measuring and testing dependence by correlation of distances distance correlation is a new measure of dependence between random vectors distance covariance and distance correlation are analogous to product moment covariance and correlation but unlike the classical definition of correlation distance correlation is zero only if the random vectors are independent the empirical distance dependence measures are based on certain euclidean distances between sample elements rather than sample moments yet have a compact representation analogous to the classical covariance and correlation asymptotic properties and applications in testing independence are discussed implementation of the test and monte carlo results are also presented

q curves and their manin ideals let c be an elliptic curve over q and let denote its neron differential uniquely determined up to sign according to a conjecture of manin stevens there is a modular parametrization x 1 n c defined over q such that the so called manin constant c is equal to 1 the constant c is defined by the equality c f q dq q where f is a normalized newform of weight 2 on 1 n in this paper we propose a generalization of the manin constant as a certain ideal we call it the manin ideal attached to modular parametrizations of elliptic curves defined over number fields we conjecture that its value is 1 generalizing manin stevens and show the manin ideal to be an integral ideal involving only primes dividing twice the conductor motivated by the manin ideal considerations we first study several aspects of building blocks of modular abelian varieties and modular parametrizations of q curves some examples are included to provide numerical evidence of the generalized conjecture and the paper also contains the analogous items on the manin ideals for modular building blocks of higher dimension

root numbers of abelian varieties we generalize a theorem of d rohrlich concerning root numbers of elliptic curves over number fields our result applies to arbitrary abelian varieties namely under certain conditions which naturally extend the conditions used by d rohrlich we show that the root number w a associated to an abelian variety a over a number field f and a complex finite dimensional irreducible representation of gal f f with real valued character is equal to 1 we also show that our result is consistent with a refined version of the conjecture of birch and swinnerton dyer

a new perturbation analysis for signal enumeration in rotational invariance techniques the esprit algorithm is a subspace based high resolution method used in source localization and spectral analysis which provides very accurate estimates of the signal parameters however the underlying theory assumes a known model order which is usually not the case in many applications in particular it is well known that underevaluating the model order biases the estimation in this paper we analyze the perturbation induced by an erroneous model order and we present an error bound for the estimated parameters based on this theoretical framework we propose a new method for selecting an appropriate modeling order which consists in minimizing the error bound this approach is applied to both synthetic and musical signals and its performance is compared to that of existing methods such as the information theoretic criteria

regulator constants and the parity conjecture the p parity conjecture for twists of elliptic curves relates multiplicities of artin representations in p selmer groups to root numbers in this paper we prove this conjecture for a class of such twists for example if e is semistable at 2 and 3 k is abelian and k is its maximal pro p extension then the p parity conjecture holds for twists of e by all orthogonal artin representations of mathop mathrm gal k infty mathbb q we also give analogous results when k is non abelian the base field is not and e is replaced by an abelian variety the heart of the paper is a study of relations between permutation representations of finite groups their regulator constants and compatibility between local root numbers and local tamagawa numbers of abelian varieties in such relations

on special zeros of p adic l functions of hilbert modular forms let e be a modular elliptic curve over a totally real number field f we prove the weak exceptional zero conjecture which links a higher derivative of the p adic l function attached to e to certain p adic periods attached to the corresponding hilbert modular form at the places above p where e has split multiplicative reduction under some mild restrictions on p and the conductor of e we deduce the exceptional zero conjecture in the strong form i e where the automorphic p adic periods are replaced by the mathcal l invariants of e defined in terms of tate periods from a special case proved earlier by mok crucial for our method is a new construction of the p adic l function of e in terms of local data

