parametric galois extensions abstract given a field k and a finite group h an h parametric extension over k is a finite galois extension of k t of galois group containing h which is regular over k and has all the galois extensions of k of group h among its specializations we are mainly interested in producing non h parametric extensions which relates to classical questions in inverse galois theory like the beckmann black problem and the existence of one parameter generic polynomials we develop a general approach started in a preceding paper and provide new non parametricity criteria and new examples

the absolute galois group acts faithfully on regular dessins and on beauville surfaces in this article we study the action of the absolute galois group gal q q on dessins d enfants and beauville surfaces a foundational result in grothendieck s theory of dessins d enfants is the fact that the absolute galois group gal q q acts faithfully on the set of all dessins however the question of whether this holds true when the action is restricted to the set of the more accessible regular dessins seems to be still an open question in the first part of this paper we give an affirmative answer to it in fact we prove the strongest result that the action is faithful on the set of quasiplatonic or triangle curves of any given hyperbolic type beauville surfaces are an important kind of algebraic surfaces introduced by catanese they are rigid surfaces of general type closely related to dessins d enfants here we prove that for any gal q q different from the identity and the complex conjugation there is a beauville surface s such that s and its galois conjugate s have non isomorphic fundamental groups this in turn easily implies that the action of gal q q on the set of beauville surfaces is faithful these results were conjectured by bauer catanese and grunewald and immediately imply that gal q q acts faithfully on the connected components of the moduli space of surfaces of general type a result due to the above mentioned authors

a conservative fourier pseudo spectral method for the nonlinear schrodinger equation a fourier pseudo spectral method that conserves mass and energy is developed for a two dimensional nonlinear schrodinger equation by establishing the equivalence between the semi norm in the fourier pseudo spectral method and that in the finite difference method we are able to extend the result in ref 56 to prove that the optimal rate of convergence of the new method is in the order of o n r 2 in the discrete l 2 norm without any restrictions on the grid ratio where n is the number of modes used in the spectral method and is the time step size a fast solver is then applied to the discrete nonlinear equation system to speed up the numerical computation for the high order method numerical examples are presented to show the efficiency and accuracy of the new method

limits of galois representations in fundamental groups along maximal degeneration of marked curves i we study gal inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i actions on the fundamental groups arising in deformation of maximally degenerate marked curves and give explicit descriptions of them in terms of the principal parameters of the grothendieck teichmuller group inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i as a special application we present galois theoretic connections between the moduli spaces m 0 4 m 0 5 m 1 1 and m 1 2 which lead to a newtype equation satisfied by the image of gal inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i

efficient and accurate numerical methods for the klein gordon schrodinger equations in this paper we present efficient unconditionally stable and accurate numerical methods for approximations of the klein gordon schrodinger kgs equations with without damping terms the key features of our methods are based on i the application of a time splitting spectral discretization for a schrodinger type equation in kgs ii the utilization of fourier pseudospectral discretization for spatial derivatives in the klein gordon equation in kgs iii the adoption of solving the ordinary differential equations odes in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying crank nicolson leap frog for linear nonlinear terms for time derivatives the numerical methods are either explicit or implicit but can be solved explicitly unconditionally stable and of spectral accuracy in space and second order accuracy in time moreover they are time reversible and time transverse invariant when there is no damping terms in kgs conserve or keep the same decay rate of the wave energy as that in kgs without or with a linear damping term keep the same dynamics of the mean value of the meson field and give exact results for the plane wave solution extensive numerical tests are presented to confirm the above properties of our numerical methods for kgs finally the methods are applied to study solitary wave collisions in one dimension 1d as well as dynamics of a 2d problem in kgs

shimura curves of genus at most two we enumerate all shimura curves x d 0 r of genus at most two there are exactly 858 such curves up to equivalence

numerical simulation of interaction between schrodinger field and klein gordon field by multisymplectic method abstract we propose a multisymplectic scheme to solve the coupled klein gordon schrodinger system the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law and can simulate the original waves well in a long time this scheme also has discrete quasi norm conservation law numerical experiments demonstrate the consistency between the theoretical analysis and the numerical results

