the integral monodromy of hyperelliptic and trielliptic curves we compute the mathbb z ell and mathbb z ell monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves in particular we provide a proof that the mathbb z ell monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group rm s p2g mathbb z ell we prove that the mathbb z ell monodromy of the moduli space of trielliptic curves with signature r s is the special unitary group rm su r s mathbb z ell otimes mathbb z zeta 3

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

strong approximation for zariski dense subgroups of semi simple algebraic groups 1 1 theorem let k be an algebraically closed field of characteristic different from 2 and 3 and g an almost simple connected and simply connected algebraic group defined over k let f be a finitely generated zariski dense subgroup of g k and a the subring of k generated by the traces tr ad y y e p then there exist b e a a subgroup f of finite index in f and a structure gab of a group scheme over ab on g such that f c ga ab and f is dense in gala b a

a conjecture of beauville and catanese revisited a theorem of green lazarsfeld and simpson formerly a conjecture of beauville and catanese states that certain naturally defined subvarieties of the picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points their proof uses analytic methods we refine and give a completely new proof of their result our proof combines galois theoretic methods and algebraic geometry in positive characteristic when the variety has a model over a function field and its picard variety has no isotrivial factors we show how to replace the galois theoretic results we need by results from model theory mathematical logic furthermore we prove partial analogs of the conjecture of beauville and catanese in positive characteristic

the existence of an abelian variety over q isogenous to no jacobian we prove the existence of an abelian variety a of dimension g over q that is not isogenous to any jacobian subject to the necessary condition g 3 recently c chai and f oort gave such a proof assuming the andr eoort conjecture we modify their proof by constructing a special sequence of cm points for which we can avoid any unproven hypotheses we make use of various techniques from the recent work of klingler yafaev et al

h sub infinity optimal control with state feedback a h sub infinity optimal control problem in which the measured outputs are the states of the plant is considered the main result shows that the infimum of the norm of the closed loop transfer function using linear static state feedback equals the infimum of the norm of the closed loop transfer function over all stabilizing dynamic even nonlinear time varying state feedback controllers

new exact travelling wave solutions of kawahara type equations abstract the auxiliary equation method is used to find analytic solutions for the kawahara and modified kawahara equations it is well known that different types of exact solutions of the given auxiliary equation produce new types of exact travelling wave solutions to nonlinear equations in this paper new exact solutions of the auxiliary equation are presented using these solutions many new exact travelling wave solutions for the kawahara type equations are obtained

abelian varieties isogenous to a jacobian we dene a notion of weyl cm points in the moduli space ag 1 of g dimensional principally polarized abelian varieties and show that the andr e oort conjecture or the grh implies the following statement for any closed subvariety x ag 1 over q a there exists a weyl special point b 2 ag 1 q a such that b is not isogenous to the abelian variety a underlying any point a 2 x the title refers to the case when g 4 and x is the torelli locus in this case tsimerman has proved the statement unconditionally the notion of weyl special points is generalized to the context of shimura varieties and we prove a corresponding conditional statement with the ambient spaceag 1 replaced by a general shimura variety

profinite completions of burnside type quotients of surface groups using quantum representations of mapping class groups we prove that profinite completions of burnside type surface group quotients are not virtually prosolvable in general further we construct infinitely many finite simple characteristic quotients of surface groups

type a images of galois representations and maximality given a compatible subsystem rho ell ell of n dimensional ell adic galois representations arising from etale cohomology of any complete non singular variety over a number field k we define gamma ell rho ell text gal bar k k and let mathbf g ell denote the zariski closure of gamma ell in mathrm gl n if mathbf g ell is of type a in the sense that all simple composition factors are of type a in the cartan killing classification then gamma ell is in a suitable sense maximal in mathbf g ell for all ell gg 0 as a corollary if rho ell is semisimple and ell is sufficiently large then mathbf g ell is unramified

walks on groups counting reducible matrices polynomials and surface and free group automorphisms we prove sharp limit theorems on random walks on graphs with values in finite groups we then apply these re sults together with some elementary algebraic geometry number theory and representation theory to finite quotients of lattices in semisimple lie groups specifically sl n z and sp 2n z to show that a random element in one of these lattices has irre ducible characteristic polynomials overz the term random can be defined in at least two ways in terms of height and also in terms of word length in terms of a generating set we show the result using both definitions we use the above results to show that a random in terms of word length element of the mapping class group of a surface is pseudo anosov and that a a random free group automorphism is irreducible with irreducible powers or strongly irreducible 1

a family of calabi yau varieties and potential automorphy we prove potential modularity theorems for l adic representations of any dimension from these results we deduce the sato tate conjecture for all elliptic curves with nonintegral j invariant defined over a totally real field

