stickelberger ideals and fitting ideals of class groups for abelian number fields in this paper we determine completely the initial fitting ideal of the minus part of the ideal class group of an abelian number field over q up to the 2 component this answers an open question of mazur and wiles invent math 76 179 330 1984 up to the 2 component and proves conjecture 0 1 in kurihara j reine angew math 561 39 86 2003 we also study brumer s conjecture and prove a stronger version for a cm field assuming certain conditions in particular on the galois group

on a conjecture of brumer in a series of papers iwasawa proposed and studied analogues for number fields of constructions due to weil for curves over finite fields cf 11 in particular he conjectured an analogue for weil s theorem relating the zeta function of a non singular curve over a finite field with the characteristic polynomial of its frobenius automorphism one strong piece of evidence for this came from the classical theorem of stickelberger however iwasawa s original conjecture was proved for f q in mw without use of stickelberger s theorem for a general totally real number field brumer proposed a generalization of stickelberger s theorem in w2 we proved iwasawa s conjecture for an arbitrary totally real field and here we use this to recover a version of brumer s conjecture we note however that iwasawa s conjecture alone is not sufficient to prove our theorem owing to the phenomenon of trivial zeroes the reason is that in iwasawa theory one habitually interpolates the special values of lfunctions which are imprimitive the key part of this paper is the development of a technique introduced in w2 of adjoining an auxiliary variable to the p adic b functions in a way that permits the study of these primitive lfunctions i am grateful to the referee for correcting some obscure arguments let f be a totally real number field and let k be a cm field which is

minimax linear observers and regulators for stochastic systems with uncertain second order statistics the problem of minimax design of linear observers and regulators for linear time varying multivariable stochastic systems with uncertain models of their second order statistics is treated in this paper general classes of allowable covariance matrices and means of the process and observation noises and of the random initial condition are considered a game formulation of the problem is adopted and it is shown that the optimal filter for the least favorable set of covariances is minimax robust for each of the filtering situations analyzed conditions satisfied by the saddle point solutions are given and their utility for finding the worst case covariances is illustrated by way of several examples of uncertainty classes of practical interest

syntomic regulators and special values of p adic l functions in this paper p adic analogs of the lichtenbaum conjectures are proven for abelian number fields f and odd prime numbers p which generalize leopoldt s p adic class number formula and express special values of p adic l functions in terms of orders of k groups and higher p adic regulators the approach uses syntomic regulator maps which are the p adic equivalent of the beilinson regulator maps they can be compared with etale regulators via the fontaine messing map and computations of bloch kato in the case that p is unramified in f lead to results about generalized coates wiles homomorphisms and cyclotomic characters

on zeta functions and iwasawa modules we study the relation between zeta functions and iwasawa modules we prove that the iwasawa modules xfor almost all p determine the zeta function k when k is a totally real field conversely we prove that the a part of the iwasawa module xk is determined by its zeta function k up to pseudo isomorphism for any number field k moreover we prove that arithmetically equivalent cm fields have also the same u invariant

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

tame kernels of cubic cyclic fields there are many results describing the structure of the tame kernels of algebraic number fields and relating them to the class numbers of appropriate fields in the present paper we give some explicit results on tame kernels of cubic cyclic fields table 1 collects the results of computations of the structure of the tame kernel for all cubic fields with only one ramified prime p 7 p 5 000 in particular we investigate the structure of the 7 primary and 13 primary parts of the tame kernels the theoretical tools we develop based on reflection theorems and singular primary units enable the determination of the structure even of 7 primary and 13 primary parts of the tame kernels for all fields as above the results are given in tables 2 and 3

on the equivariant tamagawa number conjecture for tate motives let l be a finite abelian extension of and let k be any subfield of l for each integer r with r 0 we prove the equivariant tamagawa number conjecture for the pair h0 spec l r gal l k

on certain functions for deformations of knot group representations we study the twisted knot module for the universal deformation of an sl2 representation of a knot group and introduce an associated l function which may be seen as an analogue of the algebraic p adic lfunction associated to the selmer module for the universal deformation of a galois representation we then verify affirmatively mazur s problem on the order of a zero of the l function by some examples for 2 bridge knots