large normal extension of hilbertian fields the goal of this note is to consider a certain natural family of closed normal subgroups of g q and to prove that each group in this family is free more generally consider a countable separably hilbertian field k denote the absolute galois group of k by g k then for almost all g k e the field ks is pac and e free fj2 thms 16 13 and 16 18 here ks is the separable closure of k and ks is the fixed field of in ks being pac means that every nonvoid absolutely irreducible variety defined over ks has a ks rational point we say that ks is e free if g ks i e the closed subgroup 1 e of g k generated by 1 e is free on e generators denote the largest galois extension of k which is contained in ks by ks it is the intersection of all k conjugates of ks and also the fixed field of the smallest closed normal subgroup of g k which contains 1 e if char k 0 then for almost all g k e the field ks is pac fj2 thm 16 47 lemma 1 2 below generalizes this result to arbitrary characteristic if we knew that ks is separably hilbertian then a theorem of fried volklein and pop would imply that ks is free that is g ks is isomorphic to the free profinite group fw on countably many generators unfortunately it is not clear how to prove the hilbertianity of almost all ks directly so we use instead a forerunner to the above mentioned theorem of fried volklein pop and a recent theorem of neumann neu to prove directly that g ks f for almost all g k e a theorem of roquette then implies that ks is also separably hilbertian let k be the algebraic closure of k denote the maximal purely inseparable extension of ks by k then for almost all g k e the field k

the irreducibility of certain pure cycle hurwitz spaces we study pure cycle hurwitz spaces parametrizing covers of the projective line having only one ramified point over each branch point we start with the case of genus 0 covers using a combination of limit linear series theory and group theory to show that these spaces are always irreducible in the case of four branch points we also compute the associated hurwitz numbers finally we give a conditional result in the higher genus case requiring at least 3g simply branched points these results have equivalent formulations in group theory and in this setting complement results of conway fried parker v olklein

on fields of moduli of curves the field of moduli k of a curve x a priori defined over the separable closure k5 of k need not be a field of definition this paper shows that the obstruction is essentially the same as the obstruction to k being a field of definition of the cover x x aut x using previous results of debes douai we then obtain a cohomological measure of the obstruction this yields concrete criteria for the field of moduli to be a field of definition an interesting application is the following local global principle if a curve x together with all of its automorphisms is defined over p for all primes p then it is defined over

a uniformly accurate ua multiscale time integrator fourier pseudospectral method for the klein gordon schrodinger equations in the nonrelativistic limit regime a multiscale time integrator fourier pseudospectral mti fp method is proposed and analyzed for solving the klein gordon schrodinger kgs equations in the nonrelativistic limit regime with a dimensionless parameter 0 varepsilon le 1 0 1 which is inversely proportional to the speed of light in fact the solution of the kgs equations propagates waves with wavelength at o varepsilon 2 o 2 and o 1 in time and space respectively when 0 varepsilon ll 1 0 1 which brings significantly numerical burden in practical computation the mti fp method is designed by adapting a multiscale decomposition by frequency of the solution at each time step and applying the fourier pseudospectral discretization and exponential wave integrators for spatial and temporal derivatives respectively we rigorously establish two independent error bounds for the mti fp at o tau 2 varepsilon 2 h m 0 o 2 2 hm0 and o varepsilon 2 h m 0 o 2 hm0 for varepsilon in 0 1 0 1 with tau time step size h mesh size and m 0 ge 4 m0 4 an integer depending on the regularity of the solution which imply that the mti fp converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau o for varepsilon in 0 1 0 1 in addition the mti fp method converges optimally with quadratic convergence rate at o tau 2 o 2 in the regime when 0 tau lesssim varepsilon 2 0 2 and the error is at o varepsilon 2 o 2 independent of tau in the regime when 0 varepsilon lesssim tau 1 2 0 1 2 thus the meshing strategy requirement or varepsilon scalability of the mti fp is tau o 1 o 1 and h o 1 h o 1 for 0 varepsilon ll 1 0 1 which is significantly better than that of classical methods numerical results demonstrate that our error bounds are optimal and sharp finally the mti fp method is applied to study numerically convergence rates of the kgs equations to its limiting models in the nonrelativistic limit regime

fuchsian triangle groups and grothendieck dessins variations on a theme of belyi according to a theorem of belyi a smooth projective algebraic curve is defined over a number field if and only if there exists a non constant element of its function field ramified only over 0 1 and the existence of such a belyi function is equivalent to that of a representation of the curve as a possibly compactified quotient space of the poincare upper half plane by a subgroup of finite index in a fuchsian triangle group on the other hand fuchsian triangle groups arise in many contexts such as in the theory of hypergeometric functions and certain triangular billiard problems which would appear at first sight to have no relation to the galois problems that motivated the above discovery of belyi in this note we review several results related to belyi s theorem and we develop certain aspects giving examples for preliminary accounts see the preprint wo1 the conference proceedings article bauitz and the comptes rendus note cowo2

