ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

generalized spatial dirichlet process models summary many models for the study of point referenced data explicitly introduce spatial random effects to capture residual spatial association these spatial effects are customarily modelled as a zeromean stationary gaussian process the spatial dirichlet process introduced by gelfand et al 2005 produces a random spatial process which is neither gaussian nor stationary rather it varies about a process that is assumed to be stationary and gaussian the spatial dirichlet process arises as a probability weighted collection of random surfaces this can be limiting for modelling and inferential purposes since it insists that a process realization must be one of these surfaces we introduce a random distribution for the spatial effects that allows different surface selection at different sites moreover we can specify the model so that the marginal distribution of the effect at each site still comes from a dirichlet process the development is offered constructively providing a multivariate extension of the stick breaking representation of the weights we then introduce mixing using this generalized spatial dirichlet process we illustrate with a simulated dataset of independent replications and note that we can embed the generalized process within a dynamic model specification to eliminate the independence assumption

a bayesian analysis of kriging this article is concerned with predicting for gaussian random fields in a way that appropriately deals with uncertainty in the covariance function to this end we analyze the best linear unbiased prediction procedure within a bayesian framework particular attention is paid to the treatment of parameters in the covariance structure and their effect on the quality both real and perceived of the prediction these ideas are implemented using topographical data from davis

perceptually adaptive spread transform image watermarking scheme using hadamard transform the present paper proposes a digital image watermarking scheme using the characteristics of the human visual system hvs spread transform technique and statistical information measure spread transform st scheme is implemented using the transform coefficients of both the host and the watermark signal watermark embedding strength is adaptively adjusted using frequency sensitivity luminance contrast and entropy masking of hvs model the choice of hadamard transform as watermark embedding domain offers several advantages such as low loss in image information higher image fidelity greater reliability of watermark detection and higher data hiding capacity at high degree of compression performance of the proposed method is compared with a number of recently reported watermarking schemes based on spread spectrum ss and quantization index modulation qim

selmer companion curves we say that two elliptic curves e1 e2 over a number field k are n selmer companions for a positive integer n if for every quadratic character of k there is an isomorphism seln e 1 k seln e 2 k between the nselmer groups of the quadratic twists e 1 e 2 we give sufficient conditions for two elliptic curves to be n selmer companions and give a number of examples of non isogenous pairs of companions

selmer groups of quadratic twists of elliptic curves 1 1 e y a1xy a3y x 3 a2x 2 a4x a6 where a1 a2 a3 a4 a6 z let n e denote the conductor of e j e the j invariant of e and l e s n 1 a n n s the hasse weil l function of e if e is modular then let fe z n 1 ae n q n s2 n e 1 be the associated weight 2 cusp form here 1 denotes the trivial dirichlet character throughout d will denote a square free integer and d shall denote the kronecker character for the field q d let h d denote the order of cl d the ideal class group of q d if l is prime then define h d l by

additive watermark detection in the wavelet domain using 2d garch model watermark detection can be generally achieved by statistical decision methods so selection of an accurate statistical model is one of the major issues in watermark detection usually watermark detectors assume the wavelet coefficients to be independent and identically distributed therefore they cannot take into account important characteristics of the wavelet coefficients such as dependency and heteroscedasticity this paper presents a novel detector for wavelet domain additive image watermarking we use two dimensional generalized autoregressive conditional heteroscedasticity 2d garch model for the wavelet coefficients this model can capture the dependency heteroscedasticity and heavy tailed marginal distribution of these coefficients based on 2d garch model we design a new watermark detector and derive its receiver operating characteristics experimental results demonstrate the efficiency of proposed detector and its higher performance compared to alternative watermarking methods in the literature

on the 4 rank of class groups of quadratic number fields we prove that the 4 rank of class groups of quadratic number fields behaves as predicted in an extension due to gerth of the cohen lenstra heuristics

average size of 2 selmer groups of elliptic curves i in this paper we study a class of elliptic curves over q with q torsion group z 2 z 2 and prove that the average order of the 2 selmer groups is bounded