maximality of galois actions for compatible systems in the sense of serre 19 of course the image of is not generally zariski dense in gln q so the maximality condition must be formulated relative to the zariski closure g of gal k k one might hope that the image of is a maximal compact subgroup of g q but this is too optimistic the center causes problems and we do not even know a priori that g is reductive to formulate a maximality conjecture which avoids such problems we introduce maps g g g

correction of the fi result in h sub infinity control and parameterization of h sub infinity state feedback controllers the authors correct the parameterization of the h sub infinity controller of the full information fi problem derived by j c doyle et al 1989 then they parameterize the h sub m0 state feedback controller and explain how dynamical free parameters implied in it are related to constant feedback gains different from the central solution f sub infinity

divisibility and laws in finite simple groups we provide new bounds for the divisibility function of the free group mathbf f 2 and construct short laws for the symmetric groups mathrm sym n the construction is random and relies on the classification of the finite simple groups we also give bounds on the length of laws for finite simple groups of lie type

parametrization of state feedback h controllers the state feedback h control problem is considered in this paper it is shown that under mild conditions this problem can be solved simply by using the conjugation approach proposed by kimura and kawatani with some extended treatment to the case that t3 s is tall and strictly proper a partial parametrization of state feedback controllers achieving the prescribed norm bound is obtained this is an eminent difference from previous work on the state feedback h control problem where only a static feedback control law the central solution of our result is provided both four block and one block problems are solved in this paper the results of this paper embrace those of doyle et al and petersen

synthesis of reduced order h controllers based on conjugation a new state space characterization of solutions of a class of model matching problem the so called model matching problem of the first kind by limebeer has recently been derived by kimura based o

subvarieties of shimura varieties 7 proof of the main result 19 7 3 the case where is bounded 20 7 4 the case where is not bounded 22 both authors were partially supported by the european research training network contract hprn ct 200000120 arithmetic algebraic geometry adresses irmar campus de beaulieu 35042 rennes cedex france edix maths univ rennes1 fr ayafaev maths univ rennes1 fr

a uniform spectral gap for congruence covers of a hyperbolic manifold let g be rm so n 1 or rm su n 1 and let gamma subset g denote an arithmetic lattice the hyperbolic manifold gamma backslash cal h comes with a natural family of covers coming from the congruence subgroups of gamma in many applications it is useful to have a bound for the spectral gap that is uniform for this family when gamma is itself a congruence lattice there are very good bounds coming from known results towards the ramanujan conjectures in this paper we establish an effective bound that is uniform for congruence subgroups of a non congruence lattice

on verbal subgroups of adelic groups abstract let g be a closed subgroup of p gl n z p if g is finitely generated as a profinite group and has no infinite virtually abelian quotients then every verbal subgroup in g is closed

infiniteness of double coset collections in algebraic groups abstract let g be a linear algebraic group defined over an algebraically closed field the double coset question addressed in this paper is the following given closed subgroups x and p is the double coset collection x g p finite or infinite we limit ourselves to the case where x is maximal rank and reductive and p parabolic this paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements this result is then applied to finish the classification of those x which are spherical subgroups finally excluding a case in f 4 we show that if x g p is finite then x is spherical or the levi factor of p is spherical this places great restrictions on x and p for x g p to be finite the primary method is to descend to calculations at the finite group level and then to use elementary character theory

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

new exact solutions for the kawahara equation using exp function method in this paper we applied the exp function method to solve the kawahara equation this method can be used to obtain new exact solutions and periodic solutions with parameters are obtained it is shown that the exp function method with the help of symbolic computation provides a very effective and powerful mathematical tools for discrete nonlinear evolution equations in mathematical physics

the affine sieve we establish the main saturation conjecture connected with executing a brun sieve in the setting of an orbit of a group of affine linear transformations this is carried out under the condition that the zariski closure of the group is levi semisimple it is likely that this condition is also necessary for such saturation to hold 2013 american mathematical society

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

exact solutions of coupled nonlinear klein gordon equation abstract in this paper we employ the general integral method for traveling wave solutions of coupled nonlinear klein gordon equations based on the idea of the exact jacobi elliptic function a simple and efficient method is proposed for obtaining exact solutions of nonlinear evolution equations the solutions obtained include solitons periodic solutions and jacobi elliptic function solutions

exceptional splitting of reductions of abelian surfaces heuristics based on the sato tate conjecture and the lang trotter philosophy suggest that an abelian surface defined over a number field has infinitely many places of split reduction we prove this result for abelian surfaces with real multiplication as in previous work by charles and elkies this shows that a density 0 set of primes pertaining to the reduction of abelian varieties is infinite the proof relies on the arakelov intersection theory on hilbert modular surfaces