hurwitz spaces of galois coverings of p1 whose galois groups are weyl groups abstract we prove the irreducibility of the hurwitz spaces which parametrize equivalence classes of galois coverings of p 1 whose galois group is an arbitrary weyl group and the local monodromies are reflections this generalizes a classical theorem due to luroth clebsch and hurwitz

linearly implicit conservative schemes for long term numerical simulation of klein gordon schrodinger equations abstract in this paper we propose three linearly implicit fourier pseudospectral algorithms for solving the klein gordon schrodinger equations in quantum physics all methods are linearly implicit in the sense that they do not need iterative technique to solve nonlinear equations at each time step the three algorithms are proved to admit the charge and energy conservation laws exactly numerical results reveal that all the proposed methods can provide accurate soliton solutions and simulate the collision of solitons well the numerical results also verify the theoretical analysis that the proposed methods are all charge preserving and energy preserving algorithms

analysis and comparison of numerical methods for the klein gordon equation in the nonrelativistic limit regime we analyze rigourously error estimates and compare numerically temporal spatial resolution of various numerical methods for solving the klein gordon kg equation in the nonrelativistic limit regime involving a small parameter 0 varepsilon ll 1 which is inversely proportional to the speed of light in this regime the solution is highly oscillating in time i e there are propagating waves with wavelength of o varepsilon 2 and o 1 in time and space respectively we begin with four frequently used finite difference time domain fdtd methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step as well as the small parameter varepsilon based on the error bounds in order to compute correct solutions when 0 varepsilon ll1 the four fdtd methods share the same varepsilon scalability tau o varepsilon 3 then we propose new numerical methods by using either fourier pseudospectral or finite difference approximation for spatial derivatives combined with the gautschi type exponential integrator for temporal derivatives to discretize the kg equation the new methods are unconditionally stable and their varepsilon scalability is improved to o 1 and tau o varepsilon 2 for linear and nonlinear kg equations respectively when 0 varepsilon ll1 numerical results are reported to support our error estimates

the embedding problem over a hilbertian pac field we show that the absolute galois group of a countable hilbertian p seudo a lgebraically c losed field of characteristic 0 is a free profinite group of countably infinite rank theorem a as a consequence g q q is the extension of groups with a fairly simple structure e g 2 sn by a countably free group in addition we characterize those pac fields over which every finite group is a galois group as those with the rg hilbertian property

faithful actions of the absolute galois group on connected components of moduli spaces we use a canonical procedure associating to an algebraic number a first a hyperelliptic curve c a and then a triangle curve d a g a obtained through the normal closure of an associated belyi function in this way we show that the absolute galois group mathrm gal bar mathbb q mathbb q acts faithfully on the set of isomorphism classes of marked triangle curves and on the set of connected components of marked moduli spaces of surfaces isogenous to a higher product these are the free quotients of a product c 1 times c 2 of curves of respective genera g 1 g 2 ge 2 by the action of a finite group g we show then using again the surfaces isogenous to a product first that it acts faithfully on the set of connected components of the moduli space of surfaces of general type amending an incorrect proof in a previous arxiv version of the paper and then as a consequence we obtain our main result for each element sigma in mathrm gal bar mathbb q mathbb q not in the conjugacy class of complex conjugation there exists a surface of general type x such that x and the galois conjugate surface x sigma have nonisomorphic fundamental groups using polynomials with only two critical values we can moreover exhibit infinitely many explicit examples of such a situation

