nombres de weil sommes de gauss et annulateurs galoisiens for an abelian number field k containing a primitive p rm th root of unity p an odd prime and satisfying certain technical conditions we parametrize the bbb z p rm g k bbb q annihilators of the minus part a k of the p class group by means of modules of jacobi sums using a reflection theorem and bloch kato s reciprocity law we then determine the fitting ideal of the plus part a k in terms of twisted gauss sums

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

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

a proof of the uniform boundedness of solutions to the wave equation on slowly rotating kerr backgrounds we consider kerr spacetimes with parameters a and m such that a m kerr newman spacetimes with parameters q m a m and more generally stationary axisymmetric black hole exterior spacetimes mathcal m g which are sufficiently close to a schwarzschild metric with parameter m 0 and whose killing fields span the null generator of the event horizon we show uniform boundedness on the exterior for solutions to the wave equation g 0 the most fundamental statement is at the level of energy we show that given a suitable foliation then there exists a constant c depending only on the parameter m and the choice of the foliation such that for all solutions a suitable energy flux through is bounded by c times the initial energy flux through 0 this energy flux is positive definite and does not degenerate at the horizon i e it agrees with the energy as measured by a local observer it is shown that a similar boundedness statement holds for all higher order energies again without degeneration at the horizon this leads in particular to the pointwise uniform boundedness of in terms of a higher order initial energy on 0 note that in view of the very general assumptions the separability properties of the wave equation or geodesic flow on the kerr background are not used in fact the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high frequency geometric optics regime

asymptotic symmetries and asymptotically symmetric solutions of partial differential equations symmetry methods for differential equations are a powerful tool to attack nonlinear problems in particular for determining solutions with given symmetries to nonlinear pdes since in real applications one is often interested in solutions which are asymptotically symmetric we propose here an approach to asymptotic symmetry based on the methods of lie theory we adopt translate in geometric language and develop the renormalization group approach recently proposed by bricmont and kupiainen 1992 for the ginzburg landau equation

lie equations for asymptotic solutions of perturbation problems of ordinary differential equations lie theory is applied to perturbation problems of ordinary differential equations to construct approximate solutions and invariant manifolds according to the renormalization group approach of iwasa and nozaki a method to construct asymptotic solutions invariant under the renormalization group prog theor phys 116 605 2006 it is proved that asymptotic behavior of solutions is obtained from the lie equations even if original equations have no symmetries normal forms of the lie equations are introduced to investigate the existence of invariant manifolds

renormalization group in difference systems a new singular perturbation method based on the lie symmetry group is presented to a system of difference equations this method yields consistent derivation of a renormalization group equation which gives an asymptotic solution of the difference equation the renormalization group equation is a lie differential equation of a lie group which leaves the system approximately invariant for a 2d symplectic map the renormalization group equation becomes a hamiltonian system and a long time behaviour of the symplectic map is described by the hamiltonian we study the poincar birkoff bifurcation in the 2d symplectic map by means of the hamiltonian and give a condition for the bifurcation

solution of reduced equations derived with singular perturbation methods for singular perturbation problems in dynamical systems various appropriate singular perturbation methods have been proposed to eliminate secular terms appearing in the naive expansion for example the method of multiple time scales the normal form method center manifold theory and the renormalization group method are well known it is shown that all of the solutions of the reduced equations constructed with those methods are exactly equal to the sum of the most divergent secular terms appearing in the naive expansion for the proof a method to construct a perturbation solution which differs from the conventional one is presented where we make use of the theory of the lie symmetry group

further reductions of normal forms for dynamical systems abstract we propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields with concrete and interesting applications we use lower order nonlinear terms in the normal form for the simplifications of higher order terms our approach is applicable for both the non nilpotent and the nilpotent cases for dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal or unique with respect to equivalence by formal near identity transformations we can compute at the same time a formal diffeormorphism that realizes the normalization comparisons with other methods are given for several examples

stability of an n 2 dimensional invariant torus in the kuramoto model at small coupling abstract when the natural frequencies are allocated symmetrically in the kuramoto model there exists an invariant torus of dimension n 2 1 n is the population size a global phase shift invariance allows us to reduce the model to n 1 dimensions using the phase differences and doing so the invariant torus becomes n 2 dimensional by means of perturbative calculations based on the renormalization group technique we show that this torus is asymptotically stable at small coupling if n is odd if n is even the torus can be stable or unstable depending on the natural frequencies and both possibilities persist in the small coupling limit

renormalization group and singular perturbations multiple scales boundary layers and reductive perturbation theory perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis with numerous examples we illustrate its application to ordinary differential equation problems involving multiple scales boundary layers with technically difficult asymptotic matching and wkb analysis in contrast to conventional methods the renormalization group approach requires neither ad hoc assumptions about the structure of perturbation series nor the use of asymptotic matching our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally although the latter can be reproduced if desired by appropriate expansion of the renormalization group approximant we show that the renormalization group equation may be interpreted as an amplitude equation and from this point of view develop reductive perturbation theory for partial differential equations describing spatially extended systems near bifurcation points deriving both amplitude equations and the center manifold s1063651x 96 00506 5

approximation of center manifolds on the renormalization group method the renormalization group rg method for differential equations is one of the perturbation methods for obtaining approximate solutions this article shows that the rg method is effectual for obtaining an approximate center manifold and an approximate flow on it when applied to equations having a center manifold

theorie d iwasawa des representations p adiques d un corps local let k be finite extension of qp we give a description of the iwasawa modules attached to p adic representations of the absolute galois group of k in terms of the theory of modules of fontaine when the representation is de rham this gives a natural construction of the exponential map of perrin riou which is used in the construction and the study of p adic l functions

