variations of mass formulas for definite division algebras abstract the aim of this paper is to organize some known mass formulas arising from a definite central division algebra over a global field and to deduce some more new ones

korevaar schoen s directional energy and ambrosio s regular lagrangian flows we develop korevaar schoen s theory of directional energies for metric valued sobolev maps in the case of textsf rcd source spaces to do so we crucially rely on ambrosio s concept of regular lagrangian flow our review of korevaar schoen s spaces brings new even in the smooth category insights on some aspects of the theory in particular concerning the notion of differential of a map along a vector field and about the parallelogram identity for textsf cat 0 targets to achieve these one of the ingredients we use is a new even in the euclidean setting stability result for regular lagrangian flows

triple product p adic l functions for balanced weights we construct p adic triple product l functions that interpolate square roots of central critical l values in the balanced region thus our construction complements that of harris and tilouine there are four central critical regions for the triple product l functions and two opposite settings according to the sign of the functional equation in the first case three of these regions are of interpolation having positive sign they are called the unbalanced regions and one gets three p adic l functions one for each region of interpolation this is the harris tilouine setting in the other setting there is only one region of interpolation called the balanced region we produce the corresponding p adic l function our triple product p adic l function arises as p adic period integrals interpolating normalizations of the local archimedean period integrals the latter encode information about classical representation theoretic branching laws the main step in our construction of p adic period integrals is showing that these branching laws vary in a p adic analytic fashion this relies crucially on the ash stevens theory of highest weight representations over affinoid algebras

lipschitz continuity of cheeger harmonic functions in metric measure spaces abstract we use the heat equation to establish the lipschitz continuity of cheeger harmonic functions in certain metric spaces the metric spaces under consideration are those that are endowed with a doubling measure supporting a 1 2 poincare inequality and in addition supporting a corresponding sobolev poincare type inequality for the modification of the measure obtained via the heat kernel examples are given to illustrate the necessity of our assumptions on these spaces we also provide an example to show that in the general setting the best possible regularity for the cheeger harmonic functions is lipschitz continuity

differential of metric valued sobolev maps abstract we introduce a notion of differential of a sobolev map between metric spaces the differential is given in the framework of tangent and cotangent modules of metric measure spaces developed by the first author we prove that our notion is consistent with kirchheim s metric differential when the source is a euclidean space and with the abstract differential provided by the first author when the target is r we also show compatibility with the concept of co local weak differential introduced by convent and van schaftingen

group schemes and local densities the subject matter of this paper is an old one with a rich history beginning with the work of gauss and eisenstein maturing at the hands of smith and minkowski and culminating in the fundamental results of siegel more precisely if l is a lattice over z for simplicity equipped with an integral quadratic form q the celebrated smith minkowski siegel mass formula expresses the total mass of l q which is a weighted class number of the genus of l q as a product of local factors these local factors are known as the local densities of l q subsequent work of kneser tamagawa and weil resulted in an elegant formulation of the subject in terms of tamagawa measures in particular the local density at a non archimedean place p can be expressed as the integral of a certain volume form ld over autzp l q which is an open compact subgroup of autqp l q the question that remains is whether one can find an explicit formula for the local density through the work of pall for p 6 2 and watson for p 2 such an explicit formula for the local density is in fact known for an arbitrary lattice over zp see p and wa the formula is obviously structured though cs seems to be the first to comment on this unfortunately the known proof as given in p and k does not explain this structure and involves complicated recursions on the other hand conway and sloane cs 13 have given a heuristic explanation of the formula in this paper we will give a simple and conceptual proof of the local density formula for p 6 2 the view point taken here is similar to that of our earlier work ghy and the proof is based on the observation that there exists a smooth affine group scheme g over zp with generic fiber autqp l q which satisfies g zp autzp l q this follows from general results of smoothening blr as we explain in section 3 for the purpose of obtaining an explicit formula it is necessary to have an explicit construction of g the main contribution of this paper is to give such an explicit construction of g in section 5 and to determine its special fiber in section 6 finally by comparing ld and the canonical volume form can of g we obtain the explicit formula for the local density in section 7 the smooth group schemes constructed in this paper should also be of independent interest

construction of tame supercuspidal representations the notion of depth is defined by moy prasad mp2 the notion of a generic character will be defined in 9 when g gln or g is the multiplicative group of a central division algebra of dimension n with n p 1 our generic characters are just the generic characters in my where the definition is due to kutzko moreover in these cases our construction literally specializes to howe s construction as formulated in my and it is known that the construction yields all supercuspidal representations my hm notice that the initial datum in our construction consists of very simple objects linear characters and supercuspidal representations of depth zero the latter are well understood by the work of moy prasad and morris see 3 by the work of howe irreducible supercuspidal representations of gln f with n p 1 are parametrized by certain characters of e as e varies through

harmonic maps from a simplicial complex and geometric rigidity we study harmonic maps from an admissible flat simplicial complex to a non positively curved riemannian manifold our main regularity theorem is that these maps are c 1 at the interfaces of the top dimensional simplices in addition to satisfying a balancing condition if we assume that the domain is a 2 complex then these maps are c 1 as an application we show that the regularity the balancing condition and a bochner formula lead to rigidity and vanishing theorems for harmonic maps furthermore we give an explicit relationship between our techniques and those obtained via combinatorial methods

sobolev spaces laplacian and heat kernel on alexandrov spaces we prove the compactness of the imbedding of the sobolev space w 1 2 0 omega into l 2 omega for any relatively compact open subset omega of an alexandrov space as a corollary the generator induced from the dirichlet energy form has discrete spectrum the generator can be approximated by the laplacian induced from the dc structure on the alexandrov space we also prove the existence of the locally holder continuous heat kernel