transcendence of hecke operators in the big hecke algebra we prove a transcendence result of hecke operator t l over the ring generated by diamond operators over in a non cm component of the cyclotomic ordinary big p adic hecke algebra and we discuss its conjectural implication of a modulo p version to the vanishing of the cyclotomic invariant 2014

branching random walk in random environment phase transitions for local and global growth rates let n be the infinite particle system on whose evolution is as follows at each unit of time each particle independently is replaced by a new generation the size of a new generation descending from a particle at sitex has distributionf x and each of its members independently jumps to sitex 1 with probability 1 h 2 h 0 1 the sequence f x is i i d with uniformly bounded second moment and is kept fixed during the evolution the initial configuration 0 is shift invariant and ergodic two quantities are considered

on galois structure invariants associated to tate motives we establish the equivalence of two definitions of invariants measuring the galois module structure of k groups of rings of integers in number fields due to chinburg et al on the one hand and the authors on the other we also make some remarks concerning the possibility of yet another such definition via lichtenbaum complexes

fitting ideals of l adic realizations of picard 1 motives and class groups of global function fields in previous work we proved a result on the equivariant fitting ideal of the adic realization of the picard 1 motive attached to an abelian covering of curves defined over a finite field in this paper we build upon this work to deduce results on the equivariant fitting ideal of the tate modules of the jacobian of the top curve and on the equivariant fitting ideal of the dual of the degree zero class group of the top curve at the end we discuss refinements of brumer s conjecture and consider examples

the iwasawa conjecture for totally real fields let f be a totally real number field let p be a prime number and for any integer n let fun denote the group of nth roots of unity let 41 be a p adic valued artin character for f and let f be the extension of f attached to 4 i e so that 4 is the character of a faithful representation of gal f f we will assume that f is also totally real for a number field k let k denote the cyclotomic zp extension of k following greenberg we say that 4 is of type s if f n fc f and of type w if 4 is one dimensional with f p c fcc deligne and ribet in dr following kubota and leopoldt for the case f q have proved the existence of a p adic l function associated to a one dimensional artin character 4 with f totally real this function lp s 4 is continuous for s e zp 1 and even at s 1 if 4 is not trivial and satisfies the following interpolation property

stickelberger elements fitting ideals of class groups of cm fields and dualisation in this paper we systematically construct abelian extensions of cm fields over a totally real field whose stickelberger elements are not in the fitting ideals of the class groups our evidence indicates that pontryagin duals of class groups behave better than the class groups themselves we also explore the behaviour of fitting ideals under projective limits and dualisation in a somewhat broader context

shintani cocycles and the order of vanishing of p adic hecke l series at s 0 let chi be a hecke character of finite order of a totally real number field f by using hill s shintani cocycle we provide a cohomological construction of the p adic l series l p chi s associated to chi this is used to show that l p chi s has a trivial zero at s 0 of order at least equal to the number of places of f above p where the local component of chi is trivial

quenched sub exponential tail estimates for one dimensional random walk in random environment suppose that the integers are assigned i i d random variables x taking values in the unit interval which serve as an environment this environment defines a random walk x n called a rwre which when at x moves one step to the right with probability x and one step to the left with probability 1 x solomon 1975 determined the almost sure asymptotic speed v rate of escape of a rwre greven and den hollander 1994 have proved a large deviation principle for x n n conditional upon the environment with deterministic rate function for certain environment distributions where the drifts 2 x 1 can take both positive and negative values their rate function vanisheson an interval 0 v we find the rate of decay on this interval and prove it is a stretched exponential of appropriate exponent that is the absolute value of the log of the probability that the empirical mean x n n is smaller than v v 0 v behaves roughly like a fractional power of n the annealed estimates of dembo peres and zeitouni 1996 play a crucial role in the proof we also deal with the case of positive and zero drifts and prove there a quenched decay of the form

a law of large numbers for random walks in random environment we derive a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of kalikow the approach is based on the existence of a renewal structure under an assumption of transience in the direction l this extends to a multidimensional context previous work of kesten our results also enable proving the convergence of the law of the environment viewed from the particle toward a limiting distribution

