on elliptic curves of prime power conductor over imaginary quadratic fields with class number 1 the main result of this paper is to extend from q to each of the nine imaginary quadratic fields of class number one a result of serre 1987 and mestre oesterle 1989 namely that if e is an elliptic curve of prime conductor then either e or a 2 3 or 5 isogenous curve has prime discriminant for four of the nine fields the theorem holds with no change while for the remaining five fields the discriminant of a curve with prime conductor is either prime or the square of a prime the proof is conditional in two ways first that the curves are modular so are associated to suitable bianchi newforms and second that a certain level lowering conjecture holds for bianchi newforms we also classify all elliptic curves of prime power conductor and non trivial torsion over each of the nine fields in the case of 2 torsion we find that such curves either have cm or with a small finite number of exceptions arise from a family analogous to the setzer neumann family over q

a large deviation principle for r p capacities on the wiener space we formulate and prove a large deviation principle for the r p capacity on an abstract wiener space as an application we obtain a sharpening of strassen s law of the iterated logarithm in terms of the capacity

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

comparison of semi simplifications of galois representations abstract let be the absolute galois group of a global field let 1 and 2 be two p adic finite dimensional representations of then there exists a finite number of primes q such that if the characteristic polynomials of 1 frob q and 2 frob q are equal then 1 and 2 have isomorphic semi simplifications and so the same l functions we give a method to compute a sufficient list of primes based on the ramification and the dimension of the representations we use then the result to prove a conjecture by b van geemen and j top

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

higher coherent cohomology and p adic modular forms of singular weights we investigate the p adic properties of higher coherent cohomology of automorphic vector bundles of singular weights on siegel threefolds

sobolev spaces of banach valued functions associated with a markov process we discuss sobolev spaces of banach valued functions they are extensions of sobolev spaces of scalar functions we use a gamma transform of a semigroup associated with a markov process a typical example is the ornstein uhlenbeck process on the wiener space

on serre s conjecture for mod l galois representations over totally real fields in 1987 serre conjectured that any mod two dimensional irre ducible odd representation of the absolute galois group of the rationals came from a modular form in a precise way we present a generalisation of this conjecture to 2 dimensional representations of the absolute galois group of a totally real eld where is unramied the hard work is in formulating an analogue of the weight part of serre s conjecture serre furthermore asked whether his conjecture could be rephrased in terms of a mod langlands philosophy using ideas of emerton and vign eras we formulate a mod local global principle for the group d where d is a quaternion algebra over a totally real eld split above and at 0 or 1 innite places and show how it implies the conjecture

counting zero kernel pairs over a finite field abstract helmke et al have recently given a formula for the number of reachable pairs of matrices over a finite field we give a new and elementary proof of the same formula by solving the equivalent problem of determining the number of so called zero kernel pairs over a finite field we show that the problem is equivalent to certain other enumeration problems and outline a connection with some recent results of guo and yang on the natural density of rectangular unimodular matrices over f q x we also propose a new conjecture on the density of unimodular matrix polynomials

tightness of general c1 p capacities on banach space abstract we prove tightness of c 1 p capacities on a banach space e these capacities are defined in terms of measures on e which are not necessarily gaussian our methods are purely analytic and completely different from earlier approaches which were implemented under more restrictive assumptions

on the number of controllable linear systems over a finite field it is shown that over a finite field there are more controllable systems than uncontrollable ones if and only if the number of system inputs is greater than one or the field contains at least three elements

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

modular representations of gl 3 fp symmetric squares and mod p cohomology of gl 3 z abstract we prove certain cases of nonabelian reciprocity between mod p galois representations and hecke eigenclasses in the mod p cohomology of gl 3 z using the symmetric square liftings from gl 2 and congruences on hecke eigenvalues we need to develop some modular representation theory for gl 3 z p in some cases we also need to invoke certain computer calculations to finish the proof

automorphic lifts of prescribed types we prove a variety of results on the existence of automorphic galois representations lifting a residual automorphic galois representation we prove a result on the structure of deformation rings of local galois representations and deduce from this and the method of khare and wintenberger a result on the existence of modular lifts of specified type for galois representations corresponding to hilbert modular forms of parallel weight 2 we discuss some conjectures on the weights of n dimensional mod p galois representations finally we use recent work of taylor to prove level raising and lowering results for n dimensional automorphic galois representations

