asymptotic expansions in the clt in free probability we prove edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions the proofs are based on the analytic definition of free convolution

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 note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices recently johansson and johnstone proved that the distribution of the properly rescaled largest principal component of the complex real wishart matrix x x xtx converges to the tracy widom law as n p the dimensions of x tend to in some ratio n p 0 we extend these results in two directions first of all we prove that the joint distribution of the first second third etc eigenvalues of a wishart matrix converges after a proper rescaling to the tracy widom distribution second of all we explain how the combinatorial machinery developed for wigner random matrices in refs 27 38 and 39 allows to extend the results by johansson and johnstone to the case of x with non gaussian entries provided n p o p1 3 we also prove that max n1 2 p1 2 2 o p1 2 log p a e for general 0

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

addition of freely independent random variables abstract a direct proof is given of voiculescu s addition theorem for freely independent real valued random variables using resolvents of self adjoint operators in contrast to the original proof no assumption is made on the existence of moments above the second

stable laws and domains of attraction in free probability theory in this paper we determine the distributional behavior of sums of free in the sense of voiculescu identically distributed inflnitesimal random variables the theory is shown to parallel the classical theory of independent random variables though the limit laws are usually quite difierent our work subsumes all previously known instances of weak convergence of sums of free identically distributed random variables in particular we determine the domains of attraction of stable distributions in the free theory these freely stable distributions are studied in detail in the appendix where their unimodality and duality properties are demonstrated

composition algebras over a ring of fractions abstract using some theorems on composition algebras over rings of genus zero and elementary results from algebraic geometry all composition algebras over a ring of fractions related to the projective line over a field are enumerated and partly classified

superconvergence to the central limit and failure of the cramer theorem for free random variables we show that convergence of the semicircle law in the free central limit theorem for bounded random variables is much better than expected thus the distributions which tend to the semicircle become absolutely continuous in finite time and the densities converge in a very strong sense we also show that the semicircle law is the free convolution of laws which are not semicircular thus proving that cramer s classical result for the normal distribution does not have a free counterpart

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

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

on superconvergence of sums of free random variables this paper derives sufficient conditions for superconvergence of sums of bounded free random variables and provides an estimate for the rate of superconvergence

no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it is known that as n to infty if n n converges to a positive number and the empirical distribution of the eigenvalues of t n converges to a proper probability distribution then the empirical distribution of the eigenvalues of b n converges a s to a nonrandom limit in this paper we prove that under certain conditions on the eigenvalues of t n for any closed interval outside the support of the limit with probability 1 there will be no eigenvalues in this interval for all n sufficiently large

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

the central limit theorem for free additive convolution abstract let xn n 1be a sequence of free identically distributed random variables with common distribution then there exist sequences bn n 1and an n 1of positive and real numbers respectively such that sequence of random variables formula converges in distribution to the semicircle law if and only if the function formula is slowly varying in karamata s sense in other words the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the gaussian we prove an analogous result for normal domains of attraction in the sense of linnik

composition algebras over algebraic curves of genus zero we rephrase the classical theory of composition algebras over fields particularly the cayley dickson doubling process and zorn s vector matrices in the setting of locally ringed spaces fixing an arbitrary base field we use these constructions to classify composition algebras over complete smooth curves of genus zero applications are given to composition algebras over function fields of genus zero and polynomial rings

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

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

local limit theorems in free probability theory in this paper we study the superconvergence phenomenon in the free central limit theorem for identically distributed unbounded summands we prove not only the uniform convergence of the densities to the semicircular density but also their l p convergence to the same limit for p 1 2 moreover an entropic central limit theorem is obtained as a consequence of the above results

composition algebras over rings of fractions revisited abstract let k be a field of characteristic not two let f h x 0 x 1 k x 0 x 1 be an irreducible homogeneous polynomial and denote the ring of elements of degree zero in the homogeneous localization k x 0 x 1 f h by k x 0 x 1 f h for deg f h 3 it is proved that the composition algebras over k x 0 x 1 f h not containing zero divisors are defined over k and that there is at most one split composition algebra not defined over k for deg f h 4 all composition algebras over k x 0 x 1 f h are enumerated and partly classified

the law of large numbers for free identically distributed random variables let x 1 x 2 be a sequence of free identically distributed random variables with common distribution it was shown by lindsay and pata in a more general context that a sufficient condition for the weak law of large numbers to hold for the sequence x 1 x2 is that formula math we show that this condition is necessary as well as sufficient even though the condition is identical with the corresponding one for commuting independent variables the proof of the result uses the analytical techniques of free convolution theory and it is quite different from the proof of the commutative theorem due to kolmogorov cf feller 1971 an introduction to probability theory and its applications wiley new york