precise large deviation estimates for a one dimensional random walk in a random environment suppose that the integers are assigned i i d random variables f xg taking values in the interval 1 2 1 which serve as an environment thisenvironmentdenesarandomwalk fxkg calledarwre which when at x moves one step to the right with probability x and one step to the left withprobability1 x solomon 1975 determinedthealmost sureasymp totic speed rate of escape of a rwre in a more general set up dembo peres and zeitouni 1996 following earlier work by greven and den hol lander 1994 on the quenched case have computed rough tail asymptotics for the empirical mean of the annealed rwre they conjectured the form of the rate function in a full ldp we prove in this paper their conjecture the proof is based on a icoarse graining schemei together with comparison techniques

on new examples of ballistic random walks in random environment in this article we show that random walks in random environment on zd d 3 with transition probabilities which are e perturbations of the simple random walk and such that the expectation of the local drift has size bigger than e with 52 when d 3 3 when d 4 fulfill the condition t introduced by sznitman prob theory related fields 2002 122 509 544 when e is small as a result these walks satisfy a law of large numbers with nondegenerate limiting velocity a central limit theorem and several large deviation controls in particular this provides examples of ballistic random walks in random environment which do not satisfy kalikow s condition in the terminology of sznitman and zerner ann probab 1999 27 1851 1869 an important tool in this work is the effective criterion of sznitman

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

large deviations for random walks in a random environment we prove two large deviation principles for random walks in a random environment one for quenched walks in general ergodic environments and the other for averaged walks in a product environment 2003 wiley periodicals inc

on the main conjecture for cm fields the aim of this work is to prove a divisibility relation toward the main conjecture for cm fields our main tool is the study of congruences between an eisenstein series and cusp forms on the unitary group gu 2 1 we introduce an eisenstein ideal in the nearly ordinary universal hecke algebra for gu 2 1 and show that it divides the characteristic ideal of the cm iwasawa module

an effective criterion for ballistic behavior of random walks in random environment we investigate multi dimensional random walks in random environment specifically we provide an effective criterion which can be checked by inspection of the environment in a finite box and implies a ballistic strong law of large numbers a central limit theorem and several large deviation controls with the help of this criterion we provide an example of a ballistic multi dimensional walk which does not fulfill the criterion introduced by kalikow in 4 the present work complements the results of 9 where a certain condition t was introduced and confirms the interest of this condition

restricted risk bayes linear state estimation the problem of state estimation with stochastic uncertainties in the initial state model noise and measurement noise is considered using the restricted risk bayes approach it is assumed that the a priori distributions of these quantities are not perfectly known but that some information about them may be available while offering robustness the restricted risk bayes approach incorporates the available a priori information to give less conservative state estimators than the gamma minimax approach popular in the literature when attention is restricted to linear estimators based on a quadratic loss function a systematic method to derive restricted risk bayes estimators is proposed applying to the filtering problem the restricted risk bayes approach provides us with a robust method to calibrate the kalman filter kf considering the presence of stochastic uncertainties this method is illustrated with a target tracking example and a wireless channel tracking example for which the bayes minimax and restricted risk bayes estimators are derived and their performance is compared

large deviation principle for random walk in a quenched random environment in the low speed regime we consider a one dimensional random walk x n n n in a random environment of zero or strictly positive drifts we establish a full large deviation principle for x n n of the correct order n log n 2 in the low speed regime valid for almost every environment this completes the large deviation picture obtained earlier by greven and den hollander and gantert and zeitouni in the case of zero and positive drifts the proof uses coarse graining along with concentration of measure techniques

robust filtering and prediction for linear systems with uncertain dynamics a game theoretic approach we examine the existence and behavior of game theoretic solutions for robust linear filters and predictors our basic uncertainty class includes m th order time varying discrete time systems with uncertain dynamics uncertain initial state covariance and uncertain nonstationary input and observation noise covariance our results include recursive kalman filter predictor realizations for the resulting robust procedures our approach is based on saddle point theory we emphasize the notion of a least favorable prior distribution for the uncertain parameter values to obtain a worst case design technique in this paper we highlight the role such distributions with finite support play in these decision models in particular we demonstrate that in these decision models the least favorable prior distribution is always discrete