stickelberger elements and modular parametrizations of elliptic curves in the present paper we shall give evidence to support the claim conjecture i below and 1 3 that every elliptic curve a o which can be parametrized by modular functions admits a canonical modular parametrization whose properties can be related to intrinsic properties of a in particular we will see how such a parametrizat ion can be used to prove some rather pleasant integrality properties of stickelberger elements ad p adic l functions attached to a in addition if conjecture i is true then we can give an intrinsic characterization of the isomorphism class of a special elliptic curve in the q isogeny class of a distinguished by modular considerations for most of the paper we have opted for the concrete approach and defined modular parametrizations in terms of x i n definition 1 1 however to justify our view of these parametrizations as being canonical we begin here with a more intrinsic definition recall that shimura 19 chap 6 see w 1 of this paper has defined a compatible system of canonical models of modular curves xs ses where 5 p is a certain collection of open subgroups of the group gl 2 az over the finite adeles a i of q we define the adelic upper half plane to be the pro variety ll m xs and give the q structure induced by the s field of modular functions whose q expansions at the 0 cusp have coefficients in q a modular parametrization of a is a q morphism a which sends

p adic l functions and p adic periods of modular forms let e be an elliptic curve which is defined over q and has stable reduction modulo a given prime p assuming that e is modular one can associate to e a p adic l function lp e s see mz swd a v vi mz t t for its construction in various cases this function is defined by a certain interpolation property and is analytic for sezp in this paper we will assume that e has split multiplicative reduction at p under this assumption the interpolation property implies that lp e 1 0 we will prove a formula for ep e 1 which was discovered experimentally by mazur tate and teitelbaum mz t t by tate s p adic urfiformization theory there is a p adic integer q epzp which we refer to as the tate period for e and a p adic analytic isomorphism

hida families and rational points on elliptic curves let e be an elliptic curve over q of conductor n mp having a prime p n of multiplicative reduction because e is modular it corresponds to a normalised weight two eigenform on 0 n whose q expansion is denoted f n anq letx hom z p z p z p 1 z zp which contains z as a dense subset by associating to k z the character x xk 2 denote by a u the ring of cp valued p adic analytic functions on a compact open subset u of x hida s theory associates to f a neighborhood u of 2 x which can be assumed for simplicity to be contained in the residue class of 2 modulo p 1 and a formal q expansion

cramer rao bounds for multiple poles and coefficients of quasi polynomials in colored noise in this paper we provide analytical expressions of the cramer rao bounds for the frequencies damping factors amplitudes and phases of complex exponentials in colored noise these expressions show the explicit dependence of the bounds of each distinct parameter with respect to the amplitudes and phases leading to readily interpretable formulae which are then simplified in an asymptotic context the results are presented in the general framework of the polynomial amplitude complex exponentials pace model also referred to as the quasi polynomial model in the literature which accounts for systems involving multiple poles and represents a signal as a mixture of complex exponentials modulated by polynomials this work looks further and generalizes the studies previously undertaken on the exponential and the quasi polynomial models

sampling moments and reconstructing signals of finite rate of innovation shannon meets strang fix consider the problem of sampling signals which are not bandlimited but still have a finite number of degrees of freedom per unit of time such as for example nonuniform splines or piecewise polynomials and call the number of degrees of freedom per unit of time the rate of innovation classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited recently it was shown that by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation it is possible to reconstruct such signals uniquely these sampling schemes however use kernels with infinite support and this leads to complex and potentially unstable reconstruction algorithms in this paper we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm the class of kernels that we can use is very rich and includes functions satisfying strang fix conditions exponential splines and functions with rational fourier transform this last class of kernels is quite general and includes for instance any linear electric circuit we thus show with an example how to estimate a signal of finite rate of innovation at the output of an rc circuit the case of noisy measurements is also analyzed and we present a novel algorithm that reduces the effect of noise by oversampling

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

parameter estimation for nonincreasing exponential sums by prony like methods abstract let z j e f j with f j 0 i be distinct nodes for j 1 m with complex coefficients c j 0 we consider a nonincreasing exponential sum h x c 1 e f 1 x c m e f m x x 0 many applications in electrical engineering signal processing and mathematical physics lead to the following problem determine all parameters of h if 2 n sampled values h k k 0 2 n 1 n m are given this parameter estimation problem is a nonlinear inverse problem for noiseless sampled data we describe the close connections between prony like methods namely the classical prony method the matrix pencil method and the esprit method further we present a new efficient algorithm of matrix pencil factorization based on qr decomposition of a rectangular hankel matrix the algorithms of parameter estimation are also applied to sparse fourier approximation and nonlinear approximation

integration on cal h _p times cal h and arithmetic applications this article describes a conjectural p adic analytic construction of global points on modular elliptic curves points which are defined over the ring class fields of real quadratic fields the resulting conjectures suggest that the classical heegner point construction and the theory of complex multiplication on which it is based should extend to a variety of contexts in which the underlying field is not a cm field

the rationality of stark heegner points over genus fields of real quadratic fields we study the algebraicity of stark heegner points on a modular elliptic curve e these objects are adic points on e given by the values of certain adic integrals but they are conjecturally defined over ring class fields of a real quadratic field k the present article gives some evidence for this algebraicity conjecture by showing that linear combinations of stark heegner points weighted by certain genus characters of k are defined over the predicted quadratic extensions of k the non vanishing of these combinations is also related to the appropriate twisted hasse weil l series of e over k in the spirit of the gross zagier formula for classical heegner points

modular curves of genus 2 we prove that there are exactly 149 genus two curves c defined over q such that there exists a nonconstant morphism x1 n c defined over q and the jacobian of c is q isogenous to the abelian variety af attached by shimura to a newform f s2 1 n we determine the corresponding newforms and present equations for all these curves

elementary matrix decomposition and the computation of darmon points with higher conductor we extend the algorithm of dg02 and dp06 for computing p adic darmon points on elliptic curves to the case of composite conductor we also extend the algorithm of dl03 for computing atr darmon points to treat curves of nontrivial conductor both cases involve an algorithmic decomposition into elementary matrices in congruence subgroups 1 n for ideals n in certain rings of s integers we use these extensions to provide additional evidence in support of the conjectures on the rationality of darmon points

testing mutual independence in high dimension via distance covariance summary we introduce an l2 type test for testing mutual independence and banded dependence structure for high dimensional data the test is constructed on the basis of the pairwise distance covariance and it accounts for the non linear and non monotone dependences among the data which cannot be fully captured by the existing tests based on either pearson correlation or rank correlation our test can be conveniently implemented in practice as the limiting null distribution of the test statistic is shown to be standard normal it exhibits excellent finite sample performance in our simulation studies even when the sample size is small albeit the dimension is high and is shown to identify non linear dependence in empirical data analysis successfully on the theory side asymptotic normality of our test statistic is shown under quite mild moment assumptions and with little restriction on the growth rate of the dimension as a function of sample size as a demonstration of good power properties for our distance covariance based test we further show that an infeasible version of our test statistic has the rate optimality in the class of gaussian distributions with equal correlation

extremal probabilities for gaussian quadratic forms denote by q an arbitrary positive semidefinite quadratic form in centered gaussian random variables such that e q 1 we prove that for an arbitrary x 0 inf q p q x p 2 n n x where n 2 is a chi square distributed rv with n n x degrees of freedom n x is a non increasing function of x n 1 iff x x 1 1 5364 n 2 iff x x 2 x 1 where x 2 1 2989 etc n x rank q a similar statement is not true for the supremum if 1 x 2 and z 1 z 2 are independent standard gaussian rv s then sup0 1 2p z 1 2 1 z 2 2 x is taken not at 0 or at 1 2 but at 0 x 1 2 where x is a continuous increasing function from 1 0 to 2 1 2 e g 1 5 15 applications of our theorems include asymptotic quantiles of u and v statistics signal detection and stochastic orderings of integrals of squared gaussian processes

on rigid analytic uniformizations of jacobians of shimura curves the main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the jacobian of a shimura curve over bbb q at a prime dividing exactly the level this result can be viewed as complementary to the classical theorem of v c erednik and drinfeld which provides rigid analytic uniformizations at primes dividing the discriminant as a corollary we offer a proof of a conjecture formulated by m greenberg in his paper on stark heegner points and quaternionic shimura curves thus making greenberg s construction of local points on elliptic curves over bbb q unconditional

on heegner points for primes of additive reduction ramifying in the base field let e be a rational elliptic curve and k be an imaginary quadratic field in this article we give a method to construct heegner points when e has a prime bigger than 3 of additive reduction ramifying in the field k the ideas apply to more general contexts like constructing darmon points attached to real quadratic fields which is presented in the appendix

distance multivariance new dependence measures for random vectors we introduce two new measures for the dependence of n ge2 random variables distance multivariance and total distance multivariance both measures are based on the weighted l 2 distance of quantities related to the characteristic functions of the underlying random variables these extend distance covariance introduced by szekely rizzo and bakirov from pairs of random variables to n tuplets of random variables we show that total distance multivariance can be used to detect the independence of n random variables and has a simple finite sample representation in terms of distance matrices of the sample points where distance is measured by a continuous negative definite function under some mild moment conditions this leads to a test for independence of multiple random vectors which is consistent against all alternatives

almost totally complex points on elliptic curves let f f0 be a quadratic extension of totally real number fields and let e be an elliptic curve over f which is isogenous to its galois conjugate over f0 a quadratic extension m f is said to be almost totally complex atc if all archimedean places of f but one extend to a complex place of m the main goal of this note is to provide a new construction of a supply of darmon like points on e which are conjecturally defined over certain ring class fields of m these points are constructed by means of an extension of darmon s atr method to higher dimensional modular abelian varieties from which they inherit the following features they are algebraic provided darmon s conjectures on atr points hold true and they are explicitly computable as we illustrate with a detailed example that provides numerical evidence for the validity of our conjectures

a stable numerical method for inverting shape from moments we derive a stable technique based upon matrix pencils for the reconstruction of or approximation by polygonal shapes from moments we point out that this problem can be considered the dual of 2 d numerical quadrature over polygonal domains an analysis of the sensitivity of the problem is presented along with some numerical examples illustrating the relevant points finally an application to the problem of gravimetry is explored where the shape of a gravitationally anomalous region is to be recovered from measurements of its exterior gravitational field

matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise a study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented comparison of this method to a polynomial method svd prony method shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes it is found through perturbation analysis and simulation that for signals with unknown damping factors the pencil method is less sensitive to noise than the polynomial method an expression of the cramer rao bound for the exponential signals is presented

stark heegner points and the cohomology of quaternionic shimura varieties let f be a totally real field of narrow class number one and let e f be a modular semistable elliptic curve of conductor n 1 l etk f be a non cm quadratic extension with disc k n 1 such that the sign in the functional equation of l e k s is 1 suppose further that there is a prime p n that is inert in k w e describe a p adic construction of points on e which we conjecture to be rational over ring class fields of k f and satisfy a shimura reciprocity law these points are expected to behave like classical heegner points and can be viewed as new instances of the stark heegner point construction of 5 the key idea in our construction is a reinterpretation of darmon s theory of modular symbols and mixed period integrals in terms of group cohomology

special values of l functions and the arithmetic of darmon points building on our previous work on rigid analytic uniformizations we introduce darmon points on jacobians of shimura curves attached to quaternion algebras over q and formulate conjectures about their rationality properties moreover if k is a real quadratic field e is an elliptic curve over q without complex multiplication and is a ring class character such that lk e 1 6 0 we prove a gross zagier type formula relating darmon points to a suitably defined algebraic part of lk e 1 this generalizes results of bertolini darmon and dasgupta to the case of division quaternion algebras and arbitrary characters finally as an application of this formula assuming the rationality conjectures for darmon points we obtain vanishing results for selmer groups of e over extensions of k contained in narrow ring class fields when the analytic rank of e is zero as predicted by the birch and swinnerton dyer conjecture

darmon points on elliptic curves over number fields of arbitrary signature we present new constructions of complex and pp adic darmon points on elliptic curves over base fields of arbitrary signature we conjecture that these points are global and present numerical evidence to support our conjecture

