on the eisenstein cohomology of arithmetic groups the cohomology h gamma e of an arithmetic subgroup gamma of a connected reductive algebraic group g defined over some algebraic number field f can be interpreted in terms of the automorphic spectrum of gamma with this framework in place there is a sum decomposition of the cohomology into the cuspidal cohomology i e classes represented by cuspidal automorphic forms for g and the so called eisenstein cohomology constructed as the span of appropriate residues or derivatives of eisenstein series attached to cuspidal automorphic forms on the levi components of proper parabolic f subgroups of g the main objective of this paper is to isolate a specific structural part in the eisenstein cohomology this pertains to regular eisenstein cohomology classes attached to cuspidal automorphic representations whose archimedean component is tempered it is shown that the cohomological degree of these classes is bounded from below by the constant q 0 g mathbb r 1 2 dim x g mathbb r rk g mathbb r rk k mathbb r where k mathbb r denotes a maximