on the ramanujan conjecture over number fields we extend to an arbitrary number eld the best known bounds towards ramanujan for the group gln n 2 3 4 in particular we present a technique which overcomes the analytic obstacles posed by the presence of an innite group of units

modeling of thermally assisted magnetodynamics thermomagnetic recording uses local heating of ferromagnetic media to locally decrease coercivity and change saturation magnetization of the material and alleviate magnetization reversal a modified landau lifshitz equation is proposed as a phenomenological model to allow for changes in magnetization magnitude and is studied both analytically and numerically

lower bounds for l functions at the edge of the critical strip we prove a coarse lower bound for l functions of langlands shahidi type of generic cuspidal automorphic representations on the line re s 1 we follow the path suggested by sarnak using eisenstein series and the maass selberg relations the bounds are weaker than what the method of de la vallee poussin gives for the standard l functions of gln but are applicable to more general automorphic l functions our theorem answers in a strong form a conjecture posed by gelbart and shahidi j amer math soc 14 2001 and sharpens and considerably simplifies the proof of the main result of that paper

geometrical integration of landau lifshitz gilbert equation based on the mid point rule landau lifshitz gilbert llg equation is the fundamental equation to describe magnetization vector field dynamics in microscale and nanoscale magnetic systems this equation is highly nonlinear in nature and for this reason it is generally solved by using numerical techniques in this paper the mid point rule time stepping technique is applied to the numerical time integration of llg equation and the relevant properties of the numerical scheme are discussed the mid point rule is an unconditionally stable and second order accurate scheme which preserves the fundamental geometrical properties of llg dynamics first it exactly preserves the llg property of conserving the magnetization magnitude at each spatial location second for constant in time applied fields it preserves the llg lyapunov structure namely the fact that the free energy is a decreasing function of time in addition in the case of zero damping the mid point rule preserves the conservation of the system free energy the above preservation properties are unconditionally valid i e they are fulfilled for any value of the time step finally the llg hamiltonian structure in the case of zero damping is preserved up to the third order terms with respect to the time step the main difficulty related to this scheme is the necessity of solving a large system of globally coupled nonlinear equations this problem has been circumvented by using special and reasonably fast quasi newton iterative technique the proposed numerical scheme is then tested on the standard micromagnetic problem no 4 in the numerical computations the spatial discretization is obtained by finite difference technique and the magnetostatic field is computed through the fast fourier transform method

cuspidality of symmetric powers with applications the purpose of this paper is to prove that the symmetric fourth power of a cusp form on rm gl 2 whose existence was proved earlier by the first author is cuspidal unless the corresponding automorphic representation is of dihedral tetrahedral or octahedral type as a consequence we prove a number of results toward the ramanujan petersson and sato tate conjectures in particular we establish the bound q sp 1 9 sb v for unramified hecke eigenvalues of cusp forms on rm gl 2 over an arbitrary number field this is the best bound available at present

convergence of an implicit explicit midpoint scheme for computational micromagnetics abstract based on lowest order finite elements in space we consider the numerical integration of the landau lifschitz gilbert equation llg the dynamics of llg is driven by the so called effective field which usually consists of the exchange field the external field and lower order contributions such as the stray field the latter requires the solution of an additional partial differential equation in full space following bartels and prohl 2006 we employ the implicit midpoint rule to treat the exchange field however in order to treat the lower order terms effectively we combine the midpoint rule with an explicit adams bashforth scheme the resulting integrator is formally of second order in time and we prove unconditional convergence towards a weak solution of llg numerical experiments underpin the theoretical findings

fourier coefficients of cusp forms of half integral weight let mathfrak f be a cusp form of half integral weight whose fourier coefficients mathfrak a mathfrak f n are all real we study the sign change problem of mathfrak a mathfrak f n when n runs over some specific sets of integers lower bounds of the best possible order of magnitude are established for the number of those coefficients that have the same signs these give an improvement on some recent results of bruinier and kohnen modular forms on schiermonnikoong cambridge university press cambridge 57 66 2008 and kohnen int j number theory 6 1255 1259 2010

