mazur tate elements of nonordinary modular forms we establish formulae for the iwasawa invariants of mazur tate elements of cuspidal eigenforms generalizing known results in weight 2 our first theorem deals with forms of medium weight and our second deals with forms of small slope we give examples illustrating the strange behavior which can occur in the high weight high slope case

a fast conformal mapping algorithm with no fft abstract an algorithm is presented for the computation of a conformal mapping discretized on a non uniformly spaced point set useful for the numerical solution of many problems of fluid dynamics most existing iterative techniques both those having a linear and those having a quadratic type of convergence rely on the fast fourier transform fft algorithm for calculating a convolution integral which represents the most time consuming phase of the computation the fft however definitely cannot be applied to a non uniform spacing the algorithm presented in this paper has been made possible by the construction of a calculation method for convolution integrals which despite not using an fft maintains a computation time of the same order as that of the fft the new technique is successfully applied to the problem of conformally mapping a closely spaced cascade of airfoils onto a circle which requires an exceedingly large number of points if it is solved with uniform spacing

p converse to a theorem of gross zagier kolyvagin and rubin let e be a cm elliptic curve over the rationals and p 3 a good ordinary prime for e we show that begin aligned mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 implies mathrm ord s 1 l s e mathbb q 1 end aligned for the p infty selmer group mathrm sel p infty e mathbb q and the complex l function l s e mathbb q in particular the tate shafarevich group hbox x e mathbb q is finite whenever mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 we also prove an analogous p converse for cm abelian varieties arising from weight two elliptic cm modular forms with trivial central character for non cm elliptic curves over the rationals first general results towards such a p converse theorem are independently due to skinner a converse to a theorem of gross zagier and kolyvagin arxiv 1405 7294 2014 and zhang camb j math 2 2 191 253 2014

collisions and phase vortex interactions in dissipative ginzburg landau dynamics in this paper we describe a natural framework for the vortex dynamics in the parabolic complex ginzburg landau equation in r this general setting does not rely on any assumption of well preparedness and has the advantage to be valid even after collision times we analyze carefully collisions leading to annihilation a new phenomenon is identified the phase vortex interaction related to persistence of low frequency oscillations and leading to an unexpected drift in the motion of vortices

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

a consistent metric for performance evaluation of multi object filters the concept of a miss distance or error between a reference quantity and its estimated controlled value plays a fundamental role in any filtering control problem yet there is no satisfactory notion of a miss distance in the well established field of multi object filtering in this paper we outline the inconsistencies of existing metrics in the context of multi object miss distances for performance evaluation we then propose a new mathematically and intuitively consistent metric that addresses the drawbacks of current multi object performance evaluation metrics

an iterative method for conformal mapping abstract in this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected region g onto a region h with smooth boundary the method is based on a certain riemann hilbert problem it turns out that this problem is the linearized version of a singular integral equation of the second kind hence the method is a newton method whenever the boundaries of g and h are sufficiently smooth its convergence is locally quadratic if g is the unit circle the solution of the linearized problem can be represented explicitly in terms of integral transforms from this one derives a quadratically convergent newton like numerical method that avoids the numerical solution of systems of linear equations and therefore in comparison with other methods based on integral equations is quite economical in terms of computer time and storage requirements

the iwasawa main conjecture for semistable abelian varieties over function fields we prove the iwasawa main conjecture over the arithmetic mathbb z p extension for semistable abelian varieties over function fields of characteristic p 0

plus minus p adic l functions for hilbert modular forms abstract r pollack constructed in pollack 2003 13 plus minus p adic l functions for elliptic modular forms which are p adically bounded when the hecke eigenvalues at p are zero the most supersingular case the goal of this work is to generalize his construction to hilbert modular forms we find a suitable condition for hilbert modular forms corresponding to the vanishing of p th hecke eigenvalue in elliptic modular form case which guarantees the existence of plus minus p adic l functions which are p adically bounded as an application we construct cyclotomic plus minus p adic l functions for modular elliptic curves over a totally real field f under the assumption that a p e 0 for each prime p dividing p we formulate a cyclotomic plus minus iwasawa main conjecture for such elliptic curves

