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Let Y be a smooth quasi-projective complex variety equipped with a simple normal crossings compactification. We show that integral points are potentially dense in the (relative) character varieties parametrizing $SL_2$-local systems on Y with fixed algebraic integer traces along the boundary components. The proof proceeds by using work of Corlette-Simpson to reduce to the case of Riemann surfaces, where we produce an integral point with Zariski-dense orbit under the mapping class group.

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We formulate a conjecture classifying algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the primes appearing in the denominators of the coefficients of their Taylor expansion at a non-singular point. For linear differential equations, this conjecture is a strengthening of the Grothendieck-Katz p-curvature conjecture. We prove the conjecture for many differential equations and initial conditions of algebro-geometric interest. For linear differential equations, we prove it for Picard-Fuchs equations at initial conditions corresponding to cycle classes, among other cases. For non-linear differential equations, we prove it for isomonodromy differential equations, such as the Painlevé VI equation and Schlesinger system, at initial conditions corresponding to Picard-Fuchs equations. We draw a number of algebro-geometric consequences from the proofs.

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We survey some recent developments at the interface of algebraic geometry, surface topology, and the theory of ordinary differential equations. Motivated by "non-abelian" analogues of standard conjectures on the cohomology of algebraic varieties, we study mapping class group actions on character varieties and their algebro-geometric avatar -- isomonodromy differential equations -- from the point of view of both complex and arithmetic geometry. We then collect some open questions and conjectures on these topics. These notes are an extended version of my talk at the April 2024 Current Developments in Mathematics conference at Harvard.

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Let $\Sigma_{g,n}$ be the orientable genus $g$ surface with $n$ punctures, such that $2−2g−n < 0$. Let $$\rho:\pi_1(\Sigma_{g,n})\to GL_m(\mathbb{C})$$ be a representation. Suppose that for each finite covering map $f: \Sigma_{g′,n′}\to \Sigma_{g,n}$, the orbit of (the isomorphism class of) $f_∗(\rho)$ under the mapping class group $MCG(\Sigma_{g′,n′})$ of $\Sigma_{g′,n′}$ is finite. Then we show that $\rho$ has finite image. The result is motivated by the Grothendieck-Katz $p$-curvature conjecture, and gives a reformulation of the $p$-curvature conjecture in terms of isomonodromy.

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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, connected variety over $\mathbb{C}$, the (typically infinite) set of representations of $\pi_1(X^{\text{an}})$ into $GL_n(\overline{\mathbb{Q}_\ell})$, which come from geometry, has no limit points. As a corollary, we deduce that if $L$ is a finite extension of $\mathbb{Q}_\ell$, then the set of representations of $\pi_1(X^{\text{an}})$ into $GL_n(L)$, which arise from geometry, is finite.

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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 $\rho$ of $\pi_1^{\text{et}}(X_{\bar k})$ is arithmetic if there exists a finite extension $k'$ of $k$ and a representation $\gamma$ of the etale fundamental group of $\pi_1^{\text{et}}(X_{k'})$, with $\rho$ a subquotient of $\gamma|_{\pi_1^{\text{et}}(X_{\bar k})}$. We show that there exists an integer $N=N(X, \ell)$ such that every nontrivial, semisimple arithmetic representation of $\pi_1^{\text{et}}(X_{\bar k})$ is nontrivial mod $\ell^N$. As a corollary, we prove that any nontrivial $\ell$-adic representation of $\pi_1^{\text{et}}(X_{\bar k})$ which arises from geometry is nontrivial mod $\ell^N$.

This is a streamlined and completely rewritten account of the most interesting results in the paper below.

$\qquad$+ Abstract

This is a leisurely account of the results in the paper above and some complements; some of what did not appear in the paper above will appear in its sequel, Arithmetic representations of fundamental groups II. This paper is not intended for publication.