directional rates of change under spatial process models spatial process models are now widely used for inference in many areas of application in such contexts interest is often in the rate of change of a spatial surface at a given location in a given direction examples include temperature or rainfall gradients in meteorology pollution gradients for environmental data and surface roughness assessment for digital elevation models because the spatial surface is viewed as a random realization all such rates of change are random as well we formalize the notions of directional finite difference processes and directional derivative processes building upon the concept of mean square differentiability as developed by stein and banerjee and gelfand we obtain complete distribution theory results under the assumptions of a stationary gaussian process model either for the data or for spatial random effects we present inference under a bayesian framework which in this setting presents several advantages finally we illustrate our methodology with a simulated dat

an image watermarking scheme in wavelet domain with optimized compensation of singular value decomposition via artificial bee colony digital image watermarking is the process of authenticating a digital image by embedding a watermark into it and thereby protecting the image from copyright infringement this paper proposes a novel robust image watermarking scheme developed in the wavelet domain based on the singular value decomposition svd and artificial bee colony abc algorithm the host image is transformed into an invariant wavelet domain by applying redistributed invariant wavelet transform subsequently the low frequency sub band of wavelet transformed image is segmented into non overlapping blocks the most suitable embedding blocks are selected using the human visual system for the watermark embedding the watermark bits are embedded into the target blocks by modifying the first column coefficients of the left singular vector matrix of svd decomposition with the help of a threshold and the visible distortion caused by the embedding is compensated by modifying the coefficients of the right singular vector matrix employing compensation parameters furthermore abc is employed to obtain the optimized threshold and compensation parameters experimental results compared with the related existing schemes demonstrated that the proposed scheme not only possesses the strong robustness against image manipulation attacks but also is comparable to other schemes in term of visual quality

average rank in families of quadratic twists a geometric point of view we investigate the average rank in the family of quadratic twists of a given elliptic curve defined over mathbb q when the curves are ordered using the canonical height of their lowest non torsion rational point

on svd based watermarking algorithm this short communication presents two notes for singular value decomposition svd based watermarking scheme the presented notes can increase the invisibility and capacity when embedding the watermark into u and v components of the svd

continuity properties for random fields l auteur donne pour un champ aleatoire n dimensionnel donne une condition simple sur la fonction de covariance qui garantit l existence d une version du champ aleatoire dans laquelle les realisations sont partout continues la demonstration implique une approximation assez delicate du champ aleatoire par des polynomes d interpellation d ordre eleve

average ranks of elliptic curves tension between data and conjecture rational points on elliptic curves are the gems of arithmetic they are to diophantine geometry what units in rings of integers are to algebraic number theory what algebraic cycles are to algebraic geometry a rational point in just the right context at one place in the theory can inhibit and control thanks to ideas of kolyvagin kol88 the existence of rational points and other mathematical structures elsewhere despite all that we know about these objects the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today we have a network of heuristics and conjectures regarding rational points and we have massive data accumulated to exhibit instances of the phenomena generally we would expect that our data support our conjectures and if not we lose faith in our conjectures but here there is a somewhat more surprising interrelation between data and conjecture they are not exactly in open conflict one with the other but they are no great comfort to each other either we discuss various aspects of this story including recent heuristics and data that attempt to resolve this mystery we shall try to convince the reader that despite seeming discrepancy data and conjecture are in fact in harmony

random maximal isotropic subspaces and selmer groups under suitable hypotheses we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over f p a random subspace chosen with respect to this measure is discrete with probability 1 and the dimension of its intersection with a fixed compact open maximal isotropic subspace is a certain nonnegative integer valued random variable we then prove that the p selmer group of an elliptic curve is naturally the intersection of a discrete maximal isotropic subspace with a compact open maximal isotropic subspace in a locally compact quadratic space over f p by modeling the first subspace as being random we can explain the known phenomena regarding distribution of selmer ranks such as the theorems of heath brown swinnerton dyer and kane for 2 selmer groups in certain families of quadratic twists and the average size of 2 and 3 selmer groups as computed by bhargava and shankar our model is compatible with delaunay s heuristics for p torsion in shafarevich tate groups and predicts that the average rank of elliptic curves over a fixed number field is at most 1 2 many of our results generalize to abelian varieties over global fields

