one fixed point actions on spheres ii the object of this paper is to treat the following old question of montgomery and samelson m s and some of its consequences which groups act smoothly on a closed homotopy sphere with exactly one fixed point and what are the isotropy representations of g which occur on the tangent space at the fixed point the first and only previously existing example of such an action was given by e stein for the group sl 2 z st a related question was solved by oliver o which groups act smoothly on a disk without fixed points a group which acts on a sphere with one fixed point acts on a disk without fixed points in p the author announced

p converse to a theorem of gross zagier kolyvagin and rubin let e be a cm elliptic curve over the rationals and p 3 a good ordinary prime for e we show that begin aligned mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 implies mathrm ord s 1 l s e mathbb q 1 end aligned for the p infty selmer group mathrm sel p infty e mathbb q and the complex l function l s e mathbb q in particular the tate shafarevich group hbox x e mathbb q is finite whenever mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 we also prove an analogous p converse for cm abelian varieties arising from weight two elliptic cm modular forms with trivial central character for non cm elliptic curves over the rationals first general results towards such a p converse theorem are independently due to skinner a converse to a theorem of gross zagier and kolyvagin arxiv 1405 7294 2014 and zhang camb j math 2 2 191 253 2014

the local langlands conjecture for gsp 4 we prove the local langlands conjecture for gsp4 f where f is a nonarchimedean local eld of characteristic zero

on the restricted mean survival time curve in survival analysis for a study with an event time as the endpoint its survival function contains all the information regarding the temporal stochastic profile of this outcome variable the survival probability at a specific time point say t however does not transparently capture the temporal profile of this endpoint up to t an alternative is to use the restricted mean survival time rmst at time t to summarize the profile the rmst is the mean survival time of all subjects in the study population followed up to t and is simply the area under the survival curve up to t the advantages of using such a quantification over the survival rate have been discussed in the setting of a fixed time analysis in this article we generalize this approach by considering a curve based on the rmst over time as an alternative summary to the survival function inference for instance based on simultaneous confidence bands for a single rmst curve and also the difference between two rmst curves are proposed the latter is informative for evaluating two groups under an equivalence or noninferiority setting and quantifies the difference of two groups in a time scale the proposal is illustrated with the data from two clinical trials one from oncology and the other from cardiology

averages of fourier coefficients of siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables let d be a a negative discriminant and let t vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant d we prove an asymptotic formula for d rightarrow infty for the average over t of the number of representations of t by an integral positive definite quaternary quadratic form and obtain bounds for averages of fourier coefficients of linear combinations of siegel theta series we also find an asymptotic estimate from below on the number of binary forms of fixed discriminant d which are represented by a given quaternary form in particular we can show that for growing d a positive proportion of the binary quadratic forms of discriminant d is represented by the given quaternary quadratic form

on permutation tests for comparing restricted mean survival time with small sample from randomized trials between group comparison based on the restricted mean survival time rmst is getting attention as an alternative to the conventional logrank hazard ratio approach for time to event outcomes in randomized controlled trials rcts the validity of the commonly used nonparametric inference procedure for rmst has been well supported by large sample theories however we sometimes encounter cases with a small sample size in practice where we cannot rely on the large sample properties generally the permutation approach can be useful to handle these situations in rcts however a numerical issue arises when implementing permutation tests for difference or ratio of rmst from two groups in this article we discuss the numerical issue and consider six permutation methods for comparing survival time distributions between two groups using rmst in rcts setting we conducted extensive numerical studies and assessed type i error rates of these methods our numerical studies demonstrated that the inflation of the type i error rate of the asymptotic methods is not negligible when sample size is small and that all of the six permutation methods are workable solutions although some permutation methods became a little conservative no remarkable inflation of the type i error rates were observed we recommend using permutation tests instead of the asymptotic tests especially when the sample size is less than 50 per arm

relating invariant linear form and local epsilon factors via global methods we use the recent proof of jacquet s conjecture due to harris and kudla hk and the burger sarnak principle see bs to give a proof of the relationship between the existence of trilinear forms on representations of gl 2 k u for a non archimedean local field k u and local epsilon factors which was earlier proved only in the odd residue characteristic by this author in p1 theorem 1 4 the method used is very flexible and gives a global proof of a theorem of saito and tunnell about characters of gl 2 using a theorem of waldspurger w theorem 2 about period integrals for gl 2 and also an extension of the theorem of saito and tunnell by this author in p3 theorem 1 2 which was earlier proved only in odd residue characteristic in the appendix to this article h saito gives a local proof of lemma 4 which plays an important role in the article