on a weak variant of the geometric torsion conjecture a consequence of the geometric torsion conjecture for abelian varieties over function fields is the following let k be an algebraically closed field of characteristic 0 for any integers d g 0 there exists an integer n n k d g 1 such that for any function field l k with transcendence degree 1 and genus g and any d dimensional abelian variety a l containing no nontrivial k isotrivial abelian subvariety a l tors a n in this paper we deal with a weak variant of this statement where a l runs only over abelian varieties obtained from a fixed d dimensional abelian variety by base change more precisely let k k be a function field with transcendence degree 1 and a k an abelian variety containing no nontrivial k isotrivial abelian subvariety then we show that if k has genus 1 or if a k has semistable reduction over all but possibly one place then for any integer g 0 there exists an integer n n a g 1 such that for any finite extension l k with genus g a l tors a n previous works of the authors show that this holds without any restriction on k for the l primary torsion with l a fixed prime so it is enough to prove that there exists an integer n n a g 1 such that for any finite extension l k with genus g the prime divisors of a l tors are all n

arithmetic groups have rational representation growth let be an arithmetic lattice in a semisimple algebraic group over a number field we show that if has the congruence subgroup property then the number of n dimensional irreducible representations of grows like n where is a rational number

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

geometric monodromy semisimplicity and maximality let x be a connected scheme smooth and separated over an alge braically closed field k of characteristic p 0 let f y x be a smooth proper morphism and x a geometric point on x we prove that the tensor invariants of bounded length d of 1 x x acting on the etale cohomology groups h yx fl are the reduction modulo l of those of 1 x x acting on h yx l for l greater than a constant depending only on f y x d we apply this result to show that the geometric variant with fl coefficients of the grothendieck serre semisimplicity conjecture namely that 1 x x acts semisimply on h yx fl for l 0 is equivalent to the condition that the image of 1 x x acting on h yx ql is almost maximal in a precise sense what we call almost hyperspecial with respect to the group of ql points of its zariski closure ultimately we prove the geometric variant with fl coefficients of the grothendieck serre semisimplicity conjecture

growth in finite simple groups of lie type we prove that if l is a finite simple group of lie type and a a set of generators of l then either a grows i e vert a 3 vert vert a vert 1 varepsilon where varepsilon depends only on the lie rank of l or a 3 l this implies that for simple groups of lie type of bounded rank a well known conjecture of babai holds i e the diameter of any cayley graph is polylogarithmic we also obtain new families of expanders a generalization of our proof yields the following let a be a finite subset of sl n mathbb f mathbb f an arbitrary field satisfying big vert a 3 big vert le mathcal k vert a vert then a can be covered by mathcal k m i e polynomially many cosets of a virtually soluble subgroup of sl n mathbb f which is normalized by a where m depends on n see more at http www ams org journals jams 0000 000 00 s0894 0347 2014 00821 3 home html sthash lp65mzge dpuf

congruence properties of zariski dense subgroups i this paper deals with the following general situation we are given an algebraic group g defined over a number field k and a subgroup f of the group g k of k rational points of g then what should it mean for f to be a large subgroup we might require f to be a lattice in g to be arithmetic to contain many elements of a specific kind to have a large closure in some natural topology et cetera there are many theorems proving implications between conditions of size of this kind we shall consider the case of g a k simple group usually q simple the zariski topology of g for which the closed sets are those defined by the vanishing of polynomial functions is very coarse on the other hand the various valuations v of k each give rise to a strong topology on g k if v is non archimedean and kv is the completion of k with respect to v then g kv is a non archimedean lie group and if is the ring of integers of kv the open subgroup g 5 j of g kv of integral points is defined for all but finitely many v the zariski closure f of a subgroup f of g k is a c algebraic subgroup of g while the strong closure fy is a lie subgroup of g kv for elementary reasons the dimension of tv is at most the dimension of f our results are in the opposite direction we show under suitable conditions on g that if f g then for almost all v we have that tv contains g ov to illustrate this with a specific example if ml 5 mk are matrices in sl2 q generating a group f which is zariskidense in sl2 then for all sufficiently large prime numbers p we have f p sl2 zp further this result is effective in the sense that we could in principle check the hypothesis for given mx mk and derive a bound on p beyond which the conclusion holds it is perhaps worth remarking that this special case which is the simplest application of our results may be proved with much less effort than the general theorem

affine linear sieve expanders and sum product let mathcal o be an orbit in n of a finitely generated subgroup of gl n whose zariski closure zcl is suitably large e g isomorphic to sl2 we develop a brun combinatorial sieve for estimating the number of points on mathcal o at which a fixed integral polynomial is prime or has few prime factors and discuss applications to classical problems including pythagorean triangles and integral apollonian packings a fundamental role is played by the expansion property of the congruence graphs that we associate with mathcal o this expansion property is established when zcl sl2 using crucially sum product theorem in q for q square free

the galois group of random elements of linear groups let bbb f be a finitely generated field of characteristic zero and gamma leq rm gl n bbb f a finitely generated subgroup for gamma in gamma let rm gal bbb f gamma bbb f be the galois group of the splitting field of the characteristic polynomial of gamma over bbb f we show that the structure of rm gal bbb f gamma bbb f has a typical behavior depending on bbb f and on the geometry of the zariski closure of gamma but not on gamma

on sigma lossless transfer functions and related questions abstract this paper is concerned with a systematic approach to the properties of lossless rational transfer functions in the discrete as well as in the continuous time case as a result a unifying framework is revealed where several known results fit naturally special attention is given to the embedding problem of the lyapunov equation in view of its direct application to generalized levinson algorithms

on groups of polynomial subgroup growth let be a finitely generated group andan the number of its subgroups of indexn we prove that assuming is residually nilpotent e g linear thenan grows polynomially if and only if is solvable of finite rank this answers a question of segal the proof uses a new characterization ofp adic analytic groups the theory of algebraic groups and the prime number theorem the method can be applied also to groups of polynomial word growth

on isospectral locally symmetric spaces and a theorem of von neumann 1 introduction call two lattices f1 and f2 in a locally compact group g isospectral if the representations of g on l2 g fi and l2 g f2 are unitarily equivalent here we show that isospectral lattices are quite common in semisimple real algebraic groups g we construct them using a variation of a technique of sunada su as a consequence we obtain new examples of locally symmetric spaces with isospectral laplacians in particular we get such examples with higher rank previously the only higher rank isospectral manifolds known were locally reducible these were vigneras s examples of quotients of products of hyperbolic spaces i v on the other hand brooks found irreducible higher rank locally symmetric spaces with isospectral potentials i br i however the underlying manifolds in his examples happened to be isometric this was due to mostow rigidity since the fundamental groups were isomorphic in fact this work was largely motivated by the question of whether the rigidity properties of lattices in higher rank force an isospectral rigidity

representation growth and rational singularities of the moduli space of local systems let g be a semisimple algebraic group defined over mathbb q p and let gamma be a compact open subgroup of g mathbb q p we relate the asymptotic representation theory of gamma and the singularities of the moduli space of g local systems on a smooth projective curve proving new theorems about both 1 we prove that there is a constant c independent of g such that the number of n dimensional representations of gamma grows slower than n c confirming a conjecture of larsen and lubotzky in fact we can take c 3 cdot text dim e 8 1 745 we also prove the same bounds for groups over local fields of large enough characteristic 2 we prove that the coarse moduli space of g local systems on a smooth projective curve of genus at least lceil c 2 rceil 1 374 has rational singularities for the proof we study the analytic properties of push forwards of smooth measures under algebraic maps more precisely we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities

sieve methods in group theory i powers in linear groups a general sieve method for groups is formulated it enables one to measure subsets of a finitely generated group as an application we show that if is a finitely generated non virtually solvable linear group of characteristic zero then the set of proper powers in is exponentially small this is a far reaching strengthening of the main result of hkls

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

new exact solutions of space and time fractional modified kawahara equation abstract one of the important nonlinear evolution equations in mathematical physics is the modified kawahara equation in this work by utilizing an analytic method based on the jacobi elliptic functions a group of exact solutions of the fractional modified kawahara equation had been found the method can be applied to all of the time space and space time fractional equations four problems had also been given to demonstrate the application of the method and some solutions had been illustrated by the graphics

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

the probability that the number of points on the jacobian of a genus 2 curve is prime in 2000 galbraith and mckee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points we show how their heuristics can be generalized to jacobians of curves of higher genus we then elaborate this in genus g 2 and study various related issues such as the probability of cyclicity and the probability of primality of the number of points on the curve itself finally we discuss the asymptotic behavior for g

disturbance attenuation and h optimization a design method based on the algebraic riccati equation this note presents a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system the problem under consideration involves attenuating the disturbances to a prespecified level the construction of the state feedback control law requires the solution of a certain algebraic riccati equation by applying the procedure with successively smaller values of the prespecifled disturbance level the result also provides an approach to the h infty optimization problem for the state feedback case

the andre oort conjecture in this paper we prove assuming the generalized riemann hypothesis the andre oort conjecture on the zariski closure of sets of special points in a shimura variety in the case of sets of special points satisfying an additional assumption we prove the conjecture without assuming the grh