Related papers: On a conjecture of Deligne

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. For any vector bundle $W$ on $X$, we prove that instability of…

We show that if $E/\mathbb{Q}$ is an elliptic curve without complex multiplication and for which there is a prime $q$ such that the image of $\bar{\rho}_{E,q}$ is contained in the normaliser of a split Cartan subgroup of…

We prove that the class of separably algebraically closed valued fields equipped with a distinguished Frobenius endomorphism $x \mapsto x^q$ is decidable, uniformly in $q$. The result is a simultaneous generalization of the work of…

Let $A$ be an abelian variety defined over $\mathbb{Q}$ and of dimension $g$. Assume that, for each sufficiently large prime $\ell$, $A$ has a surjective residual modulo $\ell$ Galois representation. For $t\in \mathbb{Z}$ and $x>0$, denote…

Deligne has conjectured that certain mixed Hodge theoretic invariants of complex algebraic invariants are motivic. This conjecture specializes to an algebraic construction of the Jacobian for smooth projective curves, which was done by A.…

Let R be an integral domain of finite type over Z and let f:X --> Spec R be a smooth projective morphism of relative dimension d >= 1. We investigate, for a vector bundle E on the total space X, under what arithmetical properties of a…

We consider generalizations of Szpiro's classical discriminant conjecture to hyperelliptic curves over a number field $K$, and to smooth, projective and geometrically connected curves $X$ over $K$ of genus at least one. The main results…

We prove an explicit surjectivity result for products of non-isotrivial, non-isogenous elliptic curves over a function field of arbitrary characteristic. This is by way of an isogeny degree bound in this setting, generated from bounds for…

If $X$ is a smooth toric variety over an algebraically closed field of positive characteristic and $L$ is an invertible sheaf on $X$, it is known that $F_* L$, the push-forward of $L$ along the Frobenius morphism of $X$, is a direct sum of…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication. For a prime $p$ of good reduction for $E$, we write $\#E_p(\mathbb{F}_p) = p + 1 - a_p(E)$ for the number of $\mathbb{F}_p$-rational points of the…

We provided explicit formulas for the number of stringy points over finite fields of parabolic type A character varieties with generic semisimple monodromy. This leads to formulas for their stringy E-polynomials. In particular, they satisfy…

Let $f$ be a generically finite polynomial map $f: \mathbb{C}^n\to \mathbb{C}^m$ of algebraic degree $d$. Motivated by the study of the Jacobian Conjecture, we prove that the set $S_f$ of non-properness of $f$ is covered by parametric…

Let F be a non-archimedean local field, let L be the maximal unramified extension of F, and let fr be the Frobenius automorphism. Let G be a split connected reductive group over F, and let B(1) be the Bruhat-Tits building associated to…

Let X be a smooth, connected, projective variety over an algebraically closed field of positive characteristic. In "Flat vector bundles and the fundamental group in non-zero characteristics" (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2…

Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $\mathbb R$, such that $G$ is nonabelian…

Deligne's regularity criterion for an integrable connection $\nabla$ on a smooth complex algebraic variety $X$ says that $\nabla$ is regular along the irreducible divisors at infinity in some fixed normal compactification of $X$ if and only…

Let $G$ be a connected reductive group defined and split over a non-archimedean local field $F$. We give a new geometric proof of a special case of a recent theorem of Solleveld. Namely, we show that the class of standard Iwahori-spherical…

Let $A$ be an abelian variety over $\mathbb{Q}$ of dimension $g$ such that the image of its associated absolute Galois representation $\rho_A$ is open in $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. We investigate the arithmetic of the…

The first purpose of this paper is to give the fnite transcendence of Frobenius traces for elliptic curves over $\mathbb{Q}$ without the assumption of complex multiplication (CM). This result generalizes the previous work by Luca and…

We show that if f: X --> Y is a finite, separable morphism of smooth curves defined over a finite field F_q, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(F_q) surjectively…