bessel models for gsp 4 methods of theta correspondence are used to analyze local and global bessel models for gsp 4 proving a conjecture of gross and prasad which describes these models in terms of local epsilon factors in the local case and the non vanishing of central critical l value in the global case

modular invariant of quantum tori the quantum modular invariant j qt of 2r is defined as a discontin uous pgl2 z invariant multi valued map using the distance to the nearest integer function k k for 2 q it is shown that j qt 1 and for quadratic irrationalities pari gp experiments suggest that j qt is a finite set in the case of the golden mean we produce explicit formulas involving weighted versions of the rogers ramanujan functions for the experimental supremum and infimum we then define a universal modular invariant j a mod c as a continuous and single valued map of ultrasolenoids such that 1 the classical modular invariant is a quotient of the re striction of j to a subsolenoid mod cl a mod fibering over the classical moduli space of elliptic curves and 2 the quantum modular invariant is a quotient of the restric tion of j to a subsolenoid mod qt a mod fibering over the moduli space of elliptic curves equipped with a kronecker foliation

local e factors and characters of gl 2 introduction the representation theory of gl 2 k for k a nonarchimedean local field is somewhat more complicated than the corresponding theory for gl 2 r in particular the formulas for the characters of discrete series representation of gl 2 r have a concise closed form while the published character formulas for supercuspidal representations of gl 2 k occupy several pages 10 14 the purpose of this paper is to give a concise formula for the character of certain infinite dimensional representations of gl 2 k the representations considered are admissible in the sense that the stabilizer of a vector in the representation space is open in gl 2 k and the subspace stabilized by an open compact subgroup of gl 2 k is finite dimensional irreducible admissible representations ir of gl 2 k satisfy schur s lemma so that the center k of gl 2 k acts by multiplication by a quasicharacter there exists a locally constant character function ch defined on a dense open subgroup of gl 2 k which determines the isomorphism class of ir 7 section 7 jacquet and langlands 7 section 2 introduced factors e 10 x attached to the twist of ir by quasicharacters x of k which describe the action of r in a specific model for each separable quadratic extension l of k there exists an irreducible admissible representation bcl k 1r of gl 2 l the base change of ir to l 9 section 2 this representation of gl 2 l enters into the expression for the character the main result of this paper is the following formula recall that conjugacy classes of nonsplit cartan subgroups are parameterized by quadratic separable field extensions l of k via embeddings of l in gl 2 k

the iwasawa conjecture for totally real fields let f be a totally real number field let p be a prime number and for any integer n let fun denote the group of nth roots of unity let 41 be a p adic valued artin character for f and let f be the extension of f attached to 4 i e so that 4 is the character of a faithful representation of gal f f we will assume that f is also totally real for a number field k let k denote the cyclotomic zp extension of k following greenberg we say that 4 is of type s if f n fc f and of type w if 4 is one dimensional with f p c fcc deligne and ribet in dr following kubota and leopoldt for the case f q have proved the existence of a p adic l function associated to a one dimensional artin character 4 with f totally real this function lp s 4 is continuous for s e zp 1 and even at s 1 if 4 is not trivial and satisfies the following interpolation property

multiplicity theorems modulo p for mathbf gl _ 2 mathbf q _p let f be a non archimedean local field pi an admissible irreducible mathbf gl 2 f representation with complex coefficients for a quadratic extension l f and an l times character chi a classical result of tunnell and saito establish a precise connection between the dimension of the hom space hbox hom l times big pi vert l times chi big and the normalized local factor of the pair pi chi the study of analogous hom spaces for complex valued representations has recently been generalized to mathbf gl n in aizenbud et al ann math 172 1407 1434 2010 and their connections with local factors have been established by work of moeglin and waldspurger la conjecture locale de gross prasad pour les groupes speciaux orthogonaux le cas general preprint available at http www math jussieu fr moeglin gp pdf 2009 in this paper we approach the analogous problem in the context of the p modular langlands correspondence for mathbf gl 2 mathbf q p we describe the restriction to cartan subgroups of an irreducible p modular representation pi of mathbf gl 2 mathbf q p and deduce generalized multiplicity results on the dimension of the ext spaces mathrm ext fancyscript o l times i big pi vert fancyscript o l times chi big where fancyscript o l times is the ring of integers of a quadratic extension of mathbf q p and chi a smooth character of fancyscript o l times

