heights of heegner cycles and derivatives of l series 0 introduction 99 1 intersections and heights 103 2 kuga sato varieties and cm cycles 117 3 heights of cm cycles 124 4 heights of heegner cycles 137 5 proof of the main identity and the consequences 147 references 151

finite size scaling and the crossover to mean field critical behavior in the two dimensional ising model with medium ranged interactions critical amplitudes in finite size scaling relations show a singular dependence on the range of the interactions r the respective power laws are predicted from phenomenological crossover scaling considerations these predictions are tested by monte carlo simulations for medium ranged ising square lattices it is speculated that some deviations between the simulation results and corresponding predictions may be due to logarithmic corrections

classical to critical crossovers from field theory we extend our previous determinations of nonasymptotic critical behavior of phys rev b 32 7209 1985 and 35 3585 1987 to accurate expressions of the complete classical to critical crossover in three dimensional field theory in terms of the temperaturelike scaling field i e along the critical isochore for 1 the correlation length the susceptibility and the specific heat in the homogeneous phase for the n vector model n 1 to 3 and 2 the spontaneous magnetization coexistence curve the susceptibility and the specific heat in the inhomogeneous phase for the ising model n 1 the present calculations include the seventh loop order of murray and nickel and closely account for the up to date estimates of universal asymptotic critical quantities exponents and amplitude combinations provided by guida and zinn justin j phys a 31 8103 1998

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

recursion over realizability structures abstract realizability structures play a major role in the metamathematics of intuitionistic systems and they are a basic tool in the extraction of the computational content of constructive proofs besides their rich categorical structure and effectiveness properties provide a privileged mathematical setting for the semantics of data types of programming languages in this paper we emphasize the modelling of recursive definitions of programs and types a realizability model for a language including girard s system f and an operator of recursion on types is given and some of its local properties are studied

concernant la relation de distribution satisfaite par la fonction φ associee a un reseau complexe for any pair of latticesl andl satisfying i l l and ii the indexn ofl intol is prime to 6 we construct from the usual function ofl cf no 1 some elliptic function psi psi z l underset raise0 3em hbox smash scriptscriptstyle l of the variablez with period latticel and divisor n 0 l sum limits i 1 n t i l over the torus l where the complex numberst i 1 i n describe a complete set of representatives of the quotientl l

metrics for labelled markov processes the notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions thus a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent this instability is due to the quantitative nature of probabilistic processes in a situation where the process behavior has a quantitative aspect there should be a more robust approach to process equivalence this paper studies a metric between labelled markov processes this metric has the property that processes are at zero distance if and only if they are bisimilar the metric is inspired by earlier work on logics for characterizing bisimulation and is related in spirit to the hutchinson metric

un lemme matriciel effectif we show an almost optimal effective version of masser s matrix lemma lecture notes math 1290 109 148 1987 giving a lower bound of the height of an abelian variety in terms of its period lattices

multiloop calculations in p adic string theory and bruhat tits trees we treat the openp adic string world sheet as a coset spacef t wheret is the bruhat tits tree for thep adic linear groupgl 2 p and pgl 2 p is some schottky group the boundary of this world sheet corresponds to ap adic mumford curve of finite genus the string dynamics is governed by the local gaussian action on the coset spacef the tachyon amplitudes expressed in terms ofp adic functions are proposed for the mumford curve of arbitrary genus we compare them with the corresponding usual archimedean amplitudes the sum over moduli space of the algebraic curves is conjectured to be expressed in the arithmetic surface terms we also give the necessary mathematical background including the mumford approach top adic algebraic curves the connection of the problem of closedp adic strings with the considered topics is discussed

the category theoretic solution of recursive metric space equations it is well known that one can use an adaptation of the inverse limit construction to solve recursive equations in the category of complete ultrametric spaces we show that this construction generalizes to a large class of categories with metric space structure on each set of morphisms the exact nature of the objects is less important in particular the construction immediately applies to categories where the objects are ultrametric spaces with extra structure and where the morphisms preserve this extra structure the generalization is inspired by classical domain theoretic work by smyth and plotkin for many of the categories we consider there is a natural subcategory in which each set of morphisms is required to be a compact metric space our setting allows for a proof that such a subcategory always inherits solutions of recursive equations from the full category as another application we present a construction that relates solutions of generalized domain equations in the sense of smyth and plotkin to solutions of equations in our class of categories our primary motivation for solving generalized recursive metric space equations comes from recent and ongoing work on kripke style models in which the sets of worlds must be recursively defined we show a series of examples motivated by this line of work

