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

modified subspace algorithms for doa estimation with large arrays this paper proposes the use of a new generalized asymptotic paradigm in order to analyze the performance of subspace based direction of arrival doa estimation in array signal processing applications instead of assuming that the number of samples is high whereas the number of sensors antennas remains fixed the asymptotic situation analyzed herein assumes that both quantities tend to infinity at the same rate this asymptotic situation provides a more accurate description of a potential situation where these two quantities are finite and hence comparable in magnitude it is first shown that both music and ssmusic are inconsistent when the number of antennas sensors increases without bound at the same rate as the sample size this is done by analyzing and deriving closed form expressions for the two corresponding asymptotic cost functions by examining these asymptotic cost functions one can establish the minimum number of samples per antenna needed to resolve closely spaced sources in this asymptotic regime next two alternative estimators are constructed that are strongly consistent in the new asymptotic situation i e they provide consistent doa estimates not only when the number of snapshots goes to infinity but also when the number of sensors antennas increases without bound at the same rate these estimators are inspired by the theory of g estimation and are therefore referred to as g music and g ssmusic respectively simulations show that the proposed algorithms outperform their traditional counterparts in finite sample size situations although they still present certain limitations

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

constructing semisimple p adic galois representations with prescribed properties consider a characteristic p representation of the absolute galois group of the rational numbers in this paper we show how to deform this representation to the p adics while guaranteeing that the characteristic polynomials of frobenius at a density one set of primes are algebraic and pure of specified weight the resulting representation is ramified at an infinite density zero set of primes as a consequence of the technique of proof we show that one can compatibly lift a mod pq representation again the resulting representation is ramified at infinitely many primes

spatial spectrum estimation in a location sector multiple narrowband source localization using arbitrarily configured arrays and spatial spectrum estimation is considered a new eigenspace based approach which uses projections onto a particular vector or vector set in the estimated noise only subspace is described several closest vector estimators are developed by using different measures of closeness first closest is a novel full dimensional element space approach to spatial spectrum estimation which has important performance advantages relative to pertinent established spatial spectrum estimators it incorporates a priori knowledge of the array manifold over a location sector of interest to provide snr spectral resolution thresholds which are lower than those of min norm for some arrays substantially lower second relationships between the closest approach and several established approaches to spatial spectrum estimation are established for a linear equispaced array min norm is shown to be a special case of the closest approach one which is based on projection onto a noise only subspace vector which is close to the array manifold over the entire field of view

congruences between modular forms raising the level and dropping euler factors abstract we discuss the relationship among certain generalizations of results of hida ribet and wiles on congruences between modular forms hida s result accounts for congruences in terms of the value of an l function and ribet s result is related to the behavior of the period that appears there wiles theory leads to a class number formula relating the value of the l function to the size of a galois cohomology group the behavior of the period is used to deduce that a formula at nonminimal level is obtained from one at minimal level by dropping euler factors from the l function

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

a rigid analytic gross zagier formula and arithmetic applications let f be a newform of weight 2 and squarefree level n its fourier coefficients generate a ring of whose fraction field kf has finite degree over q fix an imaginary quadratic field k of discriminant prime to n corresponding to a dirichlet character e the l series l f k s l f s l f 0 e s of f over k has an analytic continuation to the whole complex plane and a functional equation relating l f k s to l f k 2 s assume that the sign of this functional equation is 1 so that l f k s vanishes to even order at s 1 this is equivalent to saying that the number of prime factors of n which are inert in k is odd fix any such prime say p the field k determines a factorization n n nof n by taking n resp nto be the product of all the prime factors of n which are split resp

variation of iwasawa invariants in hida families let gq gl2 k be an absolutely irreducible modular galois representation over a finite field k of characteristic p assume further that is p ordinary and p distinguished in the sense that the restriction of to a decomposition group at p is reducible and non scalar the hida family h of is the set of all p ordinary p stabilized newforms f with mod p galois representation isomorphic to if is unramified at p then one must also fix an unramified line in and require that the ordinary line of f reduces to this fixed line these newforms are a dense set of points in a certain p adic analytic space of overconvergent eigenforms consisting of an intersecting system of branches i e irreducible components t a indexed by the minimal primes a of a certain hecke algebra to each modular form f h one may associate the iwasawa invariants an f an f alg f and alg f the analytic resp algebraic invariants are the number of zeroes of the p adic l function resp of the characteristic power series of the dual of the selmer group of f while the invariants are the exponents of the powers of p dividing the same objects in this paper we prove the following results on the behavior of these iwasawa invariants as f varies over h

bias analysis of the music location estimator the authors present a rigorous bias analysis of the music location estimator and they derive an accurate and concise bias expression the analysis is based on the second order taylor series expansion of the derivative of the null spectrum properties of the null spectrum and statistics of the estimated signal eigenvectors it is proven that in the derivation the remainder term in the second order taylor series can be dropped but the second order terms cannot be simulations verify that the bias expression is valid over a wide range of signal to noise ratios snrs extending down into the resolution threshold region of music although asymptotic this expression can be accurately applied to a limited number of snapshot cases the utility of the expression is shown by using it in a study of music location estimator characteristics estimate bias and standard deviation are compared for variations in snr numbers of sensors and snapshots and source correlation music resolvability and estimator performance bounds are addressed accounting for bias