overconvergent generalised eigenforms of weight one and class fields of real quadratic fields abstract this article examines the fourier expansions of certain non classical p adic modular forms of weight one the eponymous generalised eigenforms of the title so called because they lie in a generalised eigenspace for the hecke operators when this generalised eigenspace contains the theta series attached to a character of a real quadratic field k in which the prime p splits the coefficients of the attendant generalised eigenform are expressed as p adic logarithms of algebraic numbers belonging to an idoneous ring class field of k this suggests an approach to explicit class field theory for real quadratic fields which is simpler than the one based on stark s conjecture or its p adic variants and is perhaps closer in spirit to the classical theory of singular moduli

twists of hilbert modular forms the theory of newforms for hilbert modular forms is summarized including a statement of a strong multiplicity one theorem and a characterization of newforms as eigenfunctions for a certain involution whose dirichlet series has a prescribed euler product the general question of twisting hilbert modular newforms by arbitrary fecke characters is considered and the exact level of a character twist of a hilbert modular form is determined conditions under which the twist of a newform is a newform are given applications include a strengthening in the elliptic modular case of a theorem of atkin and li s regarding the characterization of imprimitive newforms as well as its generalization to the hilbert modular case and a decomposition theorem for certain spaces of newforms as the direct sum of twists of spaces of newforms of lower level

non cuspidal hida theory for siegel modular forms and trivial zeros of p adic l functions we study the derivative of the standard p adic l function associated with a p ordinary siegel modular form for p a parabolic subgroup of mathrm gl n when it presents a semi stable trivial zero this implies part of greenberg s conjecture on the order and leading coefficient of p adic l functions at such trivial zero we use the method of greenberg stevens for the construction of the improvedp adic l function we develop hida theory for non cuspidal siegel modular forms

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

factorization of p adic rankin l series we prove that the p adic l series of the tensor square of a p ordinary modular form factors as the product of the symmetric square p adic l series of the form and a kubota leopoldt p adic l series this establishes a generalization of a conjecture of citro greenberg s exceptional zero conjecture for the adjoint follows as a corollary of our theorem our method of proof follows that of gross who proved a factorization result for the katz p adic l series associated to the restriction of a dirichlet character whereas gross s method is based on comparing circular units with elliptic units our method is based on comparing these same circular units with a new family of units called beilinson flach units that we construct the beilinson flach units are constructed using bloch s intersection theory of higher chow groups applied to products of modular curves we relate these units to special values of classical and p adic l functions using work of beilinson as generalized by lei loeffler zerbes in the archimedean case and bertolini darmon rotger as generalized by kings loeffler zerbes in the p adic case central to our method are two compatibility theorems regarding bloch s intersection pairing and the classical and p adic beilinson regulators defined on higher chow groups

cube sum problem and an explicit gross zagier formula a nonzero rational number is called a it cube sum if it is of the form a 3 b 3 with a b in bbb q times in this paper we prove that for any odd integer k geq 1 there exist infinitely many cube free odd integers n with exactly k distinct prime factors such that 2n is a cube sum resp not a cube sum we present also a general construction of heegner points and obtain an explicit gross zagier formula which is used to prove the birch and swinnerton dyer conjecture for certain elliptic curves related to the cube sum problem

induction theorems of surgery obstruction groups let g be a finite group it is well known that a mackey functor h m h is a module over the burnside ring functor h h where h ranges over the set of all subgroups of g for a fixed homomorphism w g 1 1 the wall group functor h l h n z h w h is not a mackey functor if w is nontrivial in this paper we show that the wall group functor is a module over the burnside ring functor as well as over the grothendieck witt ring functor h gw 0 z h in fact we prove a more general result that the functor assigning the equivariant surgery obstruction group on manifolds with middle dimensional singular sets to each subgroup of g is a module over the burnside ring functor as well as over the special grothendieck witt ring functor as an application we obtain a computable property of the functor described with an element in the burnside ring