simplified renormalization group method for ordinary differential equations abstract the renormalization group rg method for differential equations is one of the perturbation methods which allows one to obtain invariant manifolds of a given ordinary differential equation together with approximate solutions to it this article investigates higher order rg equations which serve to refine an error estimate of approximate solutions obtained by the first order rg equations it is shown that the higher order rg equation maintains the similar theorems to those provided by the first order rg equation which are theorems on well definedness of approximate vector fields and on inheritance of invariant manifolds from those for the rg equation to those for the original equation for example since the higher order rg equation is defined by using indefinite integrals and is not unique for the reason of the undetermined integral constants the simplest form of rg equation is available by choosing suitable integral constants it is shown that this simplified rg equation is sufficient to determine whether the trivial solution to time dependent linear equations is hyperbolically stable or not and thereby a synchronous solution of a coupled oscillators is shown to be stable

on the cyclotomic main conjecture for the prime 2 we complete the proof of the equivariant tamagawa number conjecture for tate motives over absolutely abelian fields by proving a refined cyclotomic main conjecture at the prime 2

the double wedge algebra for quantum fields on schwarzschild and minkowski spacetimes we consider the klein gordon equation m 0 on the double schwarzschild wedge of the kruskal spacetime and construct the hartle hawking state h as a thermal state relative to the boulware quantization we prove that on the double wedge h is a pure state and in the corresponding representation the left and right wedgec algebras each have the reeh schlieder property while the corresponding von neumann algebras are typeiii1 factors which are dual to i e commutants of each other we discuss the extent to which these properties may generalize to non quasi free field theories

on the stability of the kerr metric the reduced in the angular coordinate wave equation and klein gordon equation are considered on a kerr background and in the framework of c 0 semigroup theory each equation is shown to have a well posed initial value problem i e to have a unique solution depending continuously on the data further it is shown that the spectrum of the semigroup s generator coincides with the spectrum of an operator polynomial whose coefficients can be read off from the equation in this way the problem of deciding stability is reduced to a spectral problem and a mathematical basis is provided for mode considerations for the wave equation it is shown that the resolvent of the semigroup s generator and the corresponding green s functions can be computed using spheroidal functions it is to be expected that analogous to the case of a schwarzschild background the quasinormal frequencies of the kerr black hole appear as resonances i e poles of the analytic continuation of this resolvent finally stability of the solutions of the reduced klein gordon equation is proven for large enough masses

group analysis and renormgroup symmetries an original regular approach to constructing special type symmetries for boundary value problems namely renormgroup symmetries is presented different methods of calculating these symmetries based on modern group analysis are described an application of the approach to boundary value problems is demonstrated with the help of a simple mathematical model

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

a generalization of the coleman map for hida deformations in this paper we give a coleman perrin riou type map for an ordinary type deformation and construct a two variable p adic l function for a hida family from the beilinson kato elements 1 introduction fix a prime p 3 we denote by q jlp the extension of the rational number field q obtained by adjoining all power roots of unity we fix a complex embedding q c and a p adic embedding q qp of an algebraic closure q of q throughout the paper where c is the field of complex numbers and qp is an algebraic closure of the field qp of p adic numbers let goo be the galois group of q jjlpo q and let dqo be the group of diamond operators for the tower of modular curves fi i see 2 we have the

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

on the structure of signed selmer groups let f be a number field unramified at an odd prime p and f infty be the mathbf z p cyclotomic extension of f generalizing kobayashi plus minus selmer groups for elliptic curves buyukboduk and lei have defined modified selmer groups called signed selmer groups for certain non ordinary mathrm gal overline f f representations in particular their construction applies to abelian varieties defined over f with good supersingular reduction at primes of f dividing p assuming that these selmer groups are cotorsion mathbf z p mathrm gal f infty f modules we show that they have no proper sub mathbf z p mathrm gal f infty f module of finite index we deduce from this a number of arithmetic applications on studying the euler poincare characteristic of these selmer groups we obtain an explicit formula on the size of the bloch kato selmer group attached to these representations furthermore for two such representations that are isomorphic modulo p we compare the iwasawa invariants of their signed selmer groups

renormalization group theory for global asymptotic analysis we show with several examples that renormalization group rg theory can be used to understand singular and reductive perturbation methods in a unified fashion amplitude equations describing slow motion dynamics in nonequilibrium phenomena are rg equations the renormalized perturbation approach may be simpler to use than other approaches because it does not require the use of asymptotic matching and yields practically superior approximations

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

the coleman map for hida families of gsp4 in this paper we construct a coleman map for ordinary hida deformations associated to the symplectic group rm gsp 4 which interpolates the dual exponential maps when arithmetic specializations of this family vary the main result theorem 3 2 will play an important role in a forthcoming work by the authors discussing a conjecture on the existence of an euler system which would give rise via our map to a three variable p adic l function

actions infinitesimales dans la correspondance de langlands locale p adique let v be a two dimensional absolutely irreducible overline mathbb qp representation of rm gal overline mathbb qp mathbb qp and let prod v be the rm gl 2 mathbb qp banach representation associated by colmez s p adic langlands correspondence we establish a link between the action of the lie algebra of rm gl 2 mathbb qp on the locally analytic vectors prod v rm an of prod v the connection nabla on the varphi gamma module associated to v and the sen polynomial of v this answers a question of harris concerning the infinitesimal character of prod v rm an using this result we give a new proof of a theorem of colmez stating that prod v has nonzero locally algebraic vectors if and only if v is potentially semi stable with distinct hodge tate weights