hida duality and the iwasawa main conjecture the central result of this paper is a refinement of hida s duality theorem between ordinary varlambda adic modular forms and the universal ordinary hecke algebra in particular we give a sufficient condition for this duality to be integral with respect to particular submodules of the space of ordinary varlambda adic modular forms this refinement allows us to give a simple proof that the universal ordinary cuspidal hecke algebra modulo eisenstein ideal is isomorphic to the iwasawa algebra modulo an ideal related to the kubota leopoldt p adic l function the motivation behind these results stems from a proof of the iwasawa main conjecture over mathbb q by ohta while simple and elegant ohta s proof requires some restrictive hypotheses which we are able to remove using our results

a note on transience versus recurrence for a branching random walk in random environment we consider a branching random walk in random environment on d where particles perform independent simple random walks and branch according to a given offspring distribution at a random subset of sites whose density tends to zero at infinity given that initially one particle starts at the origin we identify the critical rate of decay of the density of the branching sites separating transience from recurrence i e the progeny hits the origin with probability 1 resp 1 we show that for d 3 there is a dichotomy in the critical rate of decay depending on whether the mean offspring at a branching site is above or below a certain value related to the return probability of the simple random walk the dichotomy marks a transition from local to global behavior in the progeny that hits the origin we also consider the situation where the branching sites occur in two or more types with different offspring distributions and show that the classification is more subtle due to a possible interplay between the types this note is part of a series of papers by the second author and various co authors investigating the problem of transience versus recurrence for random motions in random media

tail estimates for one dimensional random walk in random environment suppose that the integers are assigned i i d random variables x taking values in the unit interval which serve as an environment this environment defines a random walk x k called a rwre which when atx moves one step to the right with probability x and one step to the left with probability 1 x solomon 1975 determined the almost sure asymptotic speed rate of escape of a rwre for certain environment distributions where the drifts 2 x 1 can take both positive and negative values we show that the chance of the rwre deviating below this speed has a polynomial rate of decay and determine the exponent in this power law for environments which allow only positive and zero drifts we show that these large deviation probabilities decay like exp cn1 3 this differs sharply from the rates derived by greven and den hollander 1994 for large deviation probabilities conditioned on the environment as a by product we also provide precise tail and moment estimates for the total population size in a branching process with random environment

the brumer stark conjecture in some families of extensions of specified degree as a starting point an important link is established between brumer s conjecture and the brumer stark conjecture which allows one to translate recent progress on the former into new results on the latter for example if k f is an abelian extension of relative degree 2p p an odd prime we prove the l part of the brumer stark conjecture for all odd primes l p with f belonging to a wide class of base fields in the same setting we study the 2 part and p part of brumer stark with no special restriction on f and are left with only two well defined specific classes of extensions that elude proof extensive computations were carried out within these two classes and a complete numerical proof of the brumer stark conjecture was obtained in all cases

slowdown and neutral pockets for a random walk in random environment we consider a d dimensional random walk in random environment for which transition probabilities at each site are either neutral or present an effective drift pointing to the right we obtain large deviation estimates on the probability that the walk moves in a too slow ballistic fashion both under the annealed and quenched measures these estimates underline the key role of large neutral pockets of the medium in the occurrence of slowdowns of the walk

annihilating selmer modules we prove that for a wide class of motives the local and global non commutative tamagawa number conjectures of fukaya and kato imply explicit restrictions on the galois structure of selmer modules in the sense of bloch and kato we thereby obtain a variety of new concrete and general conjectures concerning the structures of selmer modules in several important cases we prove these conjectures

a variational approach to branching random walk in random environment this paper considers an infinite system of particles on the integers mathbb z that 1 step to the right with a random delay and 2 split or die along the way according to a random law depending on their position the exponential growth rate of the particle density is computed in the long time limit in the form of a variational formula that can be solved explicitly the result reveals two phase transitions associated with localization vs delocalization and survival vs extinction in addition the system exhibits an intermittency effect greven and den hollander considered the more difficult situation where the particles may step both to the left and right but the analysis of the phase diagram was less complete