comparisons between sine gordon and perturbed nonlinear schrodinger equations for modeling light bullets beyond critical collapse abstract the sine gordon sg equation and perturbed nonlinear schrodinger nls equations are studied numerically for modeling the propagation of two space dimensional 2d localized pulses the so called light bullets in nonlinear dispersive optical media we begin with the 2 1 sg equation obtained as an asymptotic reduction in the two level dissipationless maxwell bloch system followed by the review on the perturbed nls equation in 2d for sg pulse envelopes which is globally well posed and has all the relevant higher order terms to regularize the collapse of standard critical cubic focusing nls the perturbed nls is approximated by truncating the nonlinearity into finite higher order terms undergoing focusing defocusing cycles efficient semi implicit sine pseudospectral discretizations for sg and perturbed nls are proposed with rigorous error estimates numerical comparison results between light bullet solutions of sg and perturbed nls as well as critical nls are reported which validate that the solution of the perturbed nls as well as its finite term truncations are in qualitative and quantitative agreement with the solution of sg for the light bullets propagation even after the critical collapse of cubic focusing nls in contrast standard critical nls is in qualitative agreement with sg only before its collapse as a benefit of such observations pulse propagations are studied via solving the perturbed nls truncated by reasonably many nonlinear terms which is a much cheaper task than solving sg equation directly

a gautschi type method for oscillatory second order differential equations we study a numerical method for second order differential equations in which high frequency oscillations are generated by a linear part for example semilinear wave equations are of this type the numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly we prove that the method admits second order error bounds which are independent of the product of the step size with the frequencies our analysis also provides new insight into the m ollified impulse method of garcia archilla sanz serna and skeel we include results of numerical experiments with the sine gordon equation

conservation of energy momentum and actions in numerical discretizations of non linear wave equations for classes of symplectic and symmetric time stepping methods trigonometric integrators and the stormer verlet or leapfrog method applied to spectral semi discretizations of semilinear wave equations in a weakly non linear setting it is shown that energy momentum and all harmonic actions are approximately preserved over long times for the case of interest where the cfl number is not a small parameter such results are outside the reach of standard backward error analysis here they are instead obtained via a modulated fourier expansion in time

orders in quaternion algebras on etudie certains ordres dans des algebres de quaternions et on developpe une formule de trace calculable pour les matrices de brandt associees a ces ordres speciaux

algebraic curves uniformized by congruence subgroups of triangle groups we construct certain subgroups of hyperbolic triangle groups which we call congruence subgroups these groups include the classical congruence subgroups of sl2 z hecke triangle groups and 19 families of arithmetic triangle groups associated to shimura curves we determine the eld of moduli of the curves associated to these groups and thereby realize the groups psl2 fq and pgl2 fq regularly as galois groups

optimal point wise error estimate of a compact difference scheme for the klein gordon schrodinger equation abstract in this paper we propose a compact finite difference scheme for computing the klein gordon schrodinger equation kgse with homogeneous dirichlet boundary conditions the proposed scheme not only conserves the total mass and energy in the discrete level but also is linearized in practical computation except for the standard energy method a new technique is introduced to obtain the optimal convergent rate without any restriction on the grid ratios at the order of o h 4 2 in the l norm with time step and mesh size h finally numerical results are reported to test the theoretical results

galois groups of unramified covers of projective curves in characteristicp abstract we show several results concerning the finite groups that occur as galois groups of unramified covers of projective curves in characteristic p in particular we prove that every finite group with g generators occurs over some curve of genus g this implies for example that every finite simple group occurs in genus 2 by similar methods we obtain several other families of groups which occur in genus 2 in addition we show that if a group g occurs over some curve of genus g then it must occur over almost all curves of genus g or greater the results are obtained using formal patching

computation of a cover of shimura curves using a hurwitz space using a hurwitz space computation we determine the canonical model of the cover of shimura curves x 0 2 to x 1 associated to the quaternion algebra over the cubic field of discriminant 13 2 which is ramified at exactly two real places and unramified at finite places then we list the coordinates of some rational cm points on x 1

a dissipative finite difference fourier pseudo spectral method for the klein gordon schrodinger equations with damping mechanism abstract we develop a semi linearized decoupled time stepping method for solving the klein gordon schrodinger equations with damping mechanism the finite difference approximation in time and fourier pseudo spectral discretization in space provide an elegant platform to deal with the physical properties of the original model we prove that the proposed numerical algorithm preserves the discrete invariant or dissipative properties of system exactly depending on the choices of the damping parameter values we establish the maximum norm error estimates by virtue of the norm equivalence between finite difference method and fourier pseudo spectral method the discrete versions of projection and interpolation estimations and mathematical induction argument ample numerical results are presented to show the effectiveness of our numerical method and to confirm our theoretical analysis

