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Thomas Scanlon

Publications and source records attributed to Thomas Scanlon.

At least 37 records · Page 2Linked to original sources
arXiv2015

On Holder-Brascamp-Lieb inequalities for torsion-free discrete Abelian groups

Hölder-Brascamp-Lieb inequalities provide upper bounds for a class of multilinear expressions, in terms of $L^p$ norms of the functions involved. They have been extensively studied for functions defined on Euclidean spaces. Bennett-Carbery-Christ-Tao have initiated the study of these inequalities for discrete Abelian groups and, in terms of suitable data, have characterized the set of all tuples of exponents for which such an inequality holds for specified data, as the convex polyhedron defined by a particular finite set of affine inequalities. In this paper we advance the theory of such inequalities for torsion-free discrete Abelian groups in three respects. The optimal constant in any such inequality is shown to equal $1$ whenever it is finite. An algorithm that computes the admissible polyhedron of exponents is developed. It is shown that nonetheless, existence of an algorithm that computes the full list of inequalities in the Bennett-Carbery-Christ-Tao description of the admissible polyhedron for all data, is equivalent to an affirmative solution of Hilbert's Tenth Problem over the rationals. That problem remains open. Applications to computer science will be explored in a forthcoming companion paper.

math.CA↗
arXiv2014

Density of orbits of endomorphisms of abelian varieties

Let $A$ be an abelian variety defined over $\bar{\mathbb{Q}}$, and let $φ$ be a dominant endomorphism of $A$ as an algebraic variety. We prove that either there exists a non-constant rational fibration preserved by $φ$, or there exists a point $x\in A(\bar{\mathbb{Q}})$ whose $φ$-orbit is Zariski dense in $A$. This provides a positive answer for abelian varieties of a question raised by Medvedev and the second author ("nvariant varieties for polynomial dynamical systems", Ann. of Math. (2) 179 (2014), no. 1, 81-177). We prove also a stronger statement of this result in which $φ$ is replaced by any commutative finitely generated monoid of dominant endomorphisms of $A$.

math.NT↗
arXiv2014

Strong minimality and the j-function

We show that the order three algebraic differential equation over ${\mathbb Q}$ satisfied by the analytic $j$-function defines a non-$\aleph_0$-categorical strongly minimal set with trivial forking geometry relative to the theory of differentially closed fields of characteristic zero answering a long-standing open problem about the existence of such sets. The theorem follows from Pila's modular Ax-Lindemann-Weierstrass with derivatives theorem using Seidenberg's embedding theorem and a theorem of Nishioka on the differential equations satisfied by automorphic functions. As a by product of this analysis, we obtain a more general version of the modular Ax-Lindemann-Weierstrass theorem, which, in particular, applies to automorphic functions for arbitrary arithmetic subgroups of $SL_2 ({\mathbb Z})$. We then apply the results to prove effective finiteness results for intersections of subvarieties of products of modular curves with isogeny classes. For example, we show that if $ψ:{\mathbb P}^1 \to {\mathbb P}^1$ is any non-identity automorphism of the projective line and $t \in {\mathbb A}^1({\mathbb C}) \smallsetminus {\mathbb A}^1({\mathbb Q}^\text{alg})$, then the set of $s \in {\mathbb A}^1({\mathbb C})$ for which the elliptic curve with $j$-invariant $s$ is isogenous to the elliptic curve with $j$-invariant $t$ and the elliptic curve with $j$-invariant $ψ(s)$ is isogenous to the elliptic curve with $j$-invariant $ψ(t)$ has size at most $36^7$. In general, we prove that if $V$ is a Kolchin-closed subset of ${\mathbb A}^n$, then the Zariski closure of the intersection of $V$ with the isogeny class of a tuple of transcendental elements is a finite union of weakly special subvarieties. We bound the sum of the degrees of the irreducible components of this union by a function of the degree and order of $V$.

math.LO↗
arXiv2014

Algebraic differential equations from covering maps

Let $Y$ be a complex algebraic variety, $G \curvearrowright Y$ an action of an algebraic group on $Y$, $U \subseteq Y({\mathbb C})$ a complex submanifold, $Γ< G({\mathbb C})$ a discrete, Zariski dense subgroup of $G({\mathbb C})$ which preserves $U$, and $π:U \to X({\mathbb C})$ an analytic covering map of the complex algebraic variety $X$ expressing $X({\mathbb C})$ as $Γ\backslash U$. We note that the theory of elimination of imaginaries in differentially closed fields produces a generalized Schwarzian derivative $\widetildeχ:Y \to Z$ (where $Z$ is some algebraic variety) expressing the quotient of $Y$ by the action of the constant points of $G$. Under the additional hypothesis that the restriction of $π$ to some set containing a fundamental domain is definable in an o-minimal expansion of the real field, we show as a consequence of the Peterzil-Starchenko o-minimal GAGA theorem that the \emph{prima facie} differentially analytic relation $χ:= \widetildeχ \circ π^{-1}$ is a well-defined, differential constructible function. The function $χ$ nearly inverts $π$ in the sense that for any differential field $K$ of meromorphic functions, if $a, b \in X(K)$ then $χ(a) = χ(b)$ if and only if after suitable restriction there is some $γ\in G({\mathbb C})$ with $π(γ\cdot π^{-1}(a)) = b$.

math.LO↗
arXiv2013

Model theory of fields with free operators in characteristic zero

Generalising and unifying the known theorems for difference and differential fields, it is shown that for every finite free ${\mathbb S}$-algebra ${\mathcal D}$ over a field $A$ of characteristic zero the theory of ${\mathcal D}$-fields has a model companion ${\mathcal D}$-CF$_0$ which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.

math.LO↗
arXiv2013

Communication lower bounds and optimal algorithms for programs that reference arrays -- Part 1

The movement of data (communication) between levels of a memory hierarchy, or between parallel processors on a network, can greatly dominate the cost of computation, so algorithms that minimize communication are of interest. Motivated by this, attainable lower bounds for the amount of communication required by algorithms were established by several groups for a variety of algorithms, including matrix computations. Prior work of Ballard-Demmel-Holtz-Schwartz relied on a geometric inequality of Loomis and Whitney for this purpose. In this paper the general theory of discrete multilinear Holder-Brascamp-Lieb (HBL) inequalities is used to establish communication lower bounds for a much wider class of algorithms. In some cases, algorithms are presented which attain these lower bounds. Several contributions are made to the theory of HBL inequalities proper. The optimal constant in such an inequality for torsion-free Abelian groups is shown to equal one whenever it is finite. Bennett-Carbery-Christ-Tao had characterized the tuples of exponents for which such an inequality is valid as the convex polyhedron defined by a certain finite list of inequalities. The problem of constructing an algorithm to decide whether a given inequality is on this list, is shown to be equivalent to Hilbert's Tenth Problem over the rationals, which remains open. Nonetheless, an algorithm which computes the polyhedron itself is constructed.

math.CA↗
arXiv2012

Invariant varieties for polynomial dynamical systems

We study algebraic dynamical systems (and, more generally, $σ$-varieties) $Φ:{\mathbb A}^n_{\mathbb C} \to {\mathbb A}^n_{\mathbb C}$ given by coordinatewise univariate polynomials by refining a theorem of Ritt. More precisely, we find a nearly canonical way to write a polynomial as a composition of "clusters". Our main result is an explicit description of the (weakly) skew-invariant varieties. As a special case, we show that if $f(x) \in {\mathbb C}[x]$ is a polynomial of degree at least two which is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and $X \subseteq {\mathbb A}^2_{\mathbb C}$ is an irreducible curve which is invariant under the action of $(x,y) \mapsto (f(x),f(y))$ and projects dominantly in both directions, then $X$ must be the graph of a polynomial which commutes with $f$ under composition. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius. We also show that in models of ACFA$_0$, a disintegrated set defined by $σ(x) = f(x)$ for a polynomial $f$ has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew-conjugacy class of $f$ is defined over a fixed field of a power of $σ$, and that nonorthogonality between two such sets is definable in families if the skew-conjugacy class of $f$ is defined over a fixed field of a power of $σ$.

math.DS↗
arXiv2012

Exponential-polynomial equations and dynamical return sets

We show that for each finite sequence of algebraic integers $α_1,...,α_n$ and polynomials $P_1(x_1,...,x_n;y_1,...,y_n),..., P_r(x_1,...,x_n;y_1,...,y_n)$ with algebraic integer coefficients, there are a natural number $N$, $n$ commuting endomorphisms $Φ_i:\Gm^N \to \Gm^N$ of the $N^\text{th}$ Cartesian power of the multiplicative group, a point $P \in \Gm^N(\QQ)$, and an algebraic subgroup $G \leq \Gm^N$ so that the return set $\{(\ell_1,...,\ell_n) \in \NN^n : Φ_1^{\circ \ell_1} \circ... \circ Φ_n^{\circ \ell_n}(P) \in G(\QQ) \}$ is identical to the set of solutions to the given exponential-polynomial equation: $\{(\ell_1,...,\ell_n) \in \NN^n : P_1(\ell_1,...,\ell_n;α_1^{\ell_1},...,α_n^{\ell_n}) = ... = P_r(\ell_1,...,\ell_n;α_1^{\ell_1},...,α_n^{\ell_n}) = 0 \}$.

math.DS↗
arXiv2011

Periods of rational maps modulo primes

Let $K$ be a number field, let $ϕ\in K(t)$ be a rational map of degree at least 2, and let $α, β\in K$. We show that if $α$ is not in the forward orbit of $β$, then there is a positive proportion of primes ${\mathfrak p}$ of $K$ such that $α\mod {\mathfrak p}$ is not in the forward orbit of $β\mod {\mathfrak p}$. Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace $α$ by a hypersurface, such as the ramification locus of a morphism $ϕ: {\mathbb P}^{n} \to {\mathbb P}^{n}$.

math.AG↗
arXiv2010

Algebraic equations on the adelic closure of a Drinfeld module

Let $k$ be a field of positive characteristic and $K = k(V)$ a function field of a variety $V$ over $k$ and let ${\mathbf A}_K$ be a ring of adéles of $K$ with respect to a cofinite set of the places on $K$ corresponding to the divisors on $V$. Given a Drinfeld module $Φ:{\mathbb F}[t] \to \operatorname{End}_K({\mathbb G}_a)$ over $K$ and a positive integer $g$ we regard both $K^g$ and ${\mathbf A}_K^g$ as $Φ({\mathbb F}_p[t])$-modules under the diagonal action induced by $Φ$. For $Γ\subseteq K^g$ a finitely generated $Φ(\F_p[t])$-submodule and an affine subvariety $X \subseteq \bG_a^g$ defined over $K$, we study the intersection of $X({\mathbf A}_K)$, the adèlic points of $X$, with $barΓ$, the closure of $Γ$ with respect to the adèlic topology, showing under various hypotheses that this intersection is no more than $X(K) \cap Γ$.

math.NT↗
arXiv2010

Generalised Hasse-Schmidt varieties and their jet spaces

This work provides a unified formalism for studying difference and (Hasse-) differential algebraic geometry, by introducing a theory of "iterative Hasse rings and schemes". As an application, Hasse jet spaces are constructed generally, allowing the development of the theory for arbitrary systems of algebraic partial difference/differential equations, where constructions by earlier authors applied only to the finite dimensional case. In particular, it is shown that under appropriate separability assumptions a Hasse variety is determined by its jet spaces at a point.

