the fundamental group of period domains over finite fields we determine the fundamental group of period domains over finite fields this answers a question of m rapoport raised in m rapoport period domains over finite and local fields in proc sympos pure math vol 62 part 1 1997 pp 361 381

moduli of finite flat group schemes and modularity we prove that under some mild conditions a two dimensional p adic galois representation which is residually modular and potentially barsotti tate at p is modular this provides a more conceptual way of establishing the shimura taniyama weil conjecture especially for elliptic curves which acquire good reduction over a wildly ramified extension of 3 the main ingredient is a new technique for analyzing flat deformation rings it involves resolving them by spaces which parametrize finite flat group scheme models of galois representations introduction

quantum scars of classical closed orbits in phase space the way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x q p through the spectral wigner function w x e this is a sum over wigner functions of eigenstates within a range of energy e in the classical limit w is concentrated on the energy surface and smoothly distributed over it closed orbits provide oscillatory corrections scars for which explicit semiclassical formulae are calculated each scar is a fringe pattern decorating the orbit as x moves off the energy surface the fringes form an airy pattern with spacing of order h as x moves off the closed orbit the fringes form a complex gaussian with spacing h

integral crystalline cohomology over very ramified valuation rings in this paper we intend to show how fontaine s comparison theory fol fo2 between crystalline and p adic etale cohomology can be extented to very ramified base rings so far it had been developed either for qp coefficients or zp coefficients if the base is unramified over the witt vectors see fa2 for example it states that for any p adic discrete valuation ring v there exists a ring b v with the following property if x denotes a smooth and proper v scheme the p adic etale cohomology of the generic fiber x xv k is related to the crystalline cohomology of x v0 vo defined below by an isomorphism

first steps towards p adic langlands functoriality by the theory of colmez and fontaine a de rham representation of the galois group of a local field roughly corresponds to a representation of the weil deligne group equipped with an admissible filtration on the underlying vector space using a modification of the classical local langlands correspondence we associate with any pair consisting of a weil deligne group representation and a type of a filtration admissible or not a specific locally algebraic representation of a general linear group we advertise the conjecture that this pair comes from a de rham representation if and only if the corresponding locally algebraic representation carries an invariant norm in the crystalline case the weil deligne group representation is unramified and the associated locally algebraic representation can be studied using the classical satake isomorphism by extending the latter to a specific norm completion of the hecke algebra we show that the existence of an invariant norm implies that our pair indeed comes from a crystalline representation we also show by using the formalism of tannakian categories that this latter fact is compatible with classical unramified langlands functoriality and therefore generalizes to arbitrary split reductive groups resume par la theorie de colmez et fontaine une representation de de rham du groupe de galois d un corps local correspond essentiellement a une representation du groupe de weil deligne dont l espace sous jacent est muni d une filtration admissible en modifiant la correspondance locale de langlands on associe a chaque couple forme d une representation du groupe de weil deligne et des poids d une filtration admissible ou pas une representation localement algebrique particuliere d un groupe lineaire general on conjecture qu un couple provient d une representation de de rham si et seulement si la representation localement algebrique correspondante possede une norme invariante dans le cas cristallin la representation du groupe de weil deligne est non ramifiee et la representation localement algebrique associee peut s etudier gr ce a l isomorphisme de satake classique en prolongeant ce dernier a une completion de l algebre de hecke on montre que l existence d une norme invariante comme ci dessus implique que le couple provient effectivement d une representation cristalline on montre aussi en utilisant le formalisme des categories tannakiennes que ce dernier fait est compatible avec la fonctorialite de langlands non ramifiee classique et donc qu il se generalise a tout groupe reductif deploye 2 c breuil p schneider

on complete congruence lattices of complete lattices the lattice of all complete congruence relations of a complete lattice is itself a complete lattice in this paper we characterize this lattice as a com plete lattice in other words for a complete lattice l we construct a complete lattice k such that l is isomorphic to the lattice of complete congruence re lations of k regarding k as an infinitary algebra this result strengthens the characterization theorem of congruence lattices of infinitary algebras of g grat zer and w a lampe in addition we show how to construct k so that it will also have a prescribed automorphism group

the cohomology of period domains for reductive groups over local fields we compute the etale cohomology of period domains over local fields in the case of a basic isocrystal for quasi split reductive groups period domains which have been introduced by rapoport and zink rz are open admissible rigid analytic subsets of generalized flag varieties they parametrize weakly admissible filtrations of a given isocrystal with additional structure of a reductive group