une preuve simple des conjectures de langlands pour gl n sur un corps p adique let f be a finite extension of p for each integer n 1 we construct a bijection from the set f 0 n of isomorphism classes of irreducible degree n representations of the absolute weil group of f onto the set f 0 n of isomorphism classes of smooth irreducible supercuspidal representations of gl n f those bijections preserve epsilon factors for pairs and hence we obtain a proof of the langlands conjectures for gl n over f which is more direct than harris and taylor s our approach is global and analogous to the derivation of local class field theory from global class field theory we start with a result of kottwitz and clozel on the good reduction of some shimura varieties and we use a trick of harris who constructs non galois automorphic induction in certain cases

on the ramanujan conjecture and finiteness of poles for certain l functions in this paper we prove a number of general results about automorphic l functions which appear in the constant terms of eisenstein series for an arbitrary quasi split connected reductive algebraic group over a number field these l functions were first introduced by langlands for chevalley groups 20 and through them he was led to make some of his deep conjectures 21 one significance of these l functions is that all the automorphic l functions studied so far are among them there are three general results first we obtain a uniform line of absolute convergence for all of these l functions theorem 5 1 a uniform estimate for the hecke eigenvalues of generic cusp forms on many absolutely simple quasi split connected reductive algebraic groups over number fields corollary 5 4 and an improvement on the best available estimate for the fourier coefficients of maass wave forms corollary 5 5 also follow next we establish a meromorphic continuation and functional equation for each of these l functions theorem 6 1 finally in theorem 6 2 we prove finiteness of poles on the whole complex plane for an important class of these l functions corollaries 6 6 through 6 10 more precisely let g be a quasi split connected reductive algebraic group over a number field f set g g a f where a f is the ring of adeles of f fix a borel subgroup b of g over f and let u be its unipotent radical let p be a maximal f parabolic subgroup of g with p d b in the context of the problems studied here nothing new will be obtained if one drops the maximality condition on p write p mn a levi decomposition n c u and let b u p m and n be the corresponding groups of adelic points for every place v of f let gv g fj similarly we have bv us pv me and nv let x xv be a generic character of u f u cf section 3 then each xv is generic let r ov 7tv be a cusp form on m we shall say 7t is

finding meaning in error terms four decades ago mikio sato and john tate predicted the shape of probability distributions to which certain error terms in number theory conform their prediction known as the sato tate conjecture has been verified for an important class of cases thanks to the recent work of laurent clozel michael harris and richard taylor 3 and of michael harris nicholas shepherd barron and richard taylor 16 combined with richard taylor s most recent 50 which establishes this advance in our understanding

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

the role of the ramanujan conjecture in analytic number theory we discuss progress towards the ramanujan conjecture for the group gln and its relation to various other topics in analytic number theory

constraint preserving implicit finite element discretization of harmonic map flow into spheres discretization of the harmonic map flow into spheres often uses a penalization or projection strategy where the first suffers from the proper choice of an additional parameter and the latter from the lack of a discrete energy law and restrictive mesh constraints we propose an implicit scheme that preserves the sphere constraint at every node enjoys a discrete energy law and unconditionally converges to weak solutions of the harmonic map heat flow

real zeros and size of rankin selberg l functions in this paper some asymptotic formula is proved for the harmonic mollified second moment of a family of rankin selberg l functions the main contribution is a substancial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average consequences new subconvexity bound exponential decay of the analytic rank non vanishing result around the real axis