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

symmetric roots and admissible pairing using the discriminant modular form and the noether formula it is possible to write the admissible self intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum over all places of a certain adelic invariant we provide a simple geometric interpretation for this invariant based on the arithmetic of symmetric roots we propose the conjecture that the invariant coincides with the invariant introduced in a recent paper by s w zhang this conjecture is true in the genus 2 case and we obtain a new proof of the bogomolov conjecture for curves of genus 2 over number fields

partiallyu 1 compactified scalar massless field on the compact riemann surface and the bosonic string amplitudes the theory of the partiallyu 1 compactified scalar massless field on the compact riemann surface with nambu goto action is defined the partition function is determined completely by a choice of the finite dimensional approximations the correlation functions are the only correctly defined objects of the theory the averages of the correlation function asymptotic values provide the amplitudes for the compact riemann surfaces of any genus the usual bosonic string amplitudes are the special cases of the above amplitudes

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

explicit arithmetic intersection theory and computation of neron tate heights we describe a general algorithm for computing intersection pairings on arithmetic surfaces we have implemented our algorithm for curves over q and we show how to use it to compute regulators for a number of jacobians of smooth plane quartics and to numerically verify the conjecture of birch and swinnerton dyer for the jacobian of the split cartan curve of level 13 up to squares

hauteur de faltings de quotients de j 0 n discriminants d algebres de hecke et congruences entre formes modulaires in this paper we compute the faltings height of the jacobian of the modular curve j 0 n we explain how it is related to the discriminant of the hecke algebra this discriminant measures congruences between modular forms as a corollary we obtain a lower bound for this discriminant the faltings height of a quotient a of j 0 n is also computed

summations over equilaterally triangulated surfaces and the critical string measure we propose a new approach to the summation over dynamically triangulated riemann surfaces which does not rely on properties of the potential in a matrix model instead we formulate a purely algebraic discretization of critical string path integral this is combined with a technique which assigns to each equilateral triangulation of a two dimensional surface a riemann surface defined over a certain finite extension of the field of rational numbers i e an arthmetic surface thus we establish a new formulation in which the sum over randomly triangulated surfaces defines an invariant measure on the moduli space of arithmetic surfaces it is shown that because of this it is far from obvious that this measure for large genera approximates the measure defined by the continuum theory i e liouville theory or critical string theory in low genus this subtlety does not exist in the case of critical string theory we explicity compute the volume of the moduli space of arithmetic surfaces in terms of the modular height function and show that for low genus it approximates correctly the continuum measure we also discuss a continuum limit which bears some resemblance with a double scaling limit in matrix models

a fixed point theorem in a category of compact metric spaces various results appear in the literature for deriving existence and uniqueness of fixed points for endofunctors on categories of complete metric spaces all these results are proved for contracting functors which satisfy some further requirements depending on the category in question following a new kind of approach based on the notion of isometry we show that the sole hypothesis of contractivity is enough for proving existence and uniqueness of fixed points for endofunctors on the category of compact metric spaces and embedding projection pairs

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

the asymptotics of the arakelov geen s function and faltings delta invariant we study the behavior of the arakelov green s function and faltings delta invariant on degenerating riemann surfaces

elements of generalized ultrametric domain theory generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces as was observed by lawvere 1973 guided by his enriched categorical view on ultra metric spaces we generalize the standard notions of cauchy sequence and limit in an ultra metric space and of adjoint pair between preorders this leads to a solution method for recursive domain equations that combines and extends the standard order theoretic smyth and plotkin 1982 and metric america and rutten 1989 approaches