seshadri constants at very general points we study the local positivity of an ample line bundle at a very general point of a smooth projective variety we obtain a slight improvement of the result of ein kuchle and lazarsfeld

a new lattice construction the box product in a recent paper the authors have proved that for lattices a and b with zero the isomorphism conc a otimes b cong conc a otimes conc b holds provided that the tensor product satis es a very natural condition of being capped implying that a otimes b is a lattice in general a otimes b is not a lattice for instance we proved that m 3 otimes f 3 is not a lattice in this paper we introduce a new lattice construction the box product for arbitrary lattices the tensor product construction for complete lattices introduced by g n raney in 1960 and by r wille in 1985 and the tensor product construction of a fraser in 1978 for semilattices bear some formal resemblance to the new construction for lattices a and b while their tensor product a otimes b as semilattices is not always a lattice the box product a bp b is always a lattice furthermore the box product and some of its ideals behave like an improved tensor product for example if a and b are lattices with unit then the isomorphism conc a bp b cong conc a otimes conc b holds there are analogous results for lattices a and b with zero and for a bounded lattice a and an arbitrary lattice b a join semilattice s with zero is called 0 representable if there exists a lattice l with zero such that conc l cong s the above isomorphism results yield the following consequence the tensor product of two 0 representable semilattices is 0 representable

eigenvalues of hermitian matrices with positive sum of bounded rank we give a minimal list of inequalities characterizing the possible eigenvalues of a set of hermitian matrices with positive semidefinite sum of bounded rank this answers a question of barvinok

conjecture de l inertie moderee de serre on considere k un corps complet pour une valuation discrete de caracteristique nulle et dont le corps residuel est suppose parfait de caracteristique p on appelle e l indice de ramification absolue de k mathcal o k son anneau des entiers et overline k une cloture algebrique soit xk un schema propre et lisse sur k admettant un modele propre et semi stable x sur mathcal o k generalisant les resultats de breuil valables pour e 1 on demontre dans cet article un isomorphisme de periodes reliant le r ieme groupe de cohomologie etale de x overline k a coefficients dans pn et un r ieme groupe de cohomologie log cristalline de la fibre speciale de x nous avons toutefois les restrictions er 1

mazur s inequality and laffaille s theorem we look at various questions related to filtrations in p adic hodge theory using a blend of building and tannakian tools specifically fontaine and rapoport used a theorem of laffaille on filtered isocrystals to establish a converse of mazur s inequality for isocrystals we generalize both results to the setting of filtered g isocrystals and also establish an analog of totaro s otimes product theorem for the harder narasimhan filtration of fargues

on the weights of mod p hilbert modular forms we prove many cases of a conjecture of buzzard diamond and jarvis on the possible weights of mod p hilbert modular forms by making use of modularity lifting theorems and computations in p adic hodge theory

complete congruence lattices of complete distributive lattices abstract in 1983 wille raised the question is every complete lattice l isomorphic to the lattice of complete congruence relations of a suitable complete lattice k in 1988 this was answered in the affirmative by the first author a number of papers have been published on this problem by freese johnson lakser teo and the authors in the present paper we prove that k can always be chosen as a complete distributive lattice in fact we prove the following more general result theorem let m be a regular cardinal 0 every m algebraic lattice l can be represented as the lattice of m complete congruence relations of an m complete distributive lattice k