the fourth power moment of automorphic l functions for gl 2 over a short interval in this paper we will prove bounds for the fourth power moment in the t aspect over a short interval of automorphic l functions l s g for gl 2 on the central critical line re s 1 2 here g is a fixed holomorphic or maass hecke eigenform for the modular group sl 2 z or in certain cases for the hecke congruence subgroup 0 n with n 1 the short interval is from a large k to k k 103 135 the proof is based on an estimate in the proof of subconvexity bounds for rankin selberg l function for maass forms by jianya liu and yangbo ye 2002 and yuk kam lau jianya liu and yangbo ye 2004 which in turn relies on the kuznetsov formula 1981 and bounds for shifted convolution sums of fourier coefficients of a cusp form proved by sarnak 2001 and by lau liu and ye 2004

on symmetric power l invariants of iwahori level hilbert modular forms we compute the arithmetic cal l invariants of greenberg benois of twists of symmetric powers of p adic galois representations attached to iwahori level hilbert modular forms under some technical conditions our method uses the automorphy of symmetric powers and the study of analytic galois representations on p adic families of automorphic forms over symplectic and unitary groups combining these families with some explicit plethysm in the representation theory of rm gl 2 we construct global galois cohomology classes with coefficients in the symmetric powers and provide formulae for the cal l invariants in terms of logarithmic derivatives of hecke eigenvalues

numerical methods for the landau lifshitz equation we discuss numerical schemes for various forms of the landau lifshitz equation a new simple projection method is introduced and is shown to be unconditionally stable the advantages over other schemes are also demonstrated numerically

an example of non normal quintic automorphic induction and modularity of symmetric powers of cusp forms of icosahedral type due to the work of arthur and clozel ac the automorphic induction for cyclic galois extensions of prime degree is well understood in great generality it is not the case for non normal extensions even for monomial representations i e the ones induced from grossencharacters the only examples we have at the moment are first non normal cubic automorphic induction due to jacquet piatetski shapiro and shalika jpss they obtained the automorphic induction as a consequence of the converse theorem on gl3 the second example is that of harris h he constructed automorphic induction for special class of algebraic hecke characters of suitable non normal extensions with solvable galois closure in this note we give an example of automorphic induction for nonnormal quintic extension whose galois closure is not solvable theorem 6 3 in fact the galois group is a5 the alternating group on 5 letters the key observation due to ramakrishnan is that symmetric fourth of the two dimensional icosahedral representation is equivalent to a suitable twist by a character of the five dimensional monomial representation of a5 our result follows immediately by combining results of k buzzard m dickinson n shepherd barron and r taylor bdst or r taylor s result ta2 on modularity of certain icosahedral representations and our result on the functoriality of symmetric fourth ki we also prove the modularity of all symmetric powers of cuspidal representations of icosahedral type theorem 6 4

performance enhancement of parameter estimators via dynamic regressor extension and mixing a new procedure to design parameter estimators with enhanced performance is proposed in the technical note for classical linear regression forms it yields a new parameter estimator whose convergence is established without the usual requirement of regressor persistency of excitation the technique is also applied to nonlinear regressions with partially monotonic parameter dependence giving rise again to estimators with enhanced performance simulation results illustrate the advantages of the proposed procedure in both scenarios

numerical scheme for augmented landau lifshitz equation in heat assisted recording the motivation of heat assisted recording is to improve the thermal stability of recorded data bits by increasing the strength of the uniaxial anisotropy during the recording process the medium is heated by a laser reducing the coercivity and allowing the head to write data bits we present a micromagnetic model based on the augmented landau lifshitz equation taking into account a phenomenological power law describing the dependence of the saturation magnetization on the temperature a full discrete numerical scheme is presented and the convergence of approximate solutions to a weak one is shown

some local global non vanishing results for theta lifts from orthogonal groups we firstly improve a theorem of b roberts which characterizes non vanishing of a global theta lift from o x to sp n in terms of non vanishing of local theta lifts in particular we will remove all the archimedean conditions imposed upon his theorem sec ondly following roberts we will apply our theorem to theta lifting of low rank similitude groups namely we characterize the non vanishing condition of a global theta lift from go 4 to gsp 2 in our improved setting also we consider non vanishing conditions of a global theta lift from go 4 to gsp 1 and explicitly compute the lift when it exists

the artin conjecture for some s_5 extensions we establish some new cases of artin s conjecture our results apply to galois representations over mathbf q with image s 5 satisfying certain local hypotheses the most important of which is that complex conjugation is conjugate to 12 34 in fact we prove the stronger claim conjectured by langlands that these representations are automorphic for the irreducible representations of dimensions 4 and 6 our result follows from known 2 dimensional cases of artin s conjecture proved by sasaki as well as the functorial properties of the asai transfer proved by ramakrishnan for the irreducible representations of dimension 5 we encounter the problem of descending an automorphic form from a quadratic extension compatibly with the galois representation this problem is partly solved by working instead with a four dimensional representation of some central extension of s 5 our modularity results in this case are contingent on the non vanishing of a certain dedekind zeta function on the real line in the critical strip a result of booker show that one can in principle explicitly verify this non vanishing and with booker s help we give an example verifying artin s conjecture for representations coming from the galois closure of the quintic field k of smallest discriminant 1609

a density theorem on automorphic l functions and some applications we establish a density theorem of automorphic l functions and give some applications on the extreme values of these l functions at s 1 and the distribution of the hecke eigenvalue of holomorphic cusp forms contents

modularity of the rankin selberg l series and multiplicity one for sl 2 let f g be primitive cusp forms holomorphic or otherwise on the upper half plane h of levels n m respectively with unitarily normalized l functions l s f equation and l s g equation when p does not divide n resp m the inverse roots p p resp p p are nonzero with sum ap resp bp for every p prime to nm set lp s f g 1 p pp s 1 p pp s 1 p pp s 1 p pp s 1 let l s f g denote the incomplete euler product of lp s f g over all p not dividing nm this is closely related to the convolution l series sum over n 1 a sub n b sub n n s whose miraculous properties were first studied by rankin and selberg

standard zero free regions for rankin selberg l functions via sieve theory we give a simple proof of a standard zero free region in the t aspect for the rankin selberg l function l s pi times widetilde pi for any unitary cuspidal automorphic representation pi of mathrm gl n mathbb a f that is tempered at every nonarchimedean place outside a set of dirichlet density zero

the explicit sato tate conjecture and densities pertaining to lehmer type questions let f z n 1 af n q n s k 0 n be a newform with squarefree level n that does not have complex multiplication for a prime p define p 0 to be the angle for which af p 2p k 1 2 cos p let i 0 be a closed subinterval and let d st 2 sin 2 d be the sato tate measure of i assuming that the symmetric power l functions of f satisfy certain analytic properties all of which follow from langlands functoriality and the generalized riemann hypothesis we prove that if x is sufficiently large then p x p i st i x 2 dt log t x log nkx log x with an implied constant of 3 33 by letting i be a short interval centered at 2 and counting the primes using a smooth cutoff we compute a lower bound for the density of positive integers n for which af n 6 0 in particular if is the ramanujan tau function then under the aforementioned hypotheses we prove that lim x n x n 6 0 x 1 1 54 10 13 we also discuss the connection between the density of positive integers n for which af n 6 0 and the number of representations of n by certain positive definite integer valued quadratic forms

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points ii we prove a general subconvex bound in the level aspect for rankin selberg l functions associated with two primitive holomorphic or maass cusp forms over q we use this bound to establish the equidistribution of incomplete galois orbits of heegner points on shimura curves associated with indefinite quaternion algebras over q

when kloosterman sums meet hecke eigenvalues by elaborating a two dimensional selberg sieve with asymptotics and equidistributions of kloosterman sums from ell adic cohomology as well as a bombieri vinogradov type mean value theorem for kloosterman sums in arithmetic progressions it is proved that for any given primitive hecke maass cusp form of trivial nebentypus the eigenvalue of the n th hecke operator does not coincide with the kloosterman sum mathrm kl 1 n for infinitely many squarefree n with at most 100 prime factors this provides a partial negative answer to a problem of katz on modular structures of kloosterman sums