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

haar measure and the artin conductor let g be a connected reductive group defined over a local non archimedean field k the group g k is locally compact and unimodular in gr a haar measure g was defined on g k using the theory of bruhat and tits in this note we give another construction of the measure g using the artin conductor of the motive m of g over k the equivalence of the two constructions is deduced from a result of prasad p 1 the root datum and motive of g in this section k is an arbitrary field and g is a connected reductive group over k we let k be an algebraic closure of k ks the separable closure of k in k and gal ks k let t b g be a maximal torus contained in a borel subgroup defined over ks let g b t be the based root datum defined by this choice we recall cf sp pg3 12 that x t t b x t t b 1 1 with x t and x t the character and cocharacter groups of t respectively and and the simple roots and coroots determined by b respectively let w ng t t be the weyl group of the finite group w acts as automorphisms of x t and is generated by the reflections s x x x 1 2 for the galois group acts as automorphisms of ie as automorphisms of the group x t preserving the finite set as follows if then we can find g g ks such that int g t g t g t and int g b g b g b with g well defined up to left multiplication by t ks hence it induces a well defined automorphism x t x t date 14th january 1997 1 2 benedict h gross and wee teck gan preserving hence we get a group homomorphism aut via acts on aut by inner automorphisms similarly if f g g is any automorphism of g over ks it induces an automorphism f of which depends only on the image of f in the quotient group outks g of outer automorphisms the resulting map outks g aut is an isomorphism which respects the respective galois actions on the two groups cf sp pg10 the galois group also acts on w via the formula s s 1 3 and the semi direct product w acts on the rational vector space e x t q 1 4 let r sym e w which is a graded q module let r be the ideal of elements of positive degree in r and define v r r 2 d 1vd 1 5 this is a graded q module and chevalley proved that dim v dim e cf ch steinberg extended the proof to show that e and v are isomorphic modules cf st pg22 we sketch a proof of this result that does not involve the classification of irreducible root systems proposition 1 6 the q modules e and v are isomorphic proof by the criterion in se pg 104 it suffices to show that for all the fixed spaces e and v have the same dimension for any graded module a am we define the poincare series of by p a t tr am t then p a b p a p b steinberg showed that there is an isomorphism of graded modules s e s vd a here a is finite dimensional with basis bw w w and action given by bw b w the degree of bw is the length l w of w with respect to the generators s furnished by this isomorphism yields the following identity of poincare series det 1 t e d 1 det 1 t vd

on the siegel weil formula for quaternionic unitary groups we extend the siegel weil formula to all quaternion dual pairs applications include the classification problem of skew hermitian forms over a quaternion algebra over a number field and a product formula for the weighted average of the representation numbers of a skew hermitian form by another skew hermitian form

conformal immersions of prescribed mean curvature in r 3 abstract we prove the existence of branched conformal immersions f s 2 r 3 with mean curvature h 0 arbitrarily prescribed up to a 3 dimensional affine indeterminacy a similar result is proved for the space forms s 3 h 3 and partial results for surfaces of higher genus

lipschitz continuity of harmonic maps between alexandrov spaces in 1997 jost calc var pde 5 1 19 1997 and lin collection of papers on geometry analysis and mathematical physics world sci publ river edge 1997 independently proved that every energy minimizing harmonic map from an alexandrov space with curvature bounded from below to an alexandrov space with non positive curvature is locally holder continuous lin 1997 proposed an open problem can the holder continuity be improved to lipschitz continuity j jost also asked a similar problem about lipschitz regularity of harmonic maps between singular spaces see page 38 in jost in jost kendall mosco rockner sturm eds new directions in dirichlet forms international press boston 1998 the main theorem of this paper gives a complete resolution to it

cheeger harmonic functions in metric measure spaces revisited let x d be a complete metric measure space with a locally doubling measure that supports a local weak l2 poincare inequality by assuming a heat semigroup type curvature condition we prove that cheeger harmonic functions are lipschitz continuous on x d gradient estimates for cheeger harmonic functions and solutions to a class of non linear poisson type equations are presented

the perron method for p harmonic functions in metric spaces we use the perron method to construct and study solutions of the dirichlet problem for p harmonic functions in proper metric measure spaces endowed with a doubling borel measure supporting a weak 1 q poincare inequality for some 1 q p the upper and lower perron solutions are constructed for functions defined on the boundary of a bounded domain and it is shown that these solutions are p harmonic in the domain it is also shown that newtonian sobolev functions and continuous functions are resolutive i e that their upper and lower perron solutions coincide and that their perron solutions are invariant under perturbations of the function on a set of capacity zero we further study the problem of resolutivity and invariance under perturbations for semicontinuous functions we also characterize removable sets for bounded p super harmonic functions

confluent hypergeometric functions on tube domains our problem specialized to the case of the siegel upper half space h of degree m concerns the fourier expansion of a series s z fl det z a det a a l here z is a variable on h l is a lattice in the space v of all real symmetric matrices of size m and fl ec it can be shown that this is convergent if re f l m and also that if re m 2 and r e f l m 2 it has a fourier expansion of the form

on an exact mass formula of shimura in a series of recent papers g shimura obtained an exact formula for the mass of a maximal lattice in a quadratic or hermitian space over a totally real number field using bruhat tits theory we obtain a quick and more conceptual proof of his formula when the form is totally definite

unrefined minimal k types for p adic groups via their restriction to compact open subgroups was begun by mautner shalika and tanaka for groups of type ai in contrast to real reductive groups where the representation theory of a maximal compact subgroup k is given in terms of slight modifications to cartan s theory of the highest weight the representation theory of a compact parahoric subgroup is quite complicated there is still no comprehensive theory for classifying the irreducible representa tions in the case of