on error bounds for the gautschi type exponential integrator applied to oscillatory second order differential equations this paper studies a numerical method for second order oscillatory differential equations in which high frequency oscillations are generated by a linear time and or solution dependent part for constant linear part it is known that the method allows second order error bounds independent of the product of the step size with the frequencies and is therefore a long time step method most real world problems are not of that kind and it is important to study more general equations the analysis in this paper shows that one obtains second order error bounds even in the case of a time and or solution dependent linear part if the matrix is evaluated at averaged positions

algorithmic enumeration of ideal classes for quaternion orders we provide algorithms to count and enumerate representatives of the right ideal classes of an eichler order in a quaternion algebra defined over a number field we analyze the run time of these algorithms and consider several related problems including the computation of two sided ideal classes isomorphism classes of orders connecting ideals for orders and ideal principalization we conclude by giving the complete list of definite eichler orders with class number at most 2

homological stability for hurwitz spaces and the cohen lenstra conjecture over function fields we prove a homological stabilization theorem for hurwitz spaces moduli spaces of branched covers of the complex projective line this has the following arithmetic consequence let 2 be prime and a a nite abelian group then there exists q q a such that for q greater than q a positive fraction of quadratic extensions of fq t have the part of their class group isomorphic to a

study and computation of a hurwitz space and totally real psl2 f8 extensions of q abstract developing on works by fried volklein matzat malle debes wewers we give a method for computing a hurwitz space and illustrate it on some example of number theoretic interest we study and compute a family of degree 9 covers of p c 1 with monodromy group psl 2 f 8 and having four branch points we deduce explicit regular psl 2 f 8 extensions of the rational function field q with totally real fibers this gives rise to totally real polynomials over q with galois group psl 2 f 8

the deligne simpson problem a survey abstract the deligne simpson problem dsp respectively the weak dsp is formulated like this give necessary and sufficient conditions for the choice of the conjugacy classes c j gl n c or c j gl n c so that there exist irreducible respectively with trivial centralizer p 1 tuples of matrices m j c j or a j c j satisfying the equality m 1 m p 1 i or a 1 a p 1 0 the matrices m j and a j are interpreted as monodromy operators of regular linear systems and as matrices residua of fuchsian ones on riemann s sphere the present paper offers a survey of the results known up to now concerning the dsp

fake congruence modular curves and subgroups of the modular group abstract we construct a special class of noncongruence modular subgroups and curves analogous in some ways to the usual congruence ones the subgroups are obtained via the burau representation and the associated quotient curves have a natural moduli space interpretation in fact they are reduced hurwitz spaces corresponding to covers with 4 branch points and monodromy group equal to semi direct products of a cyclic and an abelian group furthermore they form a modular tower in the sense of fried we study representations on the cohomology of these fake congruence modular curves and also calculate the genera of certain quotient curves

arithmetic 1 e curves and belyĭ maps using the theory of bely maps we calculate the algebraic curves associated to the fuchsian groups of signature that are commensurable with a triangle group along with the picard fuchs differential equations on these curves which are related to pullbacks of hypergeometric differential equations we focus particularly on the groups that are arithmetic

an exponential wave integrator sine pseudospectral method for the klein gordon zakharov system an exponential wave integrator sine pseudospectral method is presented and analyzed for discretizing the klein gordon zakharov kgz system with two dimensionless parameters 0 varepsilon le 1 and 0 gamma le 1 which are inversely proportional to the plasma frequency and the speed of sound respectively the main idea in the numerical method is to apply the sine pseudospectral discretization for spatial derivatives followed by using an exponential wave integrator for temporal derivatives in phase space the method is explicit symmetric in time and it is of spectral accuracy in space and second order accuracy in time for any fixed varepsilon varepsilon 0 and gamma gamma 0 in the mathcal o 1 plasma frequency and speed of sound regime i e varepsilon mathcal o 1 and gamma mathcal o 1 we establish rigorously error estimates for the numerical method in the energy space h 1 times l 2 we also study numerically the resolution of the method in the simultaneous high plasma fr

harbater mumford subvarieties of moduli spaces of covers we define harbater mumford subvarieties which are special kinds of closed subvarieties of hurwitz moduli spaces obtained by fixing some of the branch points we show that for many finite groups finding geometrically irreducible hm subvarieties defined over open image in new window is always possible this provides information on the arithmetic of hurwitz spaces and applies in particular to the regular inverse galois problem with almost all fixed branch points profinite versions of our results can also be stated providing new tools to study the geometry of modular towers and the regular inverse galois problem for profinite groups