theta dichotomy for unitary groups some recent work of gross and prasad 14 suggests that the root numbers attached to certain symplectic representations of the weil deligne group of a local field f control certain branching rules for representations of orthogonal groups over f on a global level this local phenomenon should have implications for the structure of the value at the center of symmetry for certain l functions of arithmetic interest 12 this conjectural picture is based to some extent on the now classic work of tunnell 47 and waldspurger 49 as well as on the case of the triple product l function 13 34 17 in all of these examples the local root number detects the existence of a certain type of invariant linear functional it is possible to set up analogous conjectures for the isometry groups of hermitian or quaternion hermitian forms it turns out that root numbers also play a role in the local and global theta correspondence roughly speaking certain local root numbers should control the occurrence of representations in the local theta correspondence between groups of the same size e g for dual pairs of the form sp n o 2n 1 and u n u n as will be seen the theta correspondence for such pairs is connected with a certain induced representation in s at the point s 0 on the unitary axis by contrast the correspondence for the pairs sp n o 2n and sp n o 2n 2 discussed by prasad 35 is connected to the behavior of a similar induced representation at the points 12 in which case no epsilon factor arises in this paper we consider the local theta correspondence for unitary groups in the non archimedean case let f be a non archimedean local field of characteristic not equal to 2 and let e be a quadratic extension of f for further notation see the notation section at the end of this introduction let v and w be e vector spaces of dimensions m and n equipped respectively with a hermitian form v v e and a skew hermitian form w w e then the isometry groups g v and g w of the spaces v and w form a dual reductive pair in the symplectic group sp w where w v e w viewed as an f vector space of dimension 2mn and equipped with the symplectic form

eigenform product identities for hilbert modular forms we prove that amongst all real quadratic fields and all spaces of hilbert modular forms of full level and of weight 2 or greater the product of two hecke eigenforms is not a hecke eigenform except for finitely many real quadratic fields and finitely many weights we show that for mathbb q sqrt 5 there are exactly two such identities

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

algebraic realization of manifolds with group actions abstract let g be a compact lie group and m a closed smooth g manifold with some restrictions on g or on the isotropy groups of the action we show that m is algebraically realized i e there exists a non singular real algebraic g variety which is equivariantly diffeomorphic to m

generalized heegner cycles and p adic rankin l series this article studies a distinguished collection of so called generalized heegner cycles in the product of a kuga sato variety with a power of a cm elliptic curve its main result is a p adic analogue of the gross zagier formula which relates the images of generalized heegner cycles under the p adic abel jacobi map to the special values of certain p adic rankin l series at critical points that lie outside their range of classical interpolation

efficiency of two sample tests via the restricted mean survival time for analyzing event time observations in comparing two treatments with the event time observations the hazard ratio hr estimate is routinely used to quantify the treatment difference however this model dependent estimate may be difficult to interpret clinically especially when the proportional hazards ph assumption is violated an alternative estimation procedure for treatment efficacy based on the restricted means survival time or t year mean survival time t mst has been discussed extensively in the statistical and clinical literature on the other hand a statistical test via the hr or its asymptotically equivalent counterpart the logrank test is asymptotically distribution free in this article we assess the relative efficiency of the hazard ratio and t mst tests with respect to the statistical power under various ph and non ph models theoretically and empirically when the ph assumption is valid the t mst test performs almost as well as the hr test for non ph models the t mst test can substantially outperform its hr counterpart on the other hand the hr test can be powerful when the true difference of two survival functions is quite large at end but not the beginning of the study unfortunately for this case the hr estimate may not have a simple clinical interpretation for the treatment effect due to the violation of the ph assumption

local orders root numbers and modular curves let f be a locally compact field and let wr be an irreducible admissible representation of gl2 f on an infinite dimensional complex vector space let c be the central quasicharacter of wx let k be a separable quadratic field extension of f and let x be a quasicharacter of k such that xw 1 on f the character x gives a 1 dimensional representation of the weil group wk of k via the isomorphism wab k of local classfield theory let xw be the admissible irreducible representation of gl2 f which corresponds in the sense of langlands to the 2 dimensional induced representation indwfx of the weil group of f the representation