the number of linear transformations defined on a subspace with given invariant factors abstract given a finite dimensional vector space v over the finite field f q and a subspace w of v we consider the problem of counting linear transformations t w v which have prescribed invariant factors the case w v is a well studied problem that is essentially equivalent to counting the number of square matrices over f q in a conjugacy class and an explicit formula is known in this case on the other hand the case of general w is also an interesting problem and there hasn t been substantive progress in this case for over two decades barring a special case where all the invariant factors of t are of degree zero we extend this result to the case of arbitrary w by giving an explicit counting formula as an application of our results we give new proofs of some recent enumerative results in linear control theory and derive an extension of the gerstenhaber reiner formula for the number of square matrices over f q with given characteristic polynomial

an analogue of serre s conjecture for galois representations and hecke eigenclasses in the mod p cohomology of gl n ℤ

hat g local systems on smooth projective curves are potentially automorphic dfg grants fg 1920 and spp 1489 erc grant 290766 aamot nsf grant dms 1404769 nsf grant dms 1161671 humboldt research award clay mathematics institute erc grant no 714405 gmlp

on rigid analytic uniformizations of jacobians of shimura curves the main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the jacobian of a shimura curve over bbb q at a prime dividing exactly the level this result can be viewed as complementary to the classical theorem of v c erednik and drinfeld which provides rigid analytic uniformizations at primes dividing the discriminant as a corollary we offer a proof of a conjecture formulated by m greenberg in his paper on stark heegner points and quaternionic shimura curves thus making greenberg s construction of local points on elliptic curves over bbb q unconditional

galois representations with conjectural connections to arithmetic cohomology in this paper we extend a conjecture of a ash and w sinnott relating niveau 1 galois representations to the mod p cohomology of congruence subgroups ofsln z to include galois representations of higher niveau we then present computational evidence for our conjecture in the case n 3 in the form of three dimensional galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture our examples include galois representations with nontrivial weight and level as well as irreducible three dimensional representations that are in no obvious way related to lower dimensional representations in addition we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture

the probability of rectangular unimodular matrices over fq x in this note we show that the probability that a uniformly random k n matrix over fq x can be extended to an n n invertible matrix is 1 qk n 1 qk 1 n 1 q1 n connections with dirichlet s density theorem on co prime integers and its various generalizations are also presented

strata hasse invariants hecke algebras and galois representations we construct group theoretical generalizations of the hasse invariant on strata closures of the stacks mathop g hbox mathtt zip nolimits mu restricting to zip data of hodge type we obtain a group theoretical hasse invariant on every ekedahl oort stratum closure of a general hodge type shimura variety a key tool is the construction of a stack of zip flags mathop g hbox mathtt zipflag nolimits mu fibered in flag varieties over mathop g hbox mathtt zip nolimits mu it provides a simultaneous generalization of the classical case homogeneous complex manifolds studied by griffiths schmid and the flag space for siegel varieties studied by ekedahl van der geer four applications are obtained 1 pseudo representations are attached to the coherent cohomology of hodge type shimura varieties modulo a prime power 2 galois representations are associated to many automorphic representations with non degenerate limit of discrete series archimedean component 3 it is shown that all ekedahl oort strata in the minimal compactification of a hodge type shimura variety are affine thereby proving a conjecture of oort 4 part of serre s letter to tate on mod p modular forms is generalized to general hodge type shimura varieties

geometric families of 4 dimensional galois representations with generically large images we study compatible families of four dimensional galois representations constructed in the etale cohomology of a smooth projective variety we prove a theorem asserting that the images will be generically large if certain conditions are satisfied we only consider representations with coefficients in an imaginary quadratic field we apply our result to an example constructed by jasper scholten a non selfdual 4 dimensional galois representation www math uiuc edu algebraic number theory 0183 1999 obtaining a family of linear groups and one of unitary groups as galois groups over mathbb q