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

the newton method for solving the theodorsen integral equation the newton method for the solution of the theodorsen integral equation in conformal mapping is studied one step of this method consists of solving a linear integral equation the solution of which is given explicitly as the result of a riemann hilbert problem quadratic convergence of the newton method is established under certain assumptions whereas in other methods a so called condition with 1 is required to hold our method converges also for 1 we will also present a numerical implementation in which the result of one step of the newton method is approximately by a vector in r2n which can be computed with 2n log n o n multiplications in comparison one step of the newton method for the discrete theodorsen equation requires o n3 multiplications

p adic l functions and trivial zeroes the following is adapted from the notes for the lecture it announces results and conjectures about values of the p adic l function of the symmetric square of an elliptic curve

modular representations of pgl2 and automorphic forms for shimura curves in this paper we study certain infinite dimensional representations of the general linear group gl2 over a local field k in vector spaces over k these representations are closely related to the theory of analytic functions on the p adic upper half plane over k they were originally introduced by morita mor who studied them intensively in the case when the characteristic of k is zero morita s work has subsequently been greatly extended by schneider and stuhler ss and schneider s to representations of gl k and higher p adic symmetric spaces in this paper we re consider some of morita s representations of gl2 from the point of view of integral and modular representation theory it seems quite natural to study p adic representations by reducing them modp and we supply a technique for doing this in particular suppose that ydenotes the bruhat tits tree of pglz k we consider the representations of pgl2 k on the harmonic functions on w of e v e n weight that is the harmonic functions on the edges of j taking values in even symmetric powers of the standard representation of gl2 in this paper we find an invariant integral lattice in these representations and we describe the modular representations of pgl2 k obtained by reducing this lattice modp the modular representations obtained from morita s representations are quite simple and natural objects although they seem a bit peculiar at first suppose that f is the residue field of k that v is a vector space over f and that p gl2 f end v is a finite dimensional modular representation notice that p may be viewed in a natural way as a modular representation of gl r where r is the valuation ring of k we show that a typical infinite dimensional representation of morita s when reduced mod p has a filtration whose factors are isomorphic to i cl2 r wj t n l2 t note that there are only finitely many irreducible finite dimensional modular representations of glz f the appearance of these finitely many

the main conjecture for cm elliptic curves at supersingular primes at a prime of ordinary reduction the iwasawa main conjecture for elliptic curves relates a selmer group to a p adic l function in the supersingular case the statement of the main conjecture is more complicated as neither the selmer group nor the p adic l function is well behaved recently kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case namely kobayashi s conjecture relates modified selmer groups which he defined with modified padic l functions defined by the first author in this paper we prove kobayashi s conjecture for elliptic curves with complex multiplication

iwasawa theory of elliptic curves at supersingular primes over ℤ p extensions of number fields in this paper we make a study of the iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary zp extension of a number field k in the case when p splits completely in k generalizing work of kobayashi 9 and perrin riou 17 we define restricted selmer groups and invariants we then derive asymptotic formulas describing the growth of the selmer group in terms of these invariants to be able to work with non cyclotomic zp extensions a new local result is proven that gives a complete description of the formal group of an elliptic curve at a supersingular prime along any ramified zp extension of qp

higher chern classes in iwasawa theory we begin a study of mth chern classes and mth characteristic symbols for iwasawa modules which are supported in codimension at least m this extends the classical theory of characteristic ideals and their generators for iwasawa modules which are torsion i e supported in codimension at least 1 we apply this to an iwasawa module constructed from an inverse limit of p parts of ideal class groups of abelian extensions of an imaginary quadratic field when this module is pseudo null which is conjecturally always the case we determine its second chern class and show that it has a characteristic symbol given by the steinberg symbol of two katz p adic l functions

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 heegner points in hida families given a weight two modular form f with associated p adic galois representation v f for certain quadratic imaginary fields k one can construct canonical classes in the galois cohomology of v f by taking the kummer images of heegner points on the modular abelian variety attached to f we show that these classes can be interpolated as f varies in a hida family and construct an euler system of big heegner points for hida s universal ordinary deformation of v f we show that the specialization of this big euler system to any form in the hida family is nontrivial extending results of cornut and vatsal from modular forms of weight two and trivial character to all ordinary modular forms and propose a horizontal nonvanishing conjecture for these cohomology classes the horizontal nonvanishing conjecture implies via the theory of euler systems a conjecture of greenberg on the generic ranks of selmer groups in hida families