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

the statistical performance of the music and the minimum norm algorithms in resolving plane waves in noise this paper presents an asymptotic statistical analysis of the null spectra of two eigen assisted methods music 1 and minimum norm 2 for resolving independent closely spaced plane waves in noise particular attention is paid to the average deviation of the null spectra from zero at the true angles of arrival for the plane waves these deviations are expressed as functions of signal to noise ratios number of array elements angular separation of emitters and the number of snapshots in the case of music an approximate expression is derived for the resolution threshold of two plane waves with equal power in noise this result is validated by monte carlo simulations

non hyperelliptic modular jacobians of dimension 3 we present a method to solve in an efficient way the problem of constructing the curves given by torelli s theorem in dimension 3 over the complex numbers for an absolutely simple principally polarized abelian threefold a over c given by its period matrix compute a model of the curve of genus three unique up to isomorphism whose jacobian equipped with its canonical polarization is isomorphic to a as a principally polarized abelian variety we use this method to describe the non hyperelliptic modular jacobians of dimension 3 we investigate all the non hyperelliptic new modular jacobians jac c f of dimension 3 which are isomorphic to a f where f s new 2 x 0 n n 4000

the iwasawa main conjectures for gl2 we prove the one two and three variable iwasawa greenberg main conjectures for a large class of modular forms that are ordinary with respect to an odd prime p the method of proof involves an analysis of an eisenstein ideal for ordinary hida families for gu 2 2

bias and variance of direction of arrival estimates from music min norm and fine we derive variance and bias expressions of direction of arrival doa estimates for min norm and fine these expressions for arbitrarily configured arrays are valid and accurate over wide ranges of snr and number of snapshots using these expressions we illustrate that both min norm and fine have smaller estimate bias and further fine has comparable estimate variance compared to music

some remarks on the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication we establish several results towards the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields namely i the existence of an appropriate p adic l function building on works of hida and perrin riou ii the basic structure theory of the dual selmer group following works of coates hachimori venjakob et al and iii the implications of dihedral or anticyclotomic main conjectures with basechange the result of i is deduced from the construction of hida and perrin riou which in particular is seen to give a bounded distribution the result of ii allows us to deduce a corank formula for the p primary part of the tate shafarevich group of an elliptic curve in the z p 2 extension of an imaginary quadratic field finally iii allows us to deduce a criterion for one divisibility of the two variable main conjecture in terms of specializations to cyclotomic characters following a suggestion of greenberg as well as a refinement via basechange c 2011 elsevier inc all rights reserved

on serre s conjecture for 2 dimensional mod p representations of gal q q we prove the existence in many cases of minimally ramified p adic lifts of 2 dimensional continuous odd absolutely irreducible mod p representations of the absolute galois group of q it is predicted by serre s conjecture that such representations arise from newforms of optimal level and weight

music maximum likelihood and cramer rao bound further results and comparisons the problem of determining the direction of arrival of narrowband plane waves using sensor arrays and the related problem of estimating the parameters of superimposed signals from noisy measurements are studied a number of results have been recently presented by the authors on the statistical performance of the multiple signal characterization music and the maximum likelihood ml estimators for the above problems this work extends those results in several directions first it establishes that in the class of weighted music estimators the unweighted music achieves the best performance i e the minimum variance of estimation errors in large samples next it derives the covariance matrix of the ml estimator and presents detailed analytic studies of the statistical efficiency of music and ml estimators these studies include performance comparisons of music and mle with each other as well as with the ultimate performance corresponding to the cramer rao bound finally some numerical examples are given which provide a more quantitative study of performance for the problem of finding two directions with uniform linear sensor arrays

lattices in the cohomology of shimura curves we prove the main conjectures of breuil j reine angew math 2012 including a generalisation from the principal series to the cuspidal case and dembele j reine angew math 2012 subject to a mild global hypothesis that we make in order to apply certain r mathbb t theorems more precisely we prove a multiplicity one result for the mod p cohomology of a shimura curve at iwahori level and we show that certain apparently globally defined lattices in the cohomology of shimura curves are determined by the corresponding local p adic galois representations we also indicate a new proof of the buzzard diamond jarvis conjecture in generic cases our main tools are the geometric breuil mezard philosophy developed in emerton and gee j inst math jussieu 2012 and a new and more functorial perspective on the taylor wiles kisin patching method along the way we determine the tamely potentially barsotti tate deformation rings of generic two dimensional mod p representations generalising a result of breuil and mezard bull soc math de france 2012 in the principal series case

zero cycles sur les fibrations au dessus d une courbe de genre quelconque resume soit x une variete propre et lisse sur un corps de nombres k des conjectures sur l image du groupe de chow des zero cycles de x dans le produit des memes groupes sur tous les completes de k ont ete proposees par colliot thelene kato et saito nous demontrons ces conjectures pour l espace total de fibrations en varietes rationnellement connexes verifiant l approximation faible au dessus de courbes dont le groupe de tate shafarevich est fini sous une hypothese d abelianite sur les fibres singulieres