tensor products in p adic hodge theory there is a classical relation between the p adic absolute value of the eigenvalues of frobenius on crystalline cohomology and hodge numbers for a variety in characteristic p the newton polygon lies on or above the hodge polygon 14 1 for a variety in characteristic p with a lift to characteristic 0 fontaine conjectured and faltings proved a more precise statement there is an inequality which relates the slope of frobenius on any frobenius invariant subspace of the crystalline cohomology to the hodge filtration restricted to that subspace 7 4 a vector space over a p adic field together with a linear endomorphism and a filtration which satisfies this inequality is called a weakly admissible filtered isocrystal see section 1 for the precise definition the category of such objects is one possible p adic analogue of the category of hodge structures in particular it is an abelian category we give a new proof of faltings s theorem that the tensor product of weakly admissible filtered isocrystals over a p adic field is weakly admissible 5 by a similar argument we also prove a characterization of weakly admissible filtered isocrystals with g structure in terms of geometric invariant theory which was conjectured by rapoport and zink 19 before faltings laffaille 12 had proved the tensor product theorem in the case of filtered isocrystals over an unramified extension of qp faltings s proof works by reducing this problem of linear algebra to a different problem of pure linear algebra the problem of showing that the tensor product of two vector spaces each equipped with a finite semistable set of filtrations is semistable the latter problem is solved by constructing suitable integral lattices in 5 or hermitian metrics in 20 on vector spaces with a semistable set of filtrations just as one can prove that the tensor product of semistable bundles on an algebraic curve is semistable using narasimhan seshadri s hermitian metrics 6 16 in this paper we can avoid the reduction from filtered isocrystals to filtered vector spaces the point is that ramanan and ramanathan s algebraic proof 17 that the tensor product of semistable vector bundles is semistable can be modified to apply directly to filtered isocrystals we have an inequality to prove for a class of linear subspaces s of a tensor product v w the inequality is obvious for sufficiently general subspaces s and also if s is a very special subspace say if s is a decomposable subspace s1 s2 v w but it is not clear how to prove the inequality we want if s is somewhere in the middle the solution following ramanan and ramanathan is to use geometric invariant theory to give a sharp dichotomy between general subspaces and special subspaces of v w in such a way that we get useful information in either case

roth s theorem for ruled surfaces this paper addresses conjectures of e bombieri and p vojta in the special case of ruled surfaces not birational to inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i apart from this implicit restriction to inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i bundles s over an elliptic curve the ultimate question of the arithmetic of pairs s d for a divisor d requires further restrictions on d which turn the proposed conjectures into the study of roth s theorem on approximation of algebraic numbers but for now parametrized by an elliptic curve with these restrictions best possible answers are obtained the same study may also be carried out for holomorphic maps and this is done simultaneously

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

holomorphic curves into algebraic varieties this paper establishes a defect relation for algebraically nondegenerate holomorphic mappings into an arbitrary nonsingular complex projective variety v rather than just the projective space intersecting possible nonlinear hypersurfaces extending the result of h cartan

a toric extension of faltings diophantine approximation on abelian varieties for divisors on abelian varieties faltings established an optimal bound on the proximity of rational points to the same we extend this both to the quasiprojective category where the role of abelian varieties is played by their toroidal extensions and to holomorphic maps from the line proving along the way some wholly general dynamic intersection estimates in value distribution theory of independent interest

on a general thue s equation we analyze the integral points on varieties defined by one equation of the form f 1 f r g where the f i g are polynomials in n variables with algebraic coefficients and g has small degree we shall use a method that we recently introduced in the context of siegel s theorem for integral points on curves classical very particular instances of our equations arise e g in a well known corollary of roth s theorem the case n 2 f i linear forms deg g r 2 and with the norm form equations treated by w m schmidt here we shall prove thm 1 that the integral points are not zariski dense provided deg f i n max deg f i deg g and provided the f i g satisfy certain mild assumptions which are generically verified our conclusions also cover certain complete intersection subvarieties of our hypersurface thm 2 finally we shall prove thm 3 an analogue of the schmidt s subspace theorem for arbitrary polynomials in place of linear forms

generalizations of siegel s and picard s theorems we prove new theorems that are higher dimensional generalizations of the classical theorems of siegel on integral points on affine curves and of picard on holomorphic maps from c to affine curves these include results on integral points over varying number fields of bounded degree and results on kobayashi hyperbolicity we give a number of new conjectures describing from our point of view how we expect siegel s and picard s theorems to optimally generalize to higher dimensions in this article we prove new theorems that are higher dimensional generaliza tions of the classical theorems of siegel on integral points on affine curves and of picard on holomorphic maps from c to affine curves in section 2 we will give the statements of siegel s and picard s theorems and we will recall how these two theorems from such seemingly different areas of mathematics are related we will then proceed to give a number of new conjectures describing from our point of view how we expect siegel s and picard s theorems to optimally generalize to higher dimensions these include conjectures on integral points over varying number fields of bounded degree and conjectures addressing hyperbolic questions these conjectures appear to be fundamentally new we will then summarize our progress on these conjectures we have been able to get results in all dimensions with best possible results in many cases for surfaces

construction of some families of 2 dimensional crystalline representations we construct explicitly some analytic families of etale modules which give rise to analytic families of 2 dimensional crystalline representations as an application of our constructions we verify some conjectures of breuil on the reduction modulo p of those representations and extend some results of deligne edixhoven fontaine and serre on the representations arising from modular forms