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We show that every smooth projective curve over a finite field $k$ admits a finite tame morphism to the projective line over $k$. Furthermore, we construct a curve with no such map when $k$ is an infinite perfect field of characteristic two. Our work leads to a refinement of the tame Belyi theorem in positive characteristic, building on results of Saïdi, Sugiyama-Yasuda, and Anbar-Tutdere.

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Let $\ell$ be a prime and $G$ a pro-$\ell$ group with torsion-free abelianization. We produce group-theoretic analogues of the Johnson/Morita cocycle for G -- in the case of surface groups, these cocycles appear to refine existing constructions when $\ell=2$. We apply this to the pro-$\ell$ étale fundamental groups of smooth curves to obtain Galois-cohomological analogues, and discuss their relationship to work of Hain and Matsumoto in the case the curve is proper. We analyze many of the fundamental properties of these classes and use them to give an example of a non-hyperelliptic curve whose Ceresa class has torsion image under the $\ell$-adic Abel-Jacobi map.

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Let A be a simple abelian surface over an algebraically closed field k. Let $S\subset A(k)$ be the set of torsion points of A contained in the image of a map from a genus 2 curve C to A, sending a Weierstrass point of C to the origin. The purpose of this note is to show that if k has characteristic zero, then S is finite --- this is in contrast to the situation where k is the algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and Tschinkel. We deduce that if $k=\overline{\mathbb{Q}}$, the Kummer surface associated to A has infinitely many k-points not contained in a rational curve arising from a genus 2 curve in A, again in contrast to the situation over the algebraic closure of a finite field.

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This note proves several asymptotic vanishing theorems for Frobenius twists of ample vector bundles in positive characteristic. As an application, I prove a generalization of the Bott-Danilov-Steenbrink vanishing theorem for ample vector bundles on toric varieties.

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This is the edited version of my thesis, the goal of which was to prove a Lefschetz hyperplane theorem for any representable functor. Given a smooth projective variety $X$ and an ample divisor $D$ in $X$, we study the problem of extending a map $D\to Y$ to a map $X\to Y$. This is mainly interesting when $Y$ is a moduli space. The Lefschetz hyperplane theorem for the etale fundamental group is a corollary of the main result (taking $Y=BG$ for $G$ a finite group); we get many more Lefschetz theorems as well. For example, we find Lefschetz theorems for extending families of curves from an ample divisor, extending sections to certain maps, extending maps to varieties (DM stacks) with nef cotangent bundle, and so on. The main results are in characteristic zero, but the proofs pass through positive characteristic.

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Sommese gives a conjectural classification of manifolds containing a projective bundle as an ample divisor; I prove his conjecture in many new cases, and I prove it in general assuming a natural conjectural characterization of projective space, along the lines of the famous characterization due to Andreatta and Wisniewski. (Update 11/2016: This conjectural characterization is now a theorem, due to Jie Liu (arXiv version). Thus Sommese's conjecture is now a theorem as well.)

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We prove a non-reduced analogue of the dynamical Mordell-Lang conjecture, via $p$-adic methods. We use this to place restrictions on the dynamics of the exceptional divisor of a blowup under the action of a regular automorphism. As a consequence, we show that certain blowups (e.g. blowups in high codimension) do not alter the finiteness of the automorphism group, extending results of Bayraktar-Cantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety under an automorphism.

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Kapranov showed that his motivic zeta function (a geometric analogue of the zeta function of a variety over a finite field) is rational for smooth curves which have a rational point. By analyzing the classes of certain Severi-Brauer schemes in the Grothendieck ring of varieties, I show that the same thing is true without the assumption of the existence of a rational point.

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My advisor, Ravi Vakil, conjectured that the classes of the symmetric powers of a variety "stabilize" in the Grothendieck ring, in a certain sense. I show that the answer to this question is "no" if one asks this stabilization question in a slightly different way, and conditional on various (very difficult) questions about the Grothendieck ring of varieties, that the answer to the original question is "no" as well. On the other hand, I show that these stabilization questions are controlled by the Hodge theory of the variety, and give an analogue of the famous "Newton above Hodge" theorem in the Grothendieck ring. Here is a video of me talking about this paper.