on the fibration method for zero cycles and rational points conjectures on the existence of zero cycles on arbitrary smooth projective varieties over number elds were proposed by colliot th el ene sansuc kato and saito in the 1980 s we prove that these conjectures are compatible with brations for brations into rationally connected varieties over a curve in particular they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers we prove the analogous result for rational points conditionally on a conjecture on locally split values of polynomials which a recent work of matthiesen establishes in the case of linear polynomials over the rationals

on meromorphic solutions of f2 g2 1 we show that meromorphic solutions f g of f2 g2 1 in c2 must be constant if fz2 and gz1 have the same zeros counting multiplicities we also apply the result to characterize meromorphic solutions of certain nonlinear partial differential equations

universal torsors and values of quadratic polynomials represented by norms let k k be an extension of number fields and let p t be a quadratic polynomial over k let x be the affine variety defined by p t n k k mathbf z we study the hasse principle and weak approximation for x in three cases for k k 4 and p t irreducible over k and split in k we prove the hasse principle and weak approximation for k mathbb q with arbitrary k we show that the brauer manin obstruction to the hasse principle and weak approximation is the only one for k k 4 and p t irreducible over k we determine the brauer group of smooth proper models of x in a case where it is non trivial we exhibit a counterexample to weak approximation

second order statistical analysis of totally weighted subspace fitting methods we analyze the effects of a limited number of snapshots on a general class of multidimensional signal subspace estimation methods termed totally weighted subspace fitting twsf methods this class is an extension of the weighted subspace fitting technique that includes row weighting and column weighting of the signal subspace matrix we quantify the asymptotic statistical properties of the twsf method by means of a second order statistical analysis the main contribution of this article is that it provides for the first time the estimator bias some simulation results are included to validate our analytical results

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

deformations of induced galois representations the object of this article is to study the behavior of certain deformation problems and hecke rings under base change to a real quadratic field these questions were the subject of conjectures by doi hida ishii see dhi98 and hid98 in this paper we will show that conjecture 2 2 of hid98 holds under a suitable hypothesis and that the isomorphism predicted by conjecture 3 8 of dhi98 holds up to a pseudo null cokernel the deformations studied in this paper are also interesting in light of the recent work by skinner and wiles see sw97 and sw99 on the deformation of reducible galois representations namely the ring r studied in hid98 and dhi98 turns out to be closely related to a pseudodeformation ring the explication of this connection is crucial to our results furthermore the central conjecture of hid98 turns out to be related to a certain class number condition which is exactly analogous to the condition that appears in sw97 finally we would like to point out that the deformations considered here give rise to a number of interesting examples specifically we are able to exhibit adic representations whose traces generate a nontrivial extension of and with the property that the specializations to certain arithmetic points of weight one are actually ramified over spec this is in contrast to the well known theorem of hida which asserts that arithmetic points of weight k 2 are smooth over spec to state the results more precisely we consider a real quadratic extension f q d of q with d 0 write for a generator of the group gal f q we fix a prime p 2 with d p 1 such that p pp in f with p p let fp denote the maximal algebraic extension of f unramified outside p and infinity let h gal fp f and let g gal fp q now consider a character h f

zero cycles on rational surfaces over number fields let x be a smooth projective geometrically integral surface defined over a perfect field k let kbe an algebraic closure of k and f be the function field of x k x then x is said to be rational if f is a purely transcendental extension of k the tame symbols define a map from ke f to the sum of function fields h k 7 of all irreducible divisors 7 on x let m be the cokernel of this map if k is a number field and kv the completion at a place v of k let kv be an algebraic closure of kv and m the corresponding cokernel for k x x further let i i i l k m be the kernel of the natural map from hl gal k k m to i h1 gal k v kv my in 5 w colliot th616ne and sansuc conjectured that

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

rationally connected varieties over finite fields let x be a geometrically rational or more generally separably rationally connected variety over a finite field k we prove that if k is large enough then x contains many rational curves defined over k as a consequence we prove that r equivalence is trivial on x if k is large enough these results imply the following conjecture of j l colliot thelene let y be a rationally connected variety over a number field f for a prime p let y p denote the corresponding variety over the local field f p then for almost all primes p the chow group of 0 cycles on y p is trivial and r equivalence is trivial on y p

towards the brauer manin obstruction on varieties fibred over the projective line abstract recently dasheng wei proved that the brauer manin obstruction is the only obstruction to the hasse principle for 0 cycles of degree 1 on some fibrations over the projective line defined by bi cyclic normic equations in the present paper we prove the exactness of the global to local sequence for chow groups of 0 cycles of such varieties this signifies that the brauer manin obstruction is also the only obstruction to weak approximation for 0 cycles of arbitrary degree our main theorem also generalizes several existing results

calculation of values of l functions associated to elliptic curves we calculated numerically the values of l functions of four typical elliptic curves in the critical strip in the range im s 0 which enables us to calculate for large im s furthermore we remark that a relation exists between sato tate conjecture and the generalized riemann hypothesis