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

one fixed point actions and homology 3 spheres one of the best known three dimensional manifolds is the poincare homology three sphere 1 2 3 5 this closed 3 manifold has the same homology groups as the standard sphere s3 but is not simply connected see pnc p 106 or br2 section 1 8 numerous investigations during the past century have shown that this manifold has many remarkable properties several equivalent descriptions of 1 2 3 5 are summarized in kis from the viewpoint of transformation groups one noteworthy property is that 1 2 3 5 is the only nonsimply connected homology sphere admitting a transitive action of a compact lie group brl the manifold 1 2 3 5 also figures importantly in regularity questions for group actions on spherelike manifolds linear or orthogonal actions of finite groups on spheres are generally viewed as the simplest and most regular examples of such actions and one of the themes of transformation groups is to determine the extent to which arbitrary actions on spherelike manifolds resemble linear actions one distinguishing feature of linear actions is that their fixed point sets if nonempty are also spheres the well known results of p a smith cf br2 show that continuous p group actions on homology spheres all resemble linear actions in this respect i e the fixed point sets are always zp homology cohomology spheres on the other hand outside the family of p groups there are many exotic group actions perhaps the most elementary example of this type involves 1 2 3 5 if a is the subgroup of the rotation group so3 given by the symmetries of a regular icosahedron so that a is isomorphic to the alternating group a5 then the quotient space s03 a is the poincare homology 3 sphere and the induced action of a5 a has exactly one fixed point this example suggests that general actions of finite groups on homology spheres can be quite different from linear actions and much of the work on exotic finite group actions during the past three decades was at least partially motivated by the existence of this group action

local factors of nongeneric supercuspidal representations of gsp _4 we compute the local factors of the nongeneric supercuspidal representations of gsp 4 constructed by piatetski shapiro over a nonarchemedian local field of residue characteristic not equal to two the results are in terms of the local factors of the irreducible representations of quaternion division algebra

hilbert modular forms and the gross stark conjecture let f be a totally real eld and an abelian totally odd character of f in 1988 gross stated a p adic analogue of stark s conjecture that relates the value of the derivative of the p adic l function associated to and the p adic logarithm of a p unit in the extension of f cut out by in this paper we prove gross s conjecture when f is a real quadratic eld and is a narrow ring class character the main result also applies to general totally real elds for which leopoldt s conjecture holds assuming that either there are at least two primes above p in f or that a certain condition relating the l invariants of and 1 holds this condition on l invariants is always satised when is quadratic

bessel periods and the non vanishing of yoshida lifts modulo a prime we give an explicit construction of vector valued yoshida lifts and derive a formula of the bessel periods of yoshida lifts by which we prove the non vanishing modulo a prime of yoshida lifts attached to a pair of elliptic modular newforms as a consequence we obtain a new proof of the non vanishing of yoshida lifts

nonlinear analogs of linear group actions on spheres introduction this article is based upon a principle which is so standard that it is almost a cliche the first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation to be more specific we shall describe some applications of this to the symmetry questions of topological transformation groups given a topological space x let homeo denote the set of self homeomorphisms of x this is a group under composition of mappings if g is an arbitrary group then a group action of g on i is a homomorphism homeo x frequently we wish to impose some weak assumptions on p for example if g is a topological group then we might want cp to have suitable continuity properties the usual assumption is that the map

horizontal non vanishing of heegner points and toric periods abstract let f be a totally real number field and a a modular gl 2 type abelian variety over f let k f be a cm quadratic extension let be a class group character over k such that the rankin selberg convolution l s a is self dual with root number 1 we show that the number of class group characters with bounded ramification such that l 1 a 0 increases with the absolute value of the discriminant of k we also consider a rather general rank zero situation let be a cuspidal cohomological automorphic representation over gl 2 a f let be a hecke character over k such that the rankin selberg convolution l s is self dual with root number 1 we show that the number of hecke characters with fixed type and bounded ramification such that l 1 2 0 increases with the absolute value of the discriminant of k the gross zagier formula and the waldspurger formula relate the question to horizontal non vanishing of heegner points and toric periods respectively for both situations the strategy is geometric relying on the zariski density of cm points on self products of a quaternionic shimura variety the recent result 26 31 1 on the andre oort conjecture is accordingly fundamental to the approach