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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]

Let \sigma and \tau denote a pair of absolutely irreducible p-ordinary and p-distinguished Galois representations into \operatorname {GL}_2(\overline {\mathbb {F}}_p). Given two primitive forms (f,g) such that \operatorname {wt}(f)>\operatorname {wt}(g)> 1 and where \overline {\rho }_f\cong \sigma and \overline {\rho }_g\cong \tau, we show that the Iwasawa Main Conjecture for the double product \rho _f\otimes \rho _g depends only on the residual Galois representation \sigma \otimes \tau : G_{\mathbb {Q}}\rightarrow \operatorname {GL}_4(\overline {\mathbb {F}}_p). More precisely, if IMC(f\otimes g) is true for one pair (f,g) with \overline {\rho }_f \cong \sigma and \overline {\rho }_g\cong \tau and whose \mu-invariant equals zero, then it is true for every congruent pair too. [ABSTRACT FROM AUTHOR]

In this article we define and study a zeta function ζ G —similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value ζ G (k) - 1 at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that ζ G is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζ G . We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro- C groups, where C is a class of finite groups with prescribed composition factors. We prove that every real number a ≥ 1 is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζ G are rational functions in p - s if G is virtually abelian. For finite groups G we calculate ζ G using the rational representation theory of G. [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]