We propose a conjecture on the density of arithmetic points in the deformation space of representations of the étale fundamental group in positive characteristic. This? conjecture has applications to étale cohomology theory, for example it implies a Hard Lefschetz conjecture. We prove the density conjecture in tame degree two for the curve $\mathbb{P}^1\setminus \{0,1,\infty\}$. v2: very small typos corrected.v3: final. Publication in Epiga.
Let $\ell$ be a prime, $k$ a finitely generated field of characteristic different from $\ell$, and $X$ a smooth geometrically connected curve over $k$. Say a semisimple representation of $π_1^{\mathrm{et}}(X_{\bar k})$ is arithmetic if it extends to a finite index subgroup of $π_1^{\mathrm{et}}(X)$. We show that there exists an effective constant $N=N(X,\ell)$ such that any semisimple arithmetic representation of $π_1^{\mathrm{et}}(X_{\bar k})$ into $\mathrm{GL}_n(\bar{\mathbb{Z}_\ell})$, which is trivial mod $\ell^N$, is in fact trivial. This extends a previous result of the second author fro…
We prove that any surjective self-morphism with $δ_f > 1$ on a potentially dense smooth projective surface defined over a number field $K$ has densely many $L$-rational points for a finite extension $L/K$.
Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical degree. In this article, we show that there are densely many $\overline{\mathbb Q}$-rational points with maximal arithmetic degree (i.e. whose arithmetic degree is equal to the first dynamical degree) for self-morphisms on projective varieties. For unirational varieties and abelian varieties, we show that there are densely many rational points with maximal arithme…
Let $X$ be a normal algebraic variety over a finitely generated field $k$ of characteristic zero, and let $\ell$ be a prime. Say that a continuous $\ell$-adic representation $ρ$ of $π_1^{\text{ét}}(X_{\bar k})$ is arithmetic if there exists a representation $\tilde ρ$ of a finite index subgroup of $π_1^{\text{ét}}(X)$, with $ρ$ a subquotient of $\tildeρ|_{π_1(X_{\bar k})}$. We show that there exists an integer $N=N(X, \ell)$ such that every nontrivial, semisimple arithmetic representation of $π_1^{\text{ét}}(X_{\bar k})$ is nontrivial mod $\ell^N$. As a corollary, we prove that any nontrivial …
Let $X$ be a smooth curve over a finitely generated field $k$, and let $\ell$ be a prime different from the characteristic of $k$. We analyze the dynamics of the Galois action on the deformation rings of mod $\ell$ representations of the geometric fundamental group of $X$. Using this analysis, we prove analogues of the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey-Mazur conjecture for function fields in characteristic zero. For example, we show that if $X$ is a normal, conne…
In this paper, we will give an upper bound and a lower bound of the arithmetic Hilbert-Samuel function of projective hypersurfaces, which are uniform and explicit. These two bounds have the optimal dominant terms. As an application, we use the lower bound to get an estimate of the density of rational points with small heights in a hypersurface.
Given a surjective endomorphism $f: X \to X$ on a projective variety over a number field, one can define the arithmetic degree $α_f(x)$ of $f$ at a point $x$ in $X$. The Kawaguchi - Silverman Conjecture (KSC) predicts that any forward $f$-orbit of a point $x$ in $X$ at which the arithmetic degree $α_f(x)$ is strictly smaller than the first dynamical degree $δ_f$ of $f$ is not Zariski dense. We extend the KSC to sAND (= small Arithmetic Non-Density) Conjecture that the locus $Z_f(d)$ of all points of small arithmetic degree is not Zariski dense, and verify this sAND Conjecture for endomorphisms…
We adapt methods from quiver representation theory and Hall algebra techniques to the counting of representations of virtually free groups over finite fields. This gives rise to the computation of the E-polynomials of $\mathbf{GL}_d(\mathbb{C})$-character varieties of virtually free groups. As examples we discuss the representation theory of $\mathbb{D}_\infty$ , $\mathbf{PSL}_2(\mathbb{Z})$ , $\mathbf{SL}_2(\mathbb{Z})$ , $\mathbf{GL}_2(\mathbb{Z})$ and $\mathbf{PGL}_2(\mathbb{Z})$ .
Let K be an arithmetic function field, that is, a field of finite type over the rational number field. In this note, as an application of the height theory due to Chen-Moriwaki, we would like to show that the solutions of Fermat's curve X^N + y^N = 1 of degree N over K consist of only either 0 or roots of unity for almost positive integers N. More precisely, the density of such N is 1.
We continue the study of a general class of spaces of 0-cycles on a manifold defined and begun by Farb-Wolfson-Wood. Using work of Gadish on linear subspace arrangements, we obtain representation stability for the cohomology of the ordered version of these spaces. We establish subexponential bounds on the growth of unstable cohomology, and the Grothendieck-Lefschetz trace formula then allows us to translate these topological stability phenomena to stabilization of statistics for spaces of 0-cycles over finite fields. In particular, we show that the average value of certain arithmetic quantitie…
We count points over a finite field on wild character varieties of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma-Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the possibility of a P=W conjecture for a suitable wild Hitchin system.
We contribute to the arithmetic/topology dictionary by relating asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over $\Cb$ in the example of configuration spaces of $n$ points in smooth varieties. To do this, we import the method of homological stability from the realm of topology into the theory of étale cohomology; in particular we give the first examples of stability of étale cohomology groups as Galois representations where the Galois actions are not already explicitly known. We then establish subexponential bounds …
In this paper, we apply ideas of Dijkgraaf and Witten on 2+1 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern-Simons functionals on spaces of Galois representations. In the highly speculative section 5, we consider the far-fetched possibility of using Chern-Simons theory to construct L-functions.
