lmi based h optimal control with transients in this article the worst case norm of the regulated output over all exogenous signals and initial states as a performance measure of the system is characterised in terms of linear matrix inequalities lmis optimal time invariant state and output feedback controllers are synthesised as minimising this performance measure the essential role in this synthesis plays a weighting matrix reflecting the relative importance of the uncertainty in the initial state contrary to the uncertainty in the exogenous signal h optimal control with transients is shown to be actually a trade off between h control being optimal under unknown exogenous disturbances and zero initial state and control being optimal under zero exogenous signal and unknown initial conditions if and only if the weighting matrix satisfies a fundamental inequality if this inequality is met the performance measure is achieved and the explicit formulae for the worst case disturbance and initial state are provided if this inequality fa

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 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

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

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

cohomology of units and l values at zero let klk be a finite galois extension of number fields with galois group g and let s be a finite g stable set of primes of k containing all archimedean primes we denote the g module of s units of k by e es and let as be the kernel of the augmentation map zs z which sends each basis element p c s to 1 we are concerned with invariants of k k which are associated to a g homomorphism p as e inducingq0 as cq0e these invariants were defined by tate ta2 and chinburg chl when s is large i e when s contains all ramified primes of k k and the s class group cl cls of k is trivial there are two of them and each is a function of the complex characters x of g the first aw has a x equal to the tate regulator at x divided by the leading coefficient c x of the taylor expansion at s 0 of the artin l function l s x with the euler factors at primes of s omitted according to stark s conjecture st in tate s form a takes algebraic values which satisfy a o x5 a x

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

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

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

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

universal eigenvarieties trianguline galois representations and p adic langlands functoriality using the overconvergent cohomology modules introduced by ash and stevens we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces building on work of ash stevens urban and others we also formulate a precise modularity conjecture linking trianguline galois representations with overconvergent cohomology classes in the course of giving evidence for this conjecture we establish several new instances of p adic langlands functoriality our main technical innovations are a family of universal coefficients spectral sequences for overconvergent cohomology and a generalization of chenevier s interpolation theorem

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

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

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

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

the 2 sylow subgroup of k2of for number fields f abstract let f be a quadratic number field we give a criterion via hilbert symbols for an element of order two in the tame kernel of f to be a fourth power in the tame kernel of f the result can be applied to compute the 8 rank of the tame kernel of f and the tate kernel of an imaginary quadratic number field we list the 8 ranks of k 2 o f for all quadratic number fields whose discriminants have exactly two odd prime divisors in the case when f is an imaginary quadratic number field with the 8 rank of k 2 o f 0 the tate kernel of f is given too an application of our method to the maximal real subfield of a cyclotomic field is discussed numerical examples in particular the examples of quadratic number fields f with 4 rank of k 2 o f 8 rank of k 2 o f 2 illustrate our results

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 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

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

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

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

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

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

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

hybrid iwasawa algebras and the equivariant iwasawa main conjecture let p be an odd prime we give an unconditional proof of the equivariant iwasawa main conjecture for totally real fields for an infinite class of one dimensional non abelian p adic lie extensions crucially this result does not depend on the vanishing of the relevant iwasawa mu invariant