explicit bounds for the number of p core partitions in this article we derive explicit bounds on ct n the number of t core partitions of n in the case when t p is prime we express the generating function f z as the sum

a note on maass forms of icosahedral type using ideas of ramakrishnan we consider the icosahedral analogue of the theorems of sarnak and brumley on hecke maass newforms with fourier coefficients in a quadratic order although we are unable to conclude the existence of an associated galois representation in this case we show that one can deduce some implications of such an association including weak automorphy of all symmetric powers and the value distribution of fourier coefficients predicted by the chebotarev density theorem

the effect of convolving families of l functions on the underlying group symmetries let fn and gm be families of primitive automorphic l functions for gln aq and glm aq respectively such that as n m the statistical behavior 1 level density of the low lying zeros of l functions in fn resp gm agrees with that of the eigenvalues near 1 of matrices in g1 resp g2 as the size of the matrices tend to infinity where each gi is one of the classical compact groups unitary u symplectic sp or orthogonal o so even so odd assuming that the convolved families of l functions fn gm are automorphic we study their 1 level density we also study convolved families of the form f gm for a fixed f under natural assumptions on the families which hold in many cases we can associate to each family l of l functions a symmetry constant cl equal to 0 resp 1 or 1 if the corresponding low lying zero statistics agree with those of the unitary resp symplectic or orthogonal group our main result is that cf g cf cg the symmetry type of the convolved family is the product of the symmetry types of the two families a similar statement holds for the convolved families f gm we provide examples built from dirichlet l functions and holomorphic modular forms and their symmetric powers an interesting special case is to convolve two families of elliptic curves with positive rank in this case the symmetry group of the convolution is independent of the ranks in accordance with the general principle of multiplicativity of the symmetry constants but the ranks persist before taking the limit n m as lower order terms

convergence of an implicit finite element method for the landau lifshitz gilbert equation the landau lifshitz gilbert equation describes the dynamics of ferromagnetism where strong nonlinearity and nonconvexity are hard to tackle so far existing explicit schemes to approximate weak solutions suffer from severe time step restrictions in this paper we propose an implicit fully discrete scheme and verify unconditional convergence

real zeros and size of rankin selberg l functions in the level aspect in this paper some asymptotic formulas are proved for the harmonic mollified second moment of a family of rankin selberg l functions one of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average a first consequence is a new subconvexity bound for rankin selberg l functions in the level aspect moreover infinitely many rankin selberg l functions having at most eight non trivial real zeros are produced and some new non trivial estimates for the analytic rank of the family studied are obtained

a convergent implicit finite element discretization of the maxwell landau lifshitz gilbert equation we propose an implicit fully discrete scheme for the numerical solution of the maxwell landau lifshitz gilbert equation which is based on linear finite elements and satisfies a discrete sphere constraint as well as a discrete energy law as numerical parameters tend to zero solutions weakly accumulate at weak solutions of the maxwell landau lifshitz gilbert equation a practical linearization of the nonlinear scheme is proposed and shown to converge for certain scalings of numerical parameters computational studies are presented to indicate finite time blow up behavior and to study combined electromagnetic phenomena in ferromagnets for benchmark problems

density results for automorphic forms on hilbert modular groups ii we obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region for sl2 over a totally real number field f with a discrete subgroup of hecke type 0 i for a non zero ideal i in the ring of integers of f the weights are products of fourier coefficients this implies in particular the existence of infinitely many cuspidal automorphic representations with multi eigenvalues in various regions growing to infinity for instance in the quadratic case the regions include floating boxes floating balls sectors slanted strips see 1 2 4 1 2 13 and products of prescribed small intervals for all but one of the infinite places of f the main tool in the derivation is a sum formula of kuznetsov type sum formula for sl2 over a totally real number field theorem 2 1