uniformization of modular elliptic curves via p adic periods abstract the langlands programme predicts that a weight 2 newform f over a number field k with integer hecke eigenvalues generally should have an associated elliptic curve e f over k in 19 we associated building on works of darmon 8 and greenberg 20 a p adic lattice to f under certain hypothesis and implicitly conjectured that is commensurable with the p adic tate lattice of e f in this paper we present this conjecture in detail and discuss how it can be used to compute directly from f a weierstrass equation for the conjectural e f we develop algorithms to this end and implement them in order to carry out extensive systematic computations in which we compute weierstrass equations of hundreds of elliptic curves some with huge heights over dozens of number fields the data we obtain give extensive support for the conjecture and furthermore demonstrate that the conjecture provides an efficient tool to building databases of elliptic curves over number fields

euler factors determine local weil representations we show that a frobenius semisimple weil representation over a local fieldk is determined by its euler factors over the extensions ofk the construction is explicit and we illustrate it for l adic representations attached to elliptic and genus 2 curves as an application we construct an absolutely simple 2 dimensional abelian variety over q all of whose quadratic twists must have positive rank according to the birch swinnerton dyer conjecture

kernel based tests for joint independence summary we investigate the problem of testing whether d possibly multivariate random variables which may or may not be continuous are jointly or mutually independent our method builds on ideas of the two variable hilbert schmidt independence criterion but allows for an arbitrary number of variables we embed the joint distribution and the product of the marginals in a reproducing kernel hilbert space and define the d variable hilbert schmidt independence criterion dhsic as the squared distance between the embeddings in the population case the value of dhsic is 0 if and only if the d variables are jointly independent as long as the kernel is characteristic on the basis of an empirical estimate of dhsic we investigate three non parametric hypothesis tests a permutation test a bootstrap analogue and a procedure based on a gamma approximation we apply non parametric independence testing to a problem in causal discovery and illustrate the new methods on simulated and real data sets

equivalence of distance based and rkhs based statistics in hypothesis testing we provide a unifying framework linking two classes of statistics used in two sample and independence testing on the one hand the energy distances and distance covariances from the statistics literature on the other maximum mean discrepancies mmd that is distances between embeddings of distributions to reproducing kernel hilbert spaces rkhs as established in machine learning in the case where the energy distance is computed with a semimetric of negative type a positive definite kernel termed distance kernel may be defined such that the mmd corresponds exactly to the energy distance conversely for any positive definite kernel we can interpret the mmd as energy distance with respect to some negative type semimetric this equivalence readily extends to distance covariance using kernels on the product space we determine the class of probability distributions for which the test statistics are consistent against all alternatives finally we investigate the performance of the family of distance kernels in two sample and independence tests we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels and that other choices from this family can yield more powerful tests

sparse polynomial interpolation in chebyshev bases abstract we study the problem of reconstructing a sparse polynomial in a basis of chebyshev polynomials chebyshev basis in short from given samples on a chebyshev grid of 1 1 a polynomial is called m sparse in a chebyshev basis if it can be represented by a linear combination of m chebyshev polynomials for a polynomial with known and unknown chebyshev sparsity respectively we present efficient reconstruction methods where prony like methods are used the reconstruction results are mainly presented for bases of chebyshev polynomials of first and second kind respectively but similar issues can be obtained for bases of chebyshev polynomials of third and fourth kind respectively

the teitelbaum conjecture in the indefinite setting let f be a new cusp form on gamma 0 n of even weight k 2 geq 2 suppose that there is a prime p parallel n and that we may write n pn n where n is the squarefree product of an even number of primes there is a darmon style cal l invariant cal l n f attached to this factorization which is the orton cal l invariant when n 1 we prove that cal l n f does not depend on the chosen factorization of n and it is equal to the other known cal l invariants we also give a formula for the computation of the logarithmic p adic abel jacobi image of the darmon cycles this formula is crucial for the computations of the derivatives of the p adic l functions of the weight variable attached to a real quadratic field k bbb q such that the primes dividing n are split and the primes dividing pn are inert