split embedding problems over function fields over complete domains let r be a krull domain complete with respect to a nonzero ideal let k be the quotient field of r we prove that every finite split embedding problem is solvable over every function field in one variable over k if dim r 1 then every finite split embedding problem over k is solvable

canonical models of arithmetic 1 e curves the fuchsian groups of signature 1 e are the simplest class of fuchsian groups for which the calculation of the corresponding quotient of the upper half plane presents a challenge this paper considers the finite list of arithmetic 1 e groups we define canonical models for the associated quotients by relating these to genus 1 shimura curves these models are then calculated by applying results on the mathfrak p adic uniformization of shimura curves and hilbert modular forms

convergence of a conservative difference scheme for a class of klein gordon schrodinger equations in one space dimension a conservative difference scheme is presented for the initial boundary value problem of a class of klein gordon schrodinger equations the scheme can be implicit or implicit explicit depending on the choice of a parameter on the basis of the priori estimates and an inequality about norms convergence of the difference solution is proved with order o h 2 t 2 in the energy norm

on modular representations of bar q q arising from modular forms where g is the galois group gai i q and f is a finite field of characteristic i 3 suppose that p is modular of level n i e that it arises from a weight 2 newform of level dividing n and trivial nebentypus character then p is an odd representation the matrix p c where c is a complex conjugation in g has eigenvalues 1 1 since 1 and 1 are distinct in f p is absolutely irreducible and has a model over every subfield of f containing the set trace p we assume that f has been chosen so that it is generated by this set assume that p is a prime which exactly divides n we write p i n and restrict p to a decomposition group of g for the prime p view the restriction as a representation p p of gal t p qp we say that p is finite at p 34 or 35 w if there is a finite flat f vector space scheme h over zp for which the action of gal t p qp on the f vector space h 1 p gives pp i f p 9 is finite at p if and only if pp is unramified it is clear that p is finite at p whenever p is already modular of level n p in w of 34 cf 35 serre conjectured the converse i e that if p is finite then p is modular of level n p soon thereafter mazur 1 20 proved this conjecture in the case

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

connected components of affine primitive permutation groups abstract for a finite group g the hurwitz space h r g i n g is the space of genus g covers of the riemann sphere with r branch points and the monodromy group g in this paper we give a complete list of primitive genus one systems of affine type that is we assume that g is a primitive group of affine type under this assumption we determine the braid orbits on the suitable nielsen classes which is equivalent to finding connected components in h r 1 i n g furthermore we give a new algorithm for computing large braid orbits on nielsen classes this algorithm utilizes a correspondence between the components of h r 1 i n g and h r 1 i n m where m is the point stabilizer in g

equations of hyperelliptic shimura curves we describe a method for computing explicit models of hyperelliptic shimura curves attached to an indenite quaternion algebra over q and atkin lehner quotients of them it exploits cerednik drinfeld s non archimedean uniformisation of shimura curves a formula of gross and zagier for the endomorphism ring of heegner points over artinian rings and the connection between ribet s bimodules and the specialization of heegner points as introduced in 22 as an application we provide a list of equations of shimura curves and quotients of them obtained by our method that had been conjectured by kurihara

potential theory of infinite dimensional levy processes abstract we study the potential theory of a large class of infinite dimensional levy processes including brownian motion on abstract wiener spaces the key result is the construction of compact lyapunov functions i e excessive functions with compact level sets then many techniques from classical potential theory carry over to this infinite dimensional setting thus a number of potential theoretic properties and principles can be proved answering long standing open problems even for the brownian motion on abstract wiener space as e g formulated by r carmona in 1980 in particular we prove the analog of the known result that the cameron martin space is polar in the levy case and apply the technique of controlled convergence to solve the dirichlet problem with general not necessarily continuous boundary data

the inverse galois problem and rational points on moduli spaces we reduce the regular version of the inverse galois problem for any finite group g to finding one rational point on an infinite sequence of algebraic varieties as a consequence any finite group g is the galois group of an extension l p x with l regular over any pac field p of characteristic zero a special case of this implies that g is a galois group over fp x for almost all primes p

billiards and teichmuller curves on hilbert modular surfaces this paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of riemann surfaces of genus two these teichmuller curves lie on hilbert modular surfaces parameterizing abelian varieties with real multiplication explicit examples constructed from l shaped polygons give billiard tables with optimal dynamical properties