smoothed wave functions of chaotic quantum systems abstract it is shown that wave functions of quantum systems as 0 have an extra density near unstable periodic trajectories of the classical problem the averaged wave function square is represented as the sum over a finite number of periodic trajectories the contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory the results are compared with the numerical calculations of the wave functions for the stadium billiard

exact theory for the quantum eigenstates of a strongly chaotic system we present an exact theory for the quantum eigenstates of a strongly chaotic system which consists of a point particle sliding freely on a two dimensional compact surface of constant negative curvature the main result is a general sum rule which relates an almost arbitrarily smoothed sum over the quantum wavefunctions to a sum over the lengths of the classical orbits as an example we apply this formula to the gaussian smoothed sum over wavefunctions the question of scars as imprints of individual closed orbits is discussed and the connection with bogomolny s semiclassical theory is explicitly worked out the investigation of the amplitude distribution p reveals that the higher excited wavefunctions behave as gaussian random waves

seshadri constants diophantine approximation and roth s theorem for arbitrary varieties in this paper we associate an invariant alpha x l to an algebraic point x on an algebraic variety x with an ample line bundle l the invariant alpha measures how well x can be approximated by rational points on x with respect to the height function associated to l we show that this invariant is closely related to the seshadri constant epsilon x l measuring local positivity of l at x and in particular that roth s theorem on mathbb p 1 generalizes as an inequality between these two invariants valid for arbitrary projective varieties

a finer tate duality theorem for local galois symbols abstract let k be a finite extension of q p let a b be abelian varieties over k with good reduction for any integer m 1 we consider the galois symbol k k a b m h 2 k a m b m where k k a b is the somekawa k group attached to a b this map is a generalization of the galois symbol k 2 m k m h 2 k m 2 of the bloch kato conjecture where k 2 m k is the milnor k group of k in this paper we give a geometric description of the image of this generalized galois symbol by looking at the tate duality pairing h 2 k a m b m hom g k a m b m z m where b is the dual abelian variety of b under this perfect pairing we compute the exact annihilator of the image of the galois symbol in terms of an object of integral p adic hodge theory in this way we generalize a result of tate for h 1 moreover our result has applications to zero cycles on abelian varieties

contour splitting as a criterion for surface roughness in irregular wavefunctions a measure of irregularity in a quantum state lies in the surface roughness of the corresponding eigenfunction and shows up as a splitting of contours at successive heights this phenomenon seems to be absent in integrable systems where the states are regular the authors carry out an analysis based on the semiclassical theory and extend it by a detailed numerical investigation which reveals that i a gaussian amplitude distribution and the splitting of contours occur together and ii the percentage increase in the number of contours with height is a measure of the degree of surface irregularity regular localized states in chaotic systems for which good quantum numbers can be assigned do however show contour splitting

on a conjecture of conrad diamond and taylor we prove a conjecture of conrad diamond and taylor on the size of certain deformation rings parametrizing potentially barsotti tate ga lois representations to achieve this we extend results of breuil and m ezard classifying galois lattices in semistable representations in terms of strongly divisible modules to the potentially crystalline case in hodge tate weights 0 1 we then use these strongly divisible modules to compute the desired deformation rings as a corollary we obtain new results on the modularity of potentially barsotti tate representations

a further improvement of the quantitative subspace theorem in 2002 evertse and schlickewei 11 obtained a quantitative version of the so called absolute parametric subspace theorem this result deals with a parametrized class of twisted heights one of the consequences of this result is a quantitative version of the absolute subspace theorem giving an explicit upper bound for the number of subspaces containing the solutions of the diophantine inequality under consideration in the present paper we further improve evertse s and schlickewei s quantitative version of the absolute parametric subspace theorem and deduce an improved quantitative version of the absolute subspace theorem we combine ideas from the proof of evertse and schlickewei which is basically a substantial refinement of schmidt s proof of his subspace theorem from 1972 22 with ideas from faltings and wustholz proof of the subspace theorem 14 a new feature is an interval result which gives more precise information on the distribution of the heights of the solutions of the system of inequalities considered in the subspace theorem

weight cycling and serre type conjectures for unitary groups we prove that for forms of u 3 which are compact at infinity and split at places dividing a prime p in generic situations the serre weights of a mod p modular galois representation which is irreducible when restricted to each decomposition group above p are exactly those previously predicted by herzig we do this by combining explicit computations in p adic hodge theory based on a formalism of strongly divisible modules and breuil modules with descent data which we develop here with a technique that we call weight cycling

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces