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Register a new userGood reduction and Shafarevich-type theorems for dynamical systems with portrait level structures
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Authors
Joseph H. Silverman
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Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:mathbb{P}^1_Ktomathbb{P}^1_K$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism $mathbb{P}^1_{R_S}tomathbb{P}^1_{R_S}$. A finite Galois invariant subset $Xsubsetmathbb{P}^1_K(bar{K})$ has good reduction outside $S$ if its closure in $mathbb{P}^1_{R_S}$ is etale over $R_S$. We study triples $(f,Y,X)$ with $X=Ycup f(Y)$. We prove that for a fixed $K$, $S$, and $d$, there are only finitely many $text{PGL}_2(R_S)$-equivalence classes of triples with $text{deg}(f)=d$ and $sum_{Pin Y}e_f(P)ge2d+1$ and $X$ having good reduction outside $S$. We consider refined questions in which the weighted directed graph structure on $f:Yto X$ is specified, and we give an exhaustive analysis for degree $2$ maps on $mathbb{P}^1$ when $Y=X$.
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A $textit{portrait}$ $mathcal{P}$ on $mathbb{P}^N$ is a pair of finite point sets $Ysubseteq{X}subsetmathbb{P}^N$, a map $Yto X$, and an assignment of weights to the points in $Y$. We construct a parameter space $operatorname{End}_d^N[mathcal{P}]$ whose points correspond to degree $d$ endomorphisms $f:mathbb{P}^Ntomathbb{P}^N$ such that $f:Yto{X}$ is as specified by a portrait $mathcal{P}$, and prove the existence of the GIT quotient moduli space $mathcal{M}_d^N[mathcal{P}]:=operatorname{End}_d^N//operatorname{SL}_{N+1}$ under the $operatorname{SL}_{N+1}$-action $(f,Y,X)^phi=bigl(phi^{-1}circ{f}circphi,phi^{-1}(Y),phi^{-1}(X)bigr)$ relative to an appropriately chosen line bundle. We also investigate the geometry of $mathcal{M}_d^N[mathcal{P}]$ and give two arithmetic applications.
In the present article, we define a notion of good height functions on quasi-projective varieties $V$ defined over number fields and prove an equidistribution theorem of small points for such height functions. Those good height functions are defined as limits of height functions associated with semi-positive adelic metrization on big and nef $mathbb{Q}$-line bundles on projective models of $V$ satisfying mild assumptions. Building on a recent work of the author and Vigny as well as on a classical estimate of Call and Silverman, and inspiring from recent works of Kuhne and Yuan and Zhang, we deduce the equidistribution of generic sequence of preperiodic parameters for families of polarized endomorphisms with marked points.
Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Grobner basis theory and generalized Golod-Shafarevich type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Grobner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than $8$. This answers a question of Wemyss cite{Wemyss}, related to the geometric argument of Toda cite{T}. We derive from the improved version of the Golod-Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove, that potential algebra for any homogeneous potential of degree $ngeq 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class ${cal P}_n$ of potential algebras with homogeneous potential of degree $n+1geq 4$, the minimal Hilbert series is $H_n=frac{1}{1-2t+2t^n-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but non-linear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar-Vafa invariants.
We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $phi$ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of $(bP^1)^g$ has only finite intersection with any curve contained in $(bP^1)^g$. We also show that our result holds for indecomposable polynomials $phi$ with coefficients in $bC$. Our proof uses results from $p$-adic dynamics together with an integrality argument. The extension to polynomials defined over $bC$ uses the method of specializations coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of $(phi,phi)$ on $bA^2$.
Let $K$ be a 1-dimensional function field over an algebraically closed field of characteristic $0$, and let $A/K$ be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in $A(bar{K})$. More precisely, we prove that there are constants $C_1,C_2>0$ such that the normalized Bernoulli-part of the canonical height is bounded below by $$ hat{h}_A^{mathbb{B}}(P) ge C_1bigl[K(P):Kbigr]^{-2} $$ for all points $Pin{A(bar{K})}$ whose height satisfies $0<hat{h}_A(P)le{C_2}$.
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