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If in a given rank r, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number \ell, there is an absolutely irreducible \ell-adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank r if once there is an irreducible rank r complex linear representation with determinant isomorphic to this torsion character, then for any \ell, there is an absolutely irreducible one of rank r and determinant this given character, which is defined over \bar { \mathbb {Z}}_\ell. We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth quasi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne's companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong's conjecture (Gaitsgory for \ell \ge 3). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of the Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, which shows the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple. [ABSTRACT FROM AUTHOR]

We prove that in either the convergent or overconvergent setting, an absolutely irreducible |$F$| -isocrystal on the absolute product of two or more smooth schemes over perfect fields of characteristic |$p$|⁠ , further equipped with actions of the partial Frobenius maps, is an external product of |$F$| -isocrystals over the multiplicands. The corresponding statement for lisse |$\overline{{\mathbb{Q}}}_{\ell }$| -sheaves, for |$\ell \neq p$| a prime, is a consequence of Drinfeld's lemma on the fundamental groups of absolute products of schemes in characteristic |$p$|⁠. The latter plays a key role in V. Lafforgue's approach to the Langlands correspondence for reductive groups with |$\ell $| -adic coefficients; the |$p$| -adic analogue will be considered in subsequent work with Daxin Xu. [ABSTRACT FROM AUTHOR]

For a given genus $g \geq 1$ , we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$. As a consequence of Katz–Sarnak theory, we first get for any given $g>0$ , any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q. [ABSTRACT FROM AUTHOR]

We study the class of univariate polynomials β k (X) , introduced by Carlitz, with coefficients in the algebraic function field F q (t) over the finite field F q with q elements. It is implicit in the work of Carlitz that these polynomials form an F q [ t ] -module basis of the ring Int (F q [ t ]) = { f ∈ F q (t) [ X ] | f (F q [ t ]) ⊆ F q [ t ] } of integer-valued polynomials on the polynomial ring F q [ t ]. This stands in close analogy to the famous fact that a Z -module basis of the ring Int (Z) is given by the binomial polynomials ( X k ). We prove, for k = q s , where s is a non-negative integer, that β k is irreducible in Int (F q [ t ]) and that it is even absolutely irreducible, that is, all of its powers β k m with m > 0 factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that β k is not even irreducible if k is not a power of q. [ABSTRACT FROM AUTHOR]

Over fields of characteristic zero, we determine all absolutely irreducible Yetter–Drinfeld modules over groups that have prime dimension and yield a finite-dimensional Nichols algebra. To achieve our goal, we introduce orders of braided vector spaces and study their degenerations and specializations. [ABSTRACT FROM AUTHOR]