on the images of modular and geometric three dimensional galois representations we study compatible families of three dimensional galois representations constructed in the etale cohomology of a smooth projective variety we prove a theorem asserting that the residual images will be generically large if certain easy to check conditions are satisfied we only consider representations with coefficients in an imaginary quadratic field for primes inert in this field the residual representations when irreducible are unitary we apply our result to an example constructed by van geemen and top obtaining a family of special linear groups and one of special unitary groups as galois groups over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i we also consider the case of cohomological modular forms for a congruence subgroup of sl inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i assuming clozel s conjecture stating that a geometric family of three dimensional galois representations can be attached to them we verify for three examples the conditions guaranteeing generically large images and explicitly bound the finite set of primes with non maximal image we also discuss what intrinsic conditions a modular form should verify to guarantee that the images of the attached galois representations will be generically large

matrix completion problems abstract throughout the last decades several results have been published in the area of the so called matrix completion problems in this paper we survey several results in this field in particular we describe the possible eigenvalues the characteristic polynomial the invariant polynomials or the number of nontrivial invariant polynomials of a square matrix over a field when some of its entries are prescribed and the others vary finally we present our contribution generalizing some of the previous cases to an n n matrix partitioned into k k blocks with entries in a field when some of its blocks are prescribed and the others vary

uniformization of modular elliptic curves via p adic periods abstract the langlands programme predicts that a weight 2 newform f over a number field k with integer hecke eigenvalues generally should have an associated elliptic curve e f over k in 19 we associated building on works of darmon 8 and greenberg 20 a p adic lattice to f under certain hypothesis and implicitly conjectured that is commensurable with the p adic tate lattice of e f in this paper we present this conjecture in detail and discuss how it can be used to compute directly from f a weierstrass equation for the conjectural e f we develop algorithms to this end and implement them in order to carry out extensive systematic computations in which we compute weierstrass equations of hundreds of elliptic curves some with huge heights over dozens of number fields the data we obtain give extensive support for the conjecture and furthermore demonstrate that the conjecture provides an efficient tool to building databases of elliptic curves over number fields

smith theory and hecke operators abstract of the connections between the cohomology of arithmetic groups and representations of galois groups some are known and more are conjectured this paper proves one of these conjectures for certain monomial galois representations let p be a prime we show that the representation of the absolute galois group of q induced from an f p valued ray class character of the cyclotomic field q p is attached to a hecke eigenclass in the mod p cohomology of a torsion free congruence subgroup m of gl p 1 z the method involves constructing an action of the hecke algebra on the fundamental exact sequence of smith theory arising from the action by conjugation of an element of order p in gl p 1 z on m

on a conjecture of ash abstract in this paper we prove a particular case of a conjecture of ash that states the existence of galois representations associated to hecke eigenclasses in cohomology

rigidity of p adic cohomology classes of congruence subgroups of gl n ℤ the main result of ash stevens 97 describes a framework for constructing padic analytic families of p ordinary arithmetic hecke eigenclasses in the cohomology of congruence subgroups of gl n q where the hecke eigenvalues vary p adic analytically as functions of the weight unanswered in that paper was the question of whether or not every p ordinary arithmetic eigenclass can be deformed in a positive dimensional family of arithmetic eigenclasses in this paper we make precise the notions of deformation and rigidity and investigate their properties rigidity corresponds to the non existence of positive dimensional deformations other than those coming from twisting by the powers of the determinant formal definitions are given in 3 when n 3 we will give a necessary and sufficient condition for a given p ordinary arithmetic eigenclass to be p adically rigid and we will use this criterion to give examples of p adically rigid eigenclasses on gl 3 more precisely we define a hecke eigenpacket to be a map from the hecke algebra to a coefficient ring which gives the hecke eigenvalues attached to a hecke eigenclass in the cohomology we then consider p adic analytic deformations of the hecke eigenpacket there are several advantages to this point of view as opposed to deforming the cohomology classes themselves for one thing a surjective map of coefficient modules need not induce a surjective map on the cohomology standard tools of algebra and geometry can be used to study the hecke algebra and the deformations of the eigenpackets moreover it is the hecke eigenvalues that carry the interesting arithmetic information attached to a cohomological eigenclass our main theoretical tool is the cohomology of congruence subgroups gl n z with coefficients in the space of bounded measures on the big cell of gl n see 2 in 4 we define the ring rord as a quotient of the hecke algebra acting on the p ordinary part of this cohomology in theorem 4 2 we prove that rord is universal in the sense that every p ordinary arithmetic hecke eigenpacket factors through rord we show that rord is a complete semilocal noetherian ring and study the geometry of its associated p adic rigid analytic space xord this leads