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

flow around simply and multiply connected bodies a new iterative scheme for conformal mapping discussion des techniques de representation conforme numerique des profils aerodynamiques des cascades soit pour le calcul des champs d ecoulement irrotationnel soit pour la construction de grilles de calcul numerique presentation d un algorithme iteratif a convergence quadratique

quantitative non vanishing of dirichlet l values modulo p let p be an odd prime and k a non negative integer let n be a positive integer such that p not mid n and lambda a dirichlet character modulo n we obtain quantitative lower bounds for the number of dirichlet character chi modulo f with the critical dirichlet l value l k lambda chi being p indivisible here f rightarrow infty with n f 1 and p not mid f phi f we explore the indivisibility via an algebraic and a homological approach the latter leads to a bound of the form f 1 2 the p indivisibility yields results on the distribution of the associated p selmer ranks we also consider an iwasawa variant it leads to an explicit upper bound on the lowest conductor of the characters factoring through the iwasawa mathbb z ell extension of mathbb q for an odd prime ell ne p with the corresponding critical l value twists being p indivisible

a new relaxation algorithm and passive sensor data association the static problem of associating measurements at a given time from three angle only sensors in the presence of clutter missed detections and an unknown number of targets is addressed the measurement target association problem is formulated as one of maximizing the joint likelihood function of the measurement partition mathematically this formulation leads to a generalization of the 3 d assignment matching problem which is known to be np hard the solution to the optimization problem developed is a lagrangian relaxation technique that successively solves a series of generalized two dimensional 2 d assignment problems the algorithm is illustrated by several application examples

on the birch and swinnerton dyer conjecture let f be a finite extension of the rational field q if e is an elliptic curve defined over f then the mordell weil group e f of points on e with coordinates in f is a finitely generated abelian group let l e f s be the hasseweil zeta function of e over f which at least in the case where e has complex multiplication is known to be an analytic function on the entire complex plane birch and swinnerton dyer have conjectured that l e f s has a zero at s l of order precisely equal to the rank of e f over z the strongest result known in support of this conjecture is the following theorem of coates and wiles 4

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

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

numerical conformal mapping methods based on function conjugation abstract a unifying treatment of methods for computing conformal maps from the unit disk onto a jordan region is presented integral and integro differential equations involving the conjugation operator for the boundary correspondence function are first derived using an arbitrary auxiliary function having certain properties then various iteration methods for solving these equations are described in this generality so that the basic ideas become manifest specific methods are then treated as examples of the general theory among them are in particular the successive conjugation methods of theodorsen melentiev and kulisch timman and friberg the projection method of bergstrom and the newton methods of vertgeim wegmann and hubner which make use of the easy construction of the solutions of riemann hilbert problems many of these methods are treated in greater generality than in the literature the connections with the methods of fornberg menikoff zemach chakravarthy anderson and challis burley are also outlined

critical p adic l functions we attach p adic l functions to critical modular forms and study them we prove that those l functions fit in a two variables p adic l function defined locally everywhere on the eigencurve

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

on symmetric power l invariants of iwahori level hilbert modular forms we compute the arithmetic cal l invariants of greenberg benois of twists of symmetric powers of p adic galois representations attached to iwahori level hilbert modular forms under some technical conditions our method uses the automorphy of symmetric powers and the study of analytic galois representations on p adic families of automorphic forms over symplectic and unitary groups combining these families with some explicit plethysm in the representation theory of rm gl 2 we construct global galois cohomology classes with coefficients in the symmetric powers and provide formulae for the cal l invariants in terms of logarithmic derivatives of hecke eigenvalues

an iterative method for the conformal mapping of doubly connected regions an iterative method is described for the determination of the conformal mapping of a circular annulus rq z q z 1 onto a doubly connected region g with smooth boundary curves the method is analogous to the method of 7 for simply connected regions it requires in each step the solution of a riemann hilbert problem on rq this problem can be solved explicitly in terms of a generalized conjugation operator k a degeneracy of this problem leads in a natural way to an adjustment of the parameter q in each step if the boundary curves of g admit parametrizations by functions with second derivatives which are holder continuous with exponent 0 1 then the method converges locally in the sobolev space w of 2 periodic absolutely continuous functions with square integrable derivatives the order of convergence is at least 1 the convergence is quadratic if the have lipschitz continuous second derivatives for the numerical implementation of the method the operator k can be approximated by a wittich type method which can be performed very effectively using fft a calculation with n 2m grid points on the computer requires storage of the order o n and computing time o mn the results of some test calculations are reported and compared with results of calculations with the method of theodorsen garrick

coleman adapted rubin stark kolyvagin systems and supersingular iwasawa theory of cm abelian varieties the goal of this article is to study the iwasawa theory of an abelian variety a that has complex multiplication by a cm field f that contains the reflex field of a which has supersingular reduction at every prime above p to do so we make use of the signed coleman maps constructed in our companion article bl14 to introduce signed selmer groups as well as a signed p adic l function via a reciprocity conjecture we formulate for the conjectural rubin stark elements which is a natural extension of the reciprocity conjecture for elliptic units we then prove a signed main conjecture relating these two objects to achieve this we develop along the way a theory of coleman adapted rank g euler kolyvagin systems to be applied with rubin stark elements and deduce the main conjecture for the maximal zp power extension of f for the primes failing the ordinary hypothesis of katz

algorithms for the arithmetic of elliptic curves using iwasawa theory we explain how to use results from iwasawa theory to obtain information about p parts of tate shafarevich groups of specific elliptic curves over q our method provides a practical way to compute x e q p in many cases when traditional p descent methods are completely impractical and also in situations where results of kolyvagin do not apply e g when the rank of the mordell weil group is greater than 1 we apply our results along with a computer calculation to show that x e q p 0 for the 1 534 422 pairs e p consisting of a non cm elliptic curve e over q with conductor 30 000 rank 2 and good ordinary primes p with 5 p 1000 and surjective mod p representation

on l functions of elliptic curves and cyclotomic towers in this paper we shall strengthen a recent theorem of ralph greenberg 1 concerning l functions of elliptic curves and anticyclotomic towers let k be an imaginary quadratic field and m an abelian extension of k possibly of infinite degree we say that the extension m k is anticyclotomic if m is galois over q and if the nontrivial element of gal k q acts on gal m k by inversion now fix a finite set p of rational primes and consider the compositum in an algebraic closure of k of all anticyclotomic extensions of k which are unramified outside p we denote this compositum by l and refer to it as the maximal anticyclotomic extension of k unramified outside p the field l may also be described as the union of all ring class fields of k with conductor divisible only by primes in p class field theory shows that gal l k is isomorphic to the product of a finite group and the group

a batch algorithm for estimating trajectories of point targets using expectation maximization in this paper we propose a strategy that is based on expectation maximization for tracking multiple point targets the algorithm is similar to probabilistic multi hypothesis tracking pmht but does not relax the point target model assumptions according to the point target models a target can generate at most one measurement and a measurement is generated by at most one target with this model assumption we show that the proposed algorithm can be implemented as iterations of rauch tung striebel rts smoothing for state estimation and the loopy belief propagation method for marginal data association probabilities calculation using example illustrations with tracks we compare the proposed algorithm with pmht and joint probabilistic data association jpda and show that pmht and jpda exhibit coalescence when there are closely moving targets whereas the proposed algorithm does not furthermore extensive simulations c comparing the mean optimal subpattern assignment mospa performance of the algorithm for different scenarios averaged over several monte carlo iterations show that the proposed algorithm performs better than jpda and pmht we also compare it to benchmarking algorithm n scan pruning based track oriented multiple hypothesis tracking tomht the proposed algorithm shows a good tradeoff between computational complexity and the mospa performance

uniform distribution of heegner points let e be a modular elliptic curve over q of conductor n let k denote an imaginary quadratic field of discriminant d with n d 1 if p is a prime then there exists a unique zp extension k k such that gal k q acts nontrivially on gal k k the field k is called the anticyclotomic zp extension of k let e k denote the mordell weil group of e over k then a fundamental conjecture of mazur maz predicts that the size of e k is controlled by the prime factorization of n in k equivalently mazur s conjecture relates the size of the mordell weil group to the sign in the functional equation of certain l series the conjecture was verified by greenberg rohrlich and rubin in what mazur calls the exceptional case when e has complex multiplication by k more generally they settled the conjecture for certain abelian varieties with complex multiplication for a discussion of this cm case we refer the reader to gre roh and rub our goal in this paper is to treat the generic case which occurs either when e has no cm or when the field of complexmultiplications is distinct from k under certain conditions on e and k mazur s conjecture predicts that the group e k is finitely generated our main result asserts that this is in fact the case at least when p is an ordinary prime for e or when the class number of k is prime to p the main new ingredient we introduce is that of equidistribution following ideas used by ferrero andwashington to study the cyclotomic invariant more precisely we show that theheegner points associated to definite quaternion algebras are uniformly distributed on the components of a certain curve x and that the elements of a certain galois group act independently in a suitable sense this combined with a special value formula due to gross allows us to conclude that the special values of anticyclotomic l functions are almost always nonzero so that the statement about the mordell weil groups follows from the machinery of euler systems as developed by bertolini and darmon bd

derivative of the standard p adic l function associated with a siegel form this work has greatly bene ted from an excellent long stay at columbia university founded by a fwo travel grant v4 260 14n while the author was a fwo phd fellow at ku leuven

the p adic gross zagier formula for elliptic curves at supersingular primes let p be a prime number and let e be an elliptic curve defined over of conductor n let k be an imaginary quadratic field with discriminant prime to pn such that all prime factors of n split in k b perrin riou established the p adic gross zagier formula that relates the first derivative of the p adic l function of e over k to the p adic height of the heegner point for k when e has good ordinary reduction at p in this article we prove the p adic gross zagier formula of e for the cyclotomic p extension at good supersingular prime p our result has an application for the full birch and swinnerton dyer conjecture suppose that the analytic rank of e over is 1 and assume that the iwasawa main conjecture is true for all good primes and the p adic height pairing is not identically equal to zero for all good ordinary primes then our result implies the full birch and swinnerton dyer conjecture up to bad primes in particular if e has complex multiplication and of analytic rank 1 the full birch and swinnerton dyer conjecture is true up to a power of bad primes and 2

exceptional zero formulae and a conjecture of perrin riou let a mathbf q be an elliptic curve with split multiplicative reduction at a prime p we prove an analogue of a conjecture of perrin riou relating p adic beilinson kato elements to heegner points in a mathbf q and a large part of the rank one case of the mazur tate teitelbaum exceptional zero conjecture for the cyclotomic p adic l function of a more generally let f be the weight two newform associated with a let f infty be the hida family of f and let l p f infty k s be the mazur kitagawa two variable p adic l function attached to f infty we prove a p adic gross zagier formula expressing the quadratic term of the taylor expansion of l p f infty k s at k s 2 1 as a non zero rational multiple of the extended height weight of a heegner point in a mathbf q

integral iwasawa theory of galois representations for non ordinary primes in this paper we study the iwasawa theory of a motive whose hodge tate weights are 0 or 1 thence in practice of a motive associated to an abelian variety at a non ordinary prime over the cyclotomic tower of a number field that is either totally real or cm in particular under certain technical assumptions we construct sprung type coleman maps on the local iwasawa cohomology groups and use them to define integral p adic l functions and one unconditionally and other conjecturally cotorsion selmer groups this allows us to reformulate perrin riou s main conjecture in terms of these objects in the same fashion as kobayashi s iwasawa theory for supersingular elliptic curves by the aid of the theory of coleman adapted kolyvagin systems we develop here we deduce parts of perrin riou s main conjecture from an explicit reciprocity conjecture

canonical periods and congruence formulae the purpose of this article is to show how congruences between the fourier coefficients of hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated l functions this study was initiated by b mazur in his fundamental work on the eisenstein ideal see maz77 and maz79 where it was made clear that congruences for analytic l values were closely related to the integral structure of certain hecke rings and cohomology groups the results of maz79 also showed that congruences were useful in the study of nonvanishing of l functions this idea was then further developed by stevens ste82 and rubin wiles rw82 the work of rubin and wiles in particular used congruences to study the behavior of elliptic curves in towers of cyclotomic fields a key ingredient here was a theorem of washington which states roughly that almost l values in certain families are nonzero modulo p this theme has recently been taken up again in the work of ono skinner osa osb james jam and kohnen koh97 while the earlier history was primarily concerned with cyclotomic twists the current emphasis is on families of twists by quadratic characters here one wants quantitative estimates for the number of quadratic twists of a given modular form which have nonvanishing l function at s 1 we continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three this generalizes a beautiful example due to kevin james and provides new evidence for a conjecture of goldfeld gol79 it should however be pointed out that even the study of quadratic twists may be traced back to mazur the reader is urged to look at pages 212 213 of maz79 and especially at the footnote at the bottom of page 213 the theorems of davenport heilbronn dh71 and washington was78 which are crucial in this paper are both mentioned in mazur s article we want to begin by discussing the congruences that lie at the heart of this article thus let f anq n be an elliptic modular cuspform of level m and weight k 2 assume that f is a simultaneous eigenform for all the hecke operators and that a1 f 1 the l function associated to f is defined by the dirichlet series l s f ann s which converges for the real part of s sufficiently large and has analytic continuation to s c a fundamental theorem of shimura shi76 states that l s f enjoys the following algebraicity property

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

iwasawa theory for elliptic curves at supersingular primes we give a new formulation in iwasawa theory for elliptic curves at good supersingular primes this formulation is similar to mazur s at good ordinary primes namely we define a new selmer group and show that it is of cotorsion then we formulate the iwasawa main conjecture as that the characteristic ideal is generated by pollack s p adic l function we show that this main conjecture is equivalent to kato s and perrin riou s main conjectures we also prove an inequality in the main conjecture by using kato s euler system in terms of the and the invariants of our selmer group we specify the numbers and in the asymptotic formula for the order of the tate shafarevich group by kurihara and perrin riou

on the p adic l function of a modular form at a supersingular prime in this paper we study the two p adic l functions attached to a modular form f sum a sb nq sp n at a supersingular prime p when a sb p 0 we are able to decompose both the sum and the difference of the two unbounded distributions attached to f into a bounded measure and a distribution that accounts for all of the growth moreover this distribution depends only upon the weight of f and the fact that a sb p vanishes from this description we explain how the p adic l function is controlled by two iwasawa functions and by two power series with growth which have a fixed infinite set of zeros theorem 5 1 asymptotic formulas for the p part of the analytic size of the tate shafarevich group of an elliptic curve in the cyclotomic direction are computed using this result these formulas compare favorably with results established by m kurihara in 11 and b perrin riou in 23 on the algebraic side moreover we interpret kurihara s conjectures on the galois structure of the tate shafarevich group in terms of these two iwasawa functions

bayesian sequential track formation this paper presents a theoretical framework for track building in multiple target scenarios from the bayesian point of view it is assumed that the number of targets is fixed and known we propose two optimal methods for building tracks sequentially the first one uses the labelling of the current multitarget state estimate that minimizes the mean square labeled optimal subpattern assignment error this method requires knowledge of the posterior density of the vector valued state the second assigns the labeling that maximizes the probability that the current multi target state estimate is optimally linked with the available tracks at the previous time step in this case we only require knowledge of the random finite set posterior density without labels

on the p adic height of heegner cycles 0 1 in gr za gross and zagier proved a remarkable formula which relates the first derivative of the l function of a modular form f of weight 2 on 0 n over a suitable imaginary quadratic field and the neron tate height of a heegner point over the same quadratic field on the f part of the jacobian of the modular curve x0 n later perrin riou pr 2 proved a p adic version of this formula kolyvagin s method of euler systems ko combined with the formula of gross zagier proves the conjecture of birch and swinnerton dyer up to a controlled rational factor for all modular elliptic curves over q with analytic rank 1

elliptic curves with complex multiplication and the conjecture of birch and swinnerton dyer 1 quick review of elliptic curves 2 2 elliptic curves over c 4 3 elliptic curves over local fields 6 4 elliptic curves over number fields 12 5 elliptic curves with complex multiplication 15 6 descent 22 7 elliptic units 27 8 euler systems 37 9 bounding ideal class groups 43 10 the theorem of coates and wiles 47 11 iwasawa theory and the main conjecture 50 12 computing the selmer group 61