an optimal image watermarking approach based on a multi objective genetic algorithm in accordance with the multi objective nature of image watermarking an optimal image watermarking approach using a multi objective genetic algorithm is presented in this paper both watermarking parameters and embedding positions are often important factors affecting the performance of watermarking systems the proposed multi objective watermarking method can automatically optimize system parameters and a variable length mechanism is specially designed to search the most suitable positions for embedding watermarks the method can also remove the difficult issue of determining optimal watermarking parameters from previous watermarking algorithms the proposed multi objective watermarking method directly deals with the problem of optimizing watermarking under non dominated meaning thus it can effectively avoid the difficulty of determining the optimally weighted factor in existing single objective watermarking schemes in addition a pareto optimal set generated by multi objective optimization can provide flexibility in selecting the most suitable watermarking parameters according to practical requirements

on geodetic distance computations in spatial modeling summary statisticians analyzing spatial data often need to detect and model associations based upon distances on the earth s surface accurate computation of distances are sought for exploratory and interpretation purposes as well as for developing numerically stable estimation algorithms when the data come from locations on the spherical earth application of euclidean or planar metrics for computing distances is not straightforward yet planar metrics are desirable because of their easier interpretability easy availability in software packages and well established theoretical properties while distance computations are indispensable in spatial modeling their importance and impact upon statistical estimation and prediction have gone largely unaddressed this article explores the different options in using planar metrics and investigates their impact upon spatial modeling

non vanishing of quadratic twists of modular l functions k have been the subject of much study both because of their intrinsicinterest and because of the prominent role they have played inkolyvagin s work on the birch and swinnerton dyer conjecture see b f h i j ko ma m m m1 o s p p waldspurger proved a fundamental theorem the ore me 1 w1 relating these central critical values to the fourier coe cients of half integral weight cusp forms for notational convenience if d is afundamental discriminant of a quadratic number eld then de ne d

distribution of selmer groups of quadratic twists of a family of elliptic curves we study the distribution of the size of the selmer groups arising from a 2 isogeny and its dual 2 isogeny for quadratic twists of elliptic curves with full 2 torsion points in we show that one of these selmer groups is almost always bounded while the 2 rank of the other follows a gaussian distribution this provides us with a small tate shafarevich group and a large tate shafarevich group when combined with a result obtained by yu g yu on the quadratic twists of a family of elliptic curves mathematika 52 1 2 2005 139 154 2006 this shows that the mean value of the 2 rank of the large tate shafarevich group for square free positive integers n less than x is as x 8

p ell torsion points in finite abelian groups and combinatorial identities abstract the main aim of this article is to compute all the moments of the number of p l torsion elements in some type of finite abelian groups the averages involved in these moments are those defined for the cohen lenstra heuristics for class groups and their adaptation for tate shafarevich groups in particular we prove that the heuristic model for tate shafarevich groups is compatible with the recent conjecture of poonen and rains about the moments of the orders of p selmer groups of elliptic curves for our purpose we are led to define certain polynomials indexed by integer partitions and to study them in a combinatorial way moreover from our probabilistic model we derive combinatorial identities some of which appearing to be new the others being related to the theory of symmetric functions in some sense our method therefore gives for these identities a somehow natural algebraic context

binary quartic forms having bounded invariants and the boundedness of the average rank of elliptic curves we prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants this extends to the quartic case the classical results of gauss and davenport in the quadratic and cubic cases respectively our techniques are quite general and may be applied to counting integral orbits in other representations of algebraic groups we use these counting results to prove that the average rank of elliptic curves over q when ordered by their heights is bounded in particular we show that when elliptic curves are ordered by height the mean size of the 2 selmer group is 3 this implies that the limsup of the average rank of elliptic curves is at most 1 5