contact processes in several dimensions this paper deals primarily with the basic contact processes introduced and studied by harris 12t4 these are random evolutions on the state space s a l l subsets of z a zd the d dimensional integer lattice with extremely simple local dynamics namely if we think of the contact process as representing the spread of an infection t es denoting the set of infected sites at time t then x e it becomes healthy at exponential rate 1 while x z a t is infected at a rate proport ional to the number of sites neighboring x where infection is present the proportionali ty constant 2 is called the infection parameter contact processes are perhaps the simplest s valued markov processes which exhibit a critical phenomenon infection emanating from a single site dies out with probabili ty one for small positive 2 but has positive probability of surviving for all time when is large there is a critical value 2 d where the transition occurs principal objectives of analysis are the precise formulation of the critical phenomenon and detailed description of the ergodic behavior both below and above a c our recent survey articles 10 and i 6 provide introductions to contact processes and to the general field of interacting s valued systems respectively we will make constant use of notation and techniques from the surveys assuming that the reader is familiar with thern the article 10 describes in some detail the current state of knowledge concerning one dimensional contact processes the theory for d 1 is now fairly complete with two major exceptions 1 is not even determined to one decimal place 1 18 2 e there is an

computing fitting ideals of iwasawa modules this paper determines in an equivariant sense the fitting ideals of several iwasawa modules including the most canonical one the connection between the modules themselves which are usually not of finite projective dimension and the required auxiliary modules of finite projective dimension is made rather explicit the resulting fitting ideals look like stickelberger ideals and there is a close relation with recent work of kurihara as an application we obtain new evidence in favor of the brumer stark conjecture

on multidimensional branching random walks in random environment we study branching random walks in random i i d environment in z d d geq 1 for this model the population size cannot decrease and a natural definition of recurrence is introduced we prove a dichotomy for recurrence transience depending only on the support of the environmental law we give sufficient conditions for recurrence and for transience in the recurrent case we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time

on the bloch kato conjecture for elliptic modular forms of square free level let kappa ge 6 be an even integer m an odd square free integer and f in s 2 kappa 2 gamma 0 m a newform we prove that under some reasonable assumptions that half of the lambda part of the bloch kato conjecture for the near central critical value l kappa f is true we do this by bounding the ell valuation of the order of the appropriate bloch kato selmer group below by the ell valuation of algebraic part of l kappa f we prove this by constructing a congruence between the saito kurokawa lift of f and a cuspidal siegel modular form

minus class groups of the fields of the l th roots of unity we show that for any prime number l 2 the minus class group of the field of the l th roots of unity q p l admits a finite free resolution of length 1 as a module over the ring z g 1 i here i denotes complex conjugation in g gal q p l q p z lz moreover for the primes l 509 we show that the minus class group is cyclic as a module over this ring for these primes we also determine the structure of the minus class group

non commutative fitting invariants and annihilation of class groups abstract one can associate to each finitely presented module m over a commutative ring r an r ideal fitt r m which is called the zeroth fitting ideal of m over r and which is an important natural invariant of m we generalize this notion to o orders in separable algebras where o is a complete commutative noetherian local ring as an application we construct annihilators of class groups assuming the validity of the equivariant tamagawa number conjecture for a certain motive attached to a galois cm extension of number fields

minimax robust deconvolution filters under stochastic parametric and noise uncertainties the author consider the design of robust deconvolution filters for linear discrete time systems with stochastic parameter and noise uncertainties it is assumed that some large but bounded uncertainties exist in the driving and measurement noise covariances as well as the second order statistics of stochastic parameters and initial conditions three kinds of minimax sensitivity criteria are used to develop the techniques to the synthesis of minimax deconvolution filters under uncertain linear stochastic systems their approach is based on saddle point theory and the sensitivity analysis of kalman filters the design algorithms give the recursive realization of the minimax deconvolution filters for the time varying uncertain systems under fairly general conditions for the time invariant uncertain case the existence and solutions of steady state deconvolution filters are further developed finally the utility of the minimax design approaches and the properties of the resulting minimax deconvolution filters are illustrated by a numerical example