math.AG↗
arXiv2010

A Euclidean Skolem-Mahler-Lech-Chabauty method

Using the theory of o-minimality we show that the $p$-adic method of Skolem-Mahler-Lech-Chabauty may be adapted to prove instances of the dynamical Mordell-Lang conjecture for some real analytic dynamical systems. For example, we show that if $f_1,...,f_n$ is a finite sequence of real analytic functions $f_i:(-1,1) \to (-1,1)$ for which $f_i(0) = 0$ and $|f_i'(0)| \leq 1$ (possibly zero), $a = (a_1,...,a_n)$ is an $n$-tuple of real numbers close enough to the origin and $H(x_1,...,x_n)$ is a real analytic function of $n$ variables, then the set $\{m \in {\mathbb N} : H (f_1^{\circ m} (a_1),...,f_n^{\circ m}(a_n)) = 0 \}$ is either all of ${\mathbb N}$, all of the odd numbers, all of the even numbers, or is finite.

math.AG↗
arXiv2008

Analytic relations on a dynamical orbit

Let $(K,|\cdot|)$ be a complete discretely valued field and $f:{\mathbb B}_1(K,1) \to {\mathbb B}_1(K,1)$ a nonconstant analytic map from the unit back to itself. We assume that 0 is an attracting fixed point of $f$. Let $a \in K$ with $\lim_{n \to \infty} f^n(a) = 0$ and consider the orbit ${\mathcal O}_f(a) := \{f^n(a) : n \in {\mathbb N} \}$. We show that if 0 is a \emph{superattracting} fixed point, then every irreducible analytic subvariety of ${\mathbb B}_n(K,1)$ meeting ${\mathcal O}_f(a)^n$ in an analytically Zariski dense set is defined by equations of the form $x_i = b$ and $x_j = f^\ell(x_k)$. When 0 is an attracting, non-superattracting point, we show that all analytic relations come from algebraic tori.

math.AG↗
arXiv2004

Local André-Oort conjecture for the universal abelian variety

We prove a $p$-adic analogue of the André-Oort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let $g$ and $n$ be integers with $n \geq 3$ and $p$ a prime number not dividing $n$. Let $R$ be a finite extension of $W[{\mathbb F}_p^{\mathrm alg}]$, the ring of Witt vectors of the algebraic closure of the field of $p$ elements. The moduli space $\cA = \cA_{g,1,n}$ of $g$-dimensional principally polarized abelian varieties with full level $n$-structure as well as the universal abelian variety $π:\cX \to \cA$ over $\cA$ may be defined over $R$. We call a point $ξ\in \cX(R)$ \emph{$R$-special} if $\cX_{π(ξ)}$ is a canonical lift and $ξ$ is a torsion point of its fibre. We show that an irreducible subvariety of $\cX_R$ containing a dense set of $p$-special points must be a special subvariety in the sense of mixed Shimura varieties. Our proof employs the model theory of difference fields.

math.AG↗
arXiv2003

Positive characteristic Manin-Mumford theorem

We prove a version of the Manin-Mumford conjecture for semiabelian varieties over fields of positive characteristic. The proof presented here contains the details of the proof sketched by the author in the article "Diophantine geometry from model theory," BSL 7 (2001) no. 1, 37 - 57; using the model theory of difference fields along the lines of Hrushovski. Algebraic proofs have been presented by Pink & Roessler and Pillay.

math.AG↗
arXiv2002

F-structures and integral points on semiabelian varieties over finite fields

Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination result for certain modules over finite simple extensions of the integers given together with predicates for orbits of the distinguished generator of the ring.

math.LO↗
arXiv2001

Uniformity in the Mordell-Lang conjecture

We note that Pillay's result on the stability of an algebraically closed field with a predicate for a group of Lang type implies that number uniformity follows formally from the finiteness results analogous to Faltings' Theorem.

math.LO↗