critical crossover functions for simple fluids towards the crossover modelling uniqueness based on a single non universal temperature scaling factor present in a simple fluid case a detailed analysis of non universal parameters involved in different critical to classical crossover models is given for the infinite limit of the cutoff wave number a set of three scaling parameters is defined for each model such that it shows all the shapes of the theoretical crossover functions overlap on the mean crossover function shapes close to the non trivial fixed point the analysis of corresponding links between their fluid dependent parameters opens a route to define a parametric model of crossover equation of state closely satisfying the universal features calculated from the ising like limit in the massive renormalization scheme

higher picard varieties and the height pairing let x be a smooth projective variety which is defined over a number field beilinson and bloch have defined under suitable asssumptions height pairings between chow groups of homologically trivial cycles on x beilinson has also formulated a hard lefschetz and a hodge index conjecture for these chow groups we show that the restriction of the height pairing to cycles algebraically equivalent to zero can be computed via abel jacobi maps in terms of the neron tate height pairing on the higher picard varieties of x this description is used in the case where x is an abelian variety to prove a consequence of beilinson conjectures namely we prove a hard lefschetz and a hodge index theorem for the groups of cycles algebraically equivalent to zero modulo incidence equivalence

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

standard conjectures for the arithmetic grassmannian g 2 n and racah polynomials we prove the arithmetic hodge index and hard lefschetz conjectures for the grassmannian g g 2 n parametrizing lines in projective space for the natural arithmetic lefschetz operator defined via the plucker embedding of g in projective space the analysis of the hodge index inequality involves estimates on values of certain racah polynomials

special values of canonical green s functions we give a precise formula for the value of the canonical green s function at a pair of weierstrass points on a hyperelliptic riemann surface further we express the energy of the weierstrass points in terms of a spectral invariant recently introduced by n kawazumi and s zhang it follows that the energy is strictly larger than log 2 our results generalize known formulas for elliptic curves

the category theoretic solution of recursive domain equations recursive specifications of domains plays a crucial role in denotational semantics as developed by scott and strachey and their followers the purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place the idea is to follow the well known analogy between partial orders and categories generalizing from least fixed points of continuous functions over cpos to initial ones of continuous functors over omega categories to apply these general ideas we introduce wand s bf o categories where the morphism sets have a partial order structure and which include almost all the categories occurring in semantics the idea is to find solutions in a derived category of embeddings and we give order theoretic conditions which are easy to verify and which imply the needed categorical ones the main tool is a very general form of the limit colimit coincidence remarked by scott in the concluding section we outline how compatibilit

asymptotics of determinants from functional integration the expression for the determinant of the laplace operator is used in terms of functional integration to compute the asymptotic behavior on degenerating riemann surfaces in the case of the arakelov metric the information is sufficiently precise to give a value for the absolute constants appearing in bosonization formulas

the mumford form and the polyakov measure in string theory an explicit formula is derived for the mumford form on the moduli space of algebraic curves according to the belavin knizhnik theorem this gives a formula for the polyakov bosonic string measure

heights and arakelov s intersection theory introduction this paper relates the neron tate height on jacobian varieties to an intersection theory on arithmetic surfaces constructed by s arakelov a1 here i extend and prove a statement of arakelov to the effect that his intersection pairing is a neron pairing a2i i use this to prove an analogue of the hodge index theorem for arithmetic surfaces a result due independently to g faltings the results of this paper have been extracted from my thesis written under the direction of michael artin it differs from the thesis in three ways firstly it employs the language of local height functions as defined by a neron and j tate and expounded by s lang li in the thesis i used neron s language of local intersection pairings n1 the second major difference is in the proof of theorem 1 3 in my thesis i proved this result by using work of raynaud and picard functors r 1 this method was suggested by m artin in this paper i provide a simpler proof observed by s lang and independently by b gross finally the development of arakalov s theory in the context of neron functions in section 2 is due to s lang who has kindly given me leave to publish it this exposition of my results is due in considerable part to s lang it is essentially an edited transcript of a series of conversations between us which he provided it is my great pleasure to thank both him and michael artin for their help and encouragement 1 the neron pairing neron ne has given a pairing between divisors and points on abelian varieties and also on arbitrary varieties under suitable conditions we shall give a complement to the neron theory on curves let c be a curve by which we mean a complete regular curve geometrically irreducible over a field k with an absolute value v let