the geometry of hida families i λ adic de rham cohomology we construct the lambda adic de rham analogue of hida s ordinary lambda adic etale cohomology and of ohta s lambda adic hodge cohomology and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of mathbf q p we give a purely geometric proof of the expected finiteness control and lambda adic duality theorems following ohta we then prove that our lambda adic module of differentials is canonically isomorphic to the space of ordinary lambda adic cuspforms in the sequel cais compos math to appear to this paper we construct the crystalline counterpart to hida s ordinary lambda adic etale cohomology and employ integral p adic hodge theory to prove lambda adic comparison isomorphisms between all of these cohomologies as applications of our work in this paper and cais compos math to appear we will be able to provide a cohomological construction of the family of varphi gamma modules attached to hida s ordinary lambda adic etale cohomology by dee j algebra 235 2 636 664 2001 as well as a new and purely geometric proof of hida s finiteness and control theorems we are also able to prove refinements of the main theorems in mazur and wiles compos math 59 2 231 264 1986 and ohta j reine angew math 463 49 98 1995

refined iwasawa theory and kolyvagin systems of gauss sum type in this paper we establish a refinement of the usual iwasawa main conjecture for the ideal class groups of cm fields over a totally real field using higher fitting ideals

toward equivariant iwasawa theory iii let l be an odd prime number and k k a galois extension of totally real number fields with open image in new window and k k finite where k is the cyclotomic open image in new window extension of k the main conjecture of equivariant iwasawa theory as formulated in rw2 is up to its uniqueness statement reduced to the existence of a nonabelian pseudomeasure whenever g g k k is an l group and iwasawa s invariant vanishes this follows from combining the validity of the conjecture in the maximal order case with special congruences the main tool of proof is a generalization of the taylor oliver integral group logarithm so that it applies to the setting of iwasawa theory

iwasawa theory and the eisenstein ideal we verify for each odd prime p 1000 a conjecture of w mccallum and the author s on the surjectivity of pairings on p units constructed out of the cup product on the first galois cohomology group of the maximal unramified outside p extension of q p with p coefficients in the course of the proof we relate several iwasawa theoretic and hida theoretic objects in particular we construct a canonical isomorphism between an eisenstein ideal modulo its square and the second graded piece in an augmentation filtration of a classical iwasawa module over an abelian pro p kummer extension of the cyclotomic zp extension of an abelian field this kummer extension arises from the galois representation on an inverse limit of ordinary parts of first cohomology groups of modular curves that was considered by m ohta in order to give another proof of the iwasawa main conjecture in the spirit of that of b mazur and a wiles in turn we relate the iwasawa module over the kummer extension to the quotient of the tensor product of the classical cyclotomic iwasawa module and the galois group of the kummer extension by the image of a certain reciprocity map that is constructed out of an inverse limit of cup products up the cyclotomic tower we give an application to the structure of the selmer groups of ohta s modular representation taken modulo the eisenstein ideal

on minimax estimation of a sparse normal mean vector mallows has conjectured that among distributions which are gaussian but for occasional contamination by additive noise the one having least fisher information has two sided geometric contamination a very similar problem arises in estimation of a non negative vector parameter in gaussian white noise when it is known also that most i e 1i components are zero we provide a partial asymptotic expansion of the minimax risk as 0 while the conjecture seems unlikely to be exactly true for flnite we verify it asymptotically up to the accuracy of the expansion numerical work suggests the expansion is accurate for as large as 05 the best 1 estimation rule is flrst but not second order minimax the results bear on an earlier study of maximum entropy estimation and various questions in robustness and function estimation using wavelet bases

adjoint modular galois representations and their selmer groups in the last 15 years many class number formulas and main conjectures have been proven here we discuss such formulas on the selmer groups of the three dimensional adjoint representation ad phi of a two dimensional modular galois representation phi we start with the p adic galois representation phi0 of a modular elliptic curve e and present a formula expressing in terms of l 1 ad phi0 the intersection number of the elliptic curve e and the complementary abelian variety inside the jacobian of the modular curve then we explain how one can deduce a formula for the order of the selmer group sel ad phi0 from the proof of wiles of the shimura taniyama conjecture after that we generalize the formula in an iwasawa theoretic setting of one and two variables here the first variable t is the weight variable of the universal p ordinary hecke algebra and the second variable is the cyclotomic variable s in the one variable case we let phi denote the p ordinary galois representation with values in gl2 zp t lifting phi0 and the characteristic power series of the selmer group sel ad phi is given by a p adic l function interpolating l 1 ad phik for weight k 2 specialization phik of phi in the two variable case we state a main conjecture on the characteristic power series in zp t s of sel ad phi symbol see text nu 1 where nu is the universal cyclotomic character with values in zp s finally we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p adic siegel modular forms

