influences of large height differences and overhangs on the dynamic scaling behavior of discrete models abstract in order to investigate the influences of large height differences and overhangs on the dynamic scaling behavior of discrete models meanwhile reducing the finite size effects the etching model is modified to reduce large height differences and the overhangs in ballistic deposition surfaces are removed under certain principles numerical simulations are carried out for the modified models and the results show that the modified surfaces lead to good dynamic scaling behavior even on small system length scales the values of the dynamic scaling exponents are in excellent agreement with theoretical predictions of the kardar parisi zhang equation in 1 1 dimensions

the bogomolov conjecture for totally degenerate abelian varieties we prove the bogomolov conjecture for a totally degenerate abelian variety a over a function field we adapt zhang s proof of the number field case replacing the complex analytic tools by tropical analytic geometry a key step is the tropical equidistribution theorem for a at the totally degenerate place v as a corollary we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v

big line bundles over arithmetic varieties 5 equidistribution theory 26 5 1 a generic equidistribution theorem 27 5 2 equidistribution at infinite places 30 5 3 equidistribution at finite places 32 5 4 equidistribution of small subvarieties 35 5 5 equidistribution over algebraic dynamical systems 37 5 6 equidistribution over multiplicative groups 39 5 7 equidistribution theory 41

a height gap theorem for finite subsets of mathrm gl _ d overline bbb q and nonamenable subgroups we introduce a conjugation invariant normalized heightb f on nite subsets of matrices f in gld q and describe its properties in particular we prove an analogue of the lehmer problem for this height by showing that b f whenever f generates a nonvirtually solvable subgroup of gld q where d 0 is an absolute constant this can be seen as a global adelic analog of the classical margulis lemma from hyperbolic geometry as an application we prove a uniform version of the classical burnside schur theorem on torsion linear groups in a companion paper we will apply these results to prove a strong uniform version of the tits alternative

on the spectrum of the zhang zagier height from recent work of zhang and of zagier we know that their height n is bounded away from 1 for every algebraic number a different from 0 1 1 2 3 2 the study of the related spectrum is especially interesting for it is linked to lehmer s problem and to a conjecture of bogomolov after recalling some definitions we show an improvement of the so called zhang zagier inequality to achieve this we need some algebraic numbers of small height so in the third section we describe an algorithm able to find them and we give an algebraic number with height 1 2875274z discovered in this way this search up to degree 64 suggests that the spectrum of n may have a limit point less than 1 292 we prove this fact in the fourth part

ordinal network representation representing proximities by graphs ordinal network representations are graph theoretic representations of proximity data they seek to provide parsimonious representations of the ordinal nonmetric information in observed proximity data by means of the minimum path length distance of a connected and weighted graph in contrast to traditional tree based graph theoretic approaches ordinal network representation is not limited to but includes the representation by trees asymmetry in the proximity data and violations of zero minimality are allowed for the paper explores fundamental representation and uniqueness results and discusses a method of constructing ordinal network representations a simple strategy for handling the problem of errors in the data is described and illustrated

the distribution of galois orbits of points of small height in toric varieties we study the distribution of galois orbits of points of small height on proper toric varieties and its application to the bogomolov problem we introduce the notion of monocritical toric metrized divisor we prove that a toric metrized divisor d on a proper toric variety x over a global eld k is monocritical if and only if for every generic d small sequence of algebraic points of x and every place v of k the sequence of their galois orbits on the analytic space x an converges to a measure when this is the case the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of x the key ingredient is the study of the v adic modulus distribution of ga lois orbits of generic d small sequences of algebraic points in particular we characterize all their cluster measures we generalize the bogomolov problem by asking when a subvariety of the principal orbit of a proper toric variety that has the same essential minimum than the ambient variety must be a translate of a subtorus we prove that the generalized bogomolov problem has a positive answer for monocritical toric metrized divisors and we give several examples of toric metrized divisors for which the bogomolov problem has a negative answer

the canonical arithmetic height of subvarieties of an abelian variety over a finetely generated field this paper is the sequel of 2 in 4 s zhang defined the canonical height of subvarieties of an abelian variety over a number field in terms of adelic metrics in this paper we generalize it to an abelian variety defined over a finitely generated field over q our way is slightly different from his method instead of using adelic metrics directly we introduce an adelic sequence and an adelic structure cf 3 1 let k be a finitely generated field over q with d tr degq k and b b h1 hd a polarization of k i e b is a projective arithmetic variety whose function field is k and h1 hd are nef c hermitian line bundles on b let a be an abelian variety over k and l a symmetric ample line bundle on a fix a projective arithmetic variety a over b and a nef c hermitian q line bundle l on a such that a is the generic fiber of a b and l is isomorphic to l on a then we can assign the naive height h a l x to a subvariety x of ak indeed if x is defined over k h b a l x is given by

equidistribution for torsion points of a drinfeld module we prove an equidistribution result for torsion points of drinfeld modules of generic characteristic we also show that similar equidistribution statements would provide proofs for the manin mumford and the bogomolov conjectures for drinfeld modules

second variation of zhang s λ invariant on the moduli space of curves we compute the second variation of the invariant recently introduced by s zhang on the complex moduli space mg of curves of genus g 2 using work of n kawazumi as a result we prove that 8g 4 is equal up to a constant to the invariant introduced some time ago by r hain and d reed we deduce some consequences for example we calculate the invariant for each hyperelliptic curve expressing it in terms of the petersson norm of the discriminant modular form 1 introduction recently independently s zhang 27 and n kawazumi 16 introduced a new interesting real valued function on the moduli space mg of complex curves of genus g 2 its value at a curve x m g is given as follows let h 0 x x be the space of holomorphic differentials on x equipped with the

positive line bundles on arithmetic varieties p 5 u integers is defined to count the volume of the lattice i l n of integral sections in the space j7 lon of real sections with supremum norm we want to prove that the leading term of this hilbert function as n f oo is given in terms of the height of x in as 1 this is known as a theorem of gillet and soule gs2 if x has a regular generic fiber beside this known result our proof uses hironaka s theorem on resolutions of singularities and minkowski s theorem on successive minima by hironaka s theorem we may construct

arakelov self intersection numbers of minimal regular models of modular curves x_0 p 2 x0 p2 we compute an asymptotic expression for the arakelov self intersection number of the relative dualizing sheaf of edixhoven s minimal regular model for the modular curve x 0 p 2 over mathbb q the computation of the self intersection numbers are used to prove an effective version of the bogolomov conjecture for the semi stable models of modular curves x 0 p 2 and obtain a bound on the stable faltings height for those curves in a companion article banerjee and chaudhuri in isr j math 2020

examples of torsion points on genus two curves we describe a method that sometimes determines all the torsion points lying on a curve of genus two defined over a number field and embedded in its jacobian using a weierstrass point as base point we then apply this to the examples y2 x5 x y2 x5 5x3 x and y2 y x5

the dynamical mordell lang problem for etale maps we prove a dynamical version of the mordell lang conjecture for etale endomorphisms of quasiprojective varieties we use p adic methods inspired by the work of skolem mahler and lech combined with methods from algebraic geometry as special cases of our result we obtain a new proof of the classical mordell lang conjecture for cyclic subgroups of a semiabelian variety and we also answer positively a question of keeler rogalski stafford for critically dense sequences of closed points of a noetherian integral scheme

zhang s conjecture and the effective bogomolov conjecture over function fields we prove the effective bogomolov conjecture and so the bogomolov conjecture over a function field of characteristic 0 by proving zhang s conjecture about certain invariants of metrized graphs in the function field case these conjectures were previously known to be true only for curves of good reduction for curves of genus at most 4 and a few other special cases we also either verify or improve the previous results we relate the invariants involved in zhang s conjecture to the tau constant of metrized graphs then we use and extend our previous results on the tau constant by proving another conjecture of zhang we obtain a new proof of the slope inequality for faltings heights on moduli space of curves

linear relations between polynomial orbits we study the orbits of a polynomial f 2c x namely the sets 1 f f f o with 2 c we prove that if two nonlinear complex polynomials f g have orbits with infinite intersection then f and g have a common iterate more generally we describe the intersection of any line in c with a d tuple of orbits of nonlinear polynomials and we formulate a question which generalizes both this result and the mordell lang conjecture

lower bounds on the arithmetic self intersection number of the relative dualizing sheaf on arithmetic surfaces we give an explicitly computable lower bound for the arithmetic self intersection number of the dualizing sheaf on a large class of arithmetic surfaces if some technical conditions are satisfied then this lower bound is positive in particular these technical conditions are always satisfied for minimal arithmetic surfaces with simple multiplicities and at least one reducible fiber but we have also used our techniques to obtain lower bounds for some arithmetic surfaces with non reduced fibers

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

a case of the dynamical mordell lang conjecture we prove a special case of a dynamical analogue of the classical mordell lang conjecture specifically let be a rational function with no periodic critical points other than those that are totally invariant and consider the diagonal action of on mathbb p 1 g if the coefficients of are algebraic we show that the orbit of a point outside the union of the proper preperiodic subvarieties of mathbb p 1 g has only finite intersection with any curve contained in mathbb p 1 g we also show that our result holds for indecomposable polynomials with coefficients in mathbb c our proof uses results from p adic dynamics together with an integrality argument the extension to polynomials defined over mathbb c uses the method of specialization coupled with some new results of medvedev and scanlon for describing the periodic plane curves under the action of on mathbb a 2

width distributions and the upper critical dimension of kardar parisi zhang interfaces simulations of restricted solid on solid growth models are used to build the width distributions of d 2 5 dimensional kardar parisi zhang kpz interfaces we find that the universal scaling function associated with the steady state width distribution changes smoothly as d is increased thus strongly suggesting that d 4 is not an upper critical dimension for the kpz equation the dimensional trends observed in the scaling functions indicate that the upper critical dimension is at infinity

torsion points and the lattes family we give a dynamical proof of a result of masser and zannier for any a neq b in overline bbb q setminus 0 1 there are only finitely many parameters t in bbb c for which points p a a sqrt a a 1 a t and p b b sqrt b b 1 b t are both torsion on the legendre elliptic curve e t y 2 x x 1 x t our method also gives the finiteness of parameters t where both p a and p b have small n eron tate height a key ingredient in the proof is an arithmetic equidistribution theorem on bbb p 1 for this we prove two statements about the degree 4 latt es family f t on bbb p 1 1 for each c in bbb c t the bifurcation measure mu c for the pair f t c has continuous potential across the singular parameters t 0 1 infty and 2 for distinct points a b in bbb c setminus 0 1 the bifurcation measures mu a and mu b cannot coincide combining our methods with the result of masser and zannier we extend their conclusion to points t of small height also for a b in bbb c t

canonical heights transfinite diameters and polynomial dynamics let z be a polynomial of degree at least 2 with coefficients in a number field k iterating gives rise to a dynamical system and a corresponding canonical height function as defined by call and silverman we prove a simple product formula relating the transfinite diameters of the filled julia sets of over various completions of k and we apply this formula to give a generalization of bilu s equidistribution theorem for sequences of points whose canonical heights tend to zero

intersections of polynomial orbits and a dynamical mordell lang conjecture we prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection then the polynomials have a common iterate we also prove a special case of a conjectured dynamical analogue of the mordell lang conjecture

o minimality and the andre oort conjecture for mathbb c n we give an unconditional proof of the andre oort conjecture for arbitrary products of modular curves we establish two generalizations the first includes the manin mumford conjecture for arbitrary products of elliptic curves defined over q as well as lang s conjecture for torsion points in powers of the multiplicative group the second includes the manin mumford conjecture for abelian varieties defined over q our approach uses the theory of o minimal structures a part of model theory and follows a strategy proposed by zannier and implemented in three recent papers a new proof of the manin mumford conjecture by pila zannier a proof of a special but new case of pink s relative manin mumford conjecture by masser zannier and new proofs of certain known results of andre oort manin mumford type by pila 2010 mathematics subject classification 11g18 03c64