the size of selmer groups for the congruent number problem where np is the number of solutions of the congruence y x d2x mod p then ld s has an analytic continuation as an entire function on the complex plane the conjecture of birch and swinnerton dyer then states in particular that the rank r d of ed is equal to the order r d of ld s at s 1 this being the so called analytic rank while we cannot at present find r d in all cases we can at least determine whether or not ld 1 0 and hence conjecturally whether or not r d 0 moreover one has a functional equation for ld s which relates its values at s and 2 s via a sign change ed 1 one may deduce that 1 r d ed it would then follow from the conjecture of birch and swinnerton dyer that the rank is positive whenever ed 1 according to the calculations of birch and stephens 2 one has

how many rational points does a random curve have a large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties for an elliptic curve over q the set of rational points forms a finitely generated abelian group the ranks of these groups when ranging over all elliptic curves are conjectured to be evenly distributed between rank 0 and rank 1 with higher ranks being negligible we will describe these conjectures and discuss some results on bounds for average rank highlighting the recent work of bhargava and shankar

disparity in selmer ranks of quadratic twists of elliptic curves we study the parity of 2 selmer ranks in the family of quadratic twists of an arbitrary elliptic curve e over an arbitrary number eld k we prove that the fraction of twists of a given elliptic curve over a xed number eld having even 2 selmer rank exists as a stable limit over the family of twists and we compute this fraction as an explicit product of local factors we give an example of an elliptic curve e such that as k varies these fractions are dense in 0 1 more generally our results also apply to p selmer ranks of twists of 2 dimensional self dual fp representations of the absolute galois group of k by characters of order p

l series with nonzero central critical value suppose that f zn 1 an f qtn is a cusp form of weight 2k k e n we denote by l f s the l function of f for re s sufficiently large the value of l f s is given by l f s en 1 an and one can show that l f s has analytic continuation to the entire complex plane the value of l f s at s k will be of particular interest to us and we will refer to this value as the central critical value of l f s let xd denote the dirichlet character associated to the extension q d q that is xd n va where ad denotes the discriminant of q d q define the dth quadratic twist of f to be fxd en 1 an f xd n qn for any integer d the l function of fx is the dth quadratic twist of l f s that is l fxd s zn aan f xd n we will be interested in determining how often l fxd s has nonzero central critical value as d varies over all integers since xd 2 xd we will restrict our attention to the square free integers d we expect that as we let d vary over all of the square free integers a positive proportion of the l functions l fxd s will have nonzero central critical value indeed goldfeld 7 conjectures that for newforms f of weight 2 l fxd 1 78 0 for 1 of the square free integers given an elliptic curve e y2 x3 ax2 bx c a b c e z with conductor ne and an integer d we define the dth quadratic twist of e to be the curve ed y2 x3 adx2 bd2x cd3 let l ed s denote the l function associated to ed for square free d coprime to 2ne l ed s is simply the dth quadratic twist of l ei s if f e s2 n is a newform with integer coefecients we know via the theory of eichler and shimura that there is an elliptic curve e over q having conductor n so that l e s l f s thus if d is coprime to 2n then l ed s l fxd s also one knows from the work of kolyvagin 13 as supplemented by the work of murty and murty 17 or that of bump friedberg and hoffstein 3 see also 10 for a shorter proof that if e is a modular elliptic curve and if l e 1 7 0 then the rank of e is 0 thus if f has the property that a positive proportion of the twists of l f s have nonzero central critical value then this implies that a positive density of the quadratic twists ed have rank 0 there have been many papers which have proved results in this direction for example in 2 3 6 10 16 17 19 28 one can find general theorems on