the arithmetic hodge index theorem for adelic line bundles in this paper we prove index theorems for integrable metrized line bundles on projective varieties over complete fields and number fields respectively as applications we prove a non archimedean analogue of the calabi theorem and a rigidity theorem about the preperiodic points of algebraic dynamical systems

geometric realization of conformal field theory on riemann surfaces conformal field theory on a family of riemann surfaces is formulated we derive equations of motion of vacua which are parametrized by moduli of riemann surfaces and show that these vacua are characterized uniquely by these equations our theory has a deep connection with sato s theory of kp equations

crossover scaling from classical to non classical critical behaviour interacting physical systems in the neighborhood of criticality and massive continuum field theories can often be characterized by just two physical scales a macroscopic correlation length and a microscopic interaction range related to the coupling and measured by the ginzburg number g a critical crossover limit can be defined when both scales become large while their ratio stays finite the corresponding scaling functions are universal and they are related to the standard field theory renormalization group functions the critical crossover describes the unique flow from the gaussian to the nonclassical fixed point

a dynamical pairing between two rational maps this is the publisher s final pdf the published article is copyrighted by american mathematical society and can be found at http www ams org home page

discriminants and artin conductors we study questions of multiplicities of discriminants for degenerations coming from projective duality over discrete valuation rings the main observation is a type of discriminant different formula in the sense of classical algebraic number theory and we relate it to artin conductors via bloch s conjecture in the case of discriminants of planar curves we can calculate the different precisely in general these multiplicities encode topological invariants of the singular fibers and in the case of characteristic p also wild ramification data in the form of swan conductors this builds upon results of t saito

computing canonical heights using arithmetic intersection theory the canonical height on an abelian variety a defined over a global field k is an object of fundamental importance in the study of the arithmetic of a for many applications it is required to compute p for a given point p a k for instance given generators of a subgroup of the mordell weil group a k of finite index this is necessary for most known approaches to the computation of generators of the mordell weil group a k furthermore the regulator of a k which appears in the statement of the conjecture of birch and swinnerton dyer is defined in terms of the canonical height and thus we need the ability to compute canonical heights in order to gather numerical evidence for the conjecture in the case of positive rank

crossover scaling in two dimensions we determine the scaling functions describing the crossover from ising like critical behavior to classical critical behavior in two dimensional systems with a variable interaction range since this crossover spans several decades in the reduced temperature as well as in the finite size crossover variable it has up to now largely evaded a satisfactory numerical determination using a dedicated monte carlo method we could obtain accurate results for sufficiently large interaction ranges our data cover the full crossover region both above and below the critical temperature and support the hypothesis that the crossover functions are universal also the so called effective exponents are discussed and we show that these can vary nonmonotonically in the crossover region

crossover behavior of the susceptibility and the specific heat near a second order phase transition abstract simple crossover equations for the susceptibility and the specific heat in zero field have been obtained on the basis of the renormalization group method and e expansion the equations contain the ginzburg number as a parameter at temperatures near the critical temperature scaling behavior including the first wegner corrections is reproduced at temperatures far away from the critical temperature the classical landau expansion with square root corrections is recovered for small values of the ginzburg number the crossover equations approach a universal form the equations are applied to represent experimental specific heat data for ch 4 c 2 h 6 ar o 2 and co 2 along the critical isochore in a universal form

crossover critical behavior in the three dimensional ising model the character of critical behavior in physical systems depends on the range of interactions in the limit of infinite range of the interactions systems will exhibit mean field critical behavior i e critical behavior not affected by fluctuations of the order parameter if the interaction range is finite the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class a variety of systems including fluids and anisotropic ferromagnets belongs to the three dimensional ising universality class recent numerical studies of ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation induced critical behavior and mean field type critical behavior in this work we compare these numerical results with a crossover landau model based on renormalization group matching for this purpose we consider an application of the crossover landau model to the three dimensional ising model without fitting to any adjustable parameters the crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range ten orders of the scaled distance to the critical temperature the dependence of the coupling constant on the interaction range governing the crossover critical behavior is discussed