X \times X) \otimes \Q$ whose action on algebraically trivial cycles…

We formulate and prove the Remodeling Conjecture with descendants, which is a version of all-genus equivariant descendant mirror symmetry for semi-projective toric Calabi-Yau 3-orbifolds with integral structures. We construct an isomorphism…

By the Jacquet-Langlands correspondence and Faltings' isogeny theorem, it is known that the abelian variety J_0^{Dpq}(N) (the Jacobian of a Shimura curve of discriminant Dpq with Gamma_0(N) level structure) is isogenous to the pq-new…

We discuss the existence of an absolute Chow-Kuenneth decomposition for complete degenerations of families of Abelian threefolds with complex multiplication over a particular Picard Modular Surface studied by Holzapfel. In addition to the…

The present paper is devoted to investigating the two-dimensional real Jacobian conjecture. This conjecture claims that if $F=\left(f,g\right):\mathbb{R}^2\rightarrow \mathbb{R}^2$ is a polynomial map with $\det DF\left(x,y\right)\ne0$ for…

We present a heuristic argument based on Honda-Tate theory against many conjectures in `unlikely intersections' over the algebraic closure of a finite field; notably, we conjecture that every abelian variety of dimension 4 is isogenous to a…

Jacobian conjecture states that if $F:\ \mathbb C^n(\mathbb R^n)\rightarrow \mathbb C^n(\mathbb R^n)$ is a polynomial map such that the Jacobian of $F$ is a nonzero constant, then $F$ is injective. This conjecture is still open for all…

We construct a non-proper set of two variables polynomial maps and study the nowhere vanishing Jacobian condition of the Jacobian conjecture for this set. We obtain some classes of polynomial maps satisfying the 2-dimensional Jacobian…

We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $\mathbb{F}_p(t)$, when $p$ is prime and $r\ge 2$ is an integer prime to $p$. When $q$ is a…

This is a short addendum to a note of Beauville on the subject of the title. We prove an inequality that takes into account the constant part of the Jacobian.

For every odd prime number p, we give examples of non-constant smooth families of genus 2 curves over fields of characteristic p which have pro-Galois (pro-\'etale) covers of infinite degree with geometrically connected fibers. The…

Using the Galois theory over function field, and the holomorphy of algebroids defined via irreducible polynomial at singular points, we prove the injectivity of any kellerian mapping. The famous Jacobian conjecture is true.

The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In…