a new reliable optimized image watermarking scheme based on the integer wavelet transform and singular value decomposition for copyright protection although image watermarking schemes based on singular value decomposition svd demonstrate high robustness and imperceptibility they are exposed to the false positive problem fpp this drawback mostly occurs when embedding steps depend on singular values while singular vectors are used as secret keys in this study a new reliable svd based image watermarking scheme that uses integer wavelet transform iwt is proposed to overcome fpp and fulfil all watermarking requirements unlike in other schemes the s and v matrices of the watermark are used as secret keys whereas the s singular vector of the watermark is embedded into the singular values of the host image the additional secret key is obtained from the watermarked image during the embedding process to increase security and avoid fpp completely to improve the robustness as well as achieve balance between robustness and imperceptibility multi objective ant colony optimization moaco is utilized to find the optimal scaling factors namely multiple zooming factors results of the robustness imperceptibility and reliability tests demonstrate that the proposed iwt svd moaco scheme outperforms several previous schemes and avoids fpp completely

bayesian design and analysis of computer experiments use of derivatives in surface prediction this article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain t given values of yo and all of its first derivatives at a set of design sites points in t the intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in t represents a particular configuration of the input parameters it is assumed that the first derivatives are already available e g from a sensitivity analysis or can be produced by the code that implements the model a bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary gaussian stochastic process is used the calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example the issue of experimental design is also discussed in particular the criterion of maximizing the reduction in entropy

distributed kriged kalman filter for spatial estimation this paper considers robotic sensor networks performing spatially distributed estimation tasks a robotic sensor network is deployed in an environment of interest and takes successive point measurements of a dynamic physical process modeled as a spatio temporal random field taking a bayesian perspective on the kriging interpolation technique from geostatistics we design the distributed kriged kalman filter for predictive inference of the random field and of its gradient the proposed algorithm makes use of a novel distributed strategy to compute weighted least squares estimates when measurements are spatially correlated this strategy results from the combination of the jacobi overrelaxation method with dynamic average consensus algorithms as an application of the proposed algorithm we design a gradient ascent cooperative strategy and analyze its convergence properties in the absence of measurement errors via stochastic lyapunov functions we illustrate our results in simulation

the square free sieve and the rank of elliptic curves let e be an elliptic curve over q a celebrated theorem of mordell asserts that e q the abelian group of rational points of e is finitely generated by the rank of e we mean the rank of e q thus the rank of e is positive if and only if e possesses an infinity of rational points relatively few general qualitative assertions can be made about the rank as e varies how large can the rank get although we expect that there are elliptic curves over q with arbitrarily high ranks this is presently unknown what is the average size of the rank recently brumer and mcguinness have reported on their study of 310 716 elliptic curves of prime conductor less than 108 where they have found that 20 06 of those curves have even rank 2 cf bm also see forthcoming work of brumer where subject to a number of standard conjectures he shows that 2 3 is an upper bound for the average rank for all elliptic curves over q ordered in terms of their faltings height what is the behavior of the rank over the family of twists of a given elliptic curve here one has three natural kinds of families of twists 1 quadratic twists one can take any elliptic curve and systematically twist it by all quadratic characters this is the type of family of elliptic curves we are concerned with in this paper specifically if e is an elliptic curve over q given by the weierstrass equation y2 x a x b and d is any squarefree integer the quadratic twist of e by d ed is given by the equation

the parity of the period of the continued fraction of d we call a positive square free integer d special if d is not divisible by primes congruent to 3 mod 4 we show that the period of the expansion of ad in continued fractions is asymptotically more often odd than even when we restrict to special integers we note that this period is always even for a non special square free integer d it is well known that the above period is odd if and only if the negative pell equation x2 dy2 1 is solvable the latter problem is solvable if and only if the narrow and the ordinary class groups of q ad are equal in a prior work we fully described the asymptotics of the 4 ranks of those class groups here we get the first non trivial results about the asymptotic behavior of the 8 rank of the narrow class group for example we show that more than 76 of the quadratic fields q ad where d is special have the property that the 8 rank of the narrow class group is zero

a unified theory of surface roughness the properties of the peaks and summits of a rough surface are predicted on the assumption that the surface is two dimensional random noise the important result is that in non dimensional form the answers depend only to a minor degree on the parameters describing the surface or on the sampling interval used on the other hand the absolute values are strongly dependent on the sampling interval experimental results on a real surface agree remarkably well with the predictions