inductive construction of the p adic zeta functions for noncommutative p extensions of exponent p of totally real fields we construct the p adic zeta function for a one dimensional as a p adic lie extension noncommutative p extension f of a totally real number field f such that the finite part of its galois group g is a p group of exponent p we first calculate the whitehead groups of the iwasawa algebra g and its canonical ore localization g s by using oliver and taylor s theory of integral logarithms this calculation reduces the existence of the noncommutative p adic zeta function to certain congruences between abelian p adic zeta pseudomeasures then we finally verify these congruences by using deligne and ribet s theory and a certain inductive technique as an application we prove a special case of the p part of the noncommutative equivariant tamagawa number conjecture for critical tate motives

on the equivariant tamagawa number conjecture in tame cm extensions let l k be a finite galois cm extension with galois group g the equivariant tamagawa number conjecture etnc for the pair h 0 rm spec l 0 mathbb z g naturally decomposes into p parts where p runs over all rational primes if p is odd these p parts in turn decompose into a plus and a minus part let l k be tame above p we show that a certain ray class group of l defines an element in k 0 mathbb z p g mathbb q p which is determined by a corresponding stickelberger element if and only if the minus part of the etnc at p holds for this we use the lifted root number conjecture for small sets of places which is equivalent to the etnc in the number field case for abelian g we show that the minus part of the etnc at p implies the strong brumer stark conjecture at p we prove the minus part of the etnc at p for almost all primes p

the structure of selmer groups our objective in this paper is to prove a rather broad generalization of some classical theorems in iwasawa theory

hilbert modular forms and the gross stark conjecture let f be a totally real eld and an abelian totally odd character of f in 1988 gross stated a p adic analogue of stark s conjecture that relates the value of the derivative of the p adic l function associated to and the p adic logarithm of a p unit in the extension of f cut out by in this paper we prove gross s conjecture when f is a real quadratic eld and is a narrow ring class character the main result also applies to general totally real elds for which leopoldt s conjecture holds assuming that either there are at least two primes above p in f or that a certain condition relating the l invariants of and 1 holds this condition on l invariants is always satised when is quadratic

integrality of stickelberger elements and the equivariant tamagawa number conjecture let l k be a nite galois cm extension of number elds with galois group g in an earlier paper the author has de ned a module sku l k over the center of the group ring zg which coincides with the sinnott kurihara ideal if g is abelian and in particular contains many stickelberger elements it was shown that a certain conjecture on the integrality of sku l k implies the minus part of the equivariant tamagawa number conjecture at an odd prime p for an in nite class of non abelian galois cm extensions of number elds which are at most tamely rami ed above p provided that iwasawa s invariant vanishes here we prove a relevant part of this integrality conjecture which enables us to deduce the equivariant tamagawa number conjecture from the vanishing of for the same class of extensions

lyapounov exponents and quenched large deviations for multidimensional random walk in random environment assign to the lattice sizes z z d i i d random 2d dimensional vectors z z e e 1 whose entries take values in the open unit interval and add up to one given a realization of this environment let x n n 0 be a markov chain on z d which when at z moves one step to its neighbor z e with transition probability z z e we derive a large deviation principle for x n n by means of a result similar to the shape theorem of first passage percolation and related models this result produces certain constants that are the analogue of the lyapounov exponents known from brownian motion in poissonian potential or random walk in random potential we follow a strategy similar to sznitman

state estimation with initial state uncertainty the problem of state estimation with initial state uncertainty is approached from a statistical decision theory point of view the initial state is regarded as deterministic and unknown it is only known that the initial state vector belongs to a specified parameter set the frequentist risk is considered as the performance measure and the minimax approach is adopted minimax estimators are derived for some important cases of unbounded parameter sets if the parameter set is bounded a method of finding estimators whose maximum risk is arbitrarily close to that of a minimax estimator is provided this method is illustrated with an example in which an estimator whose maximum risk is at most 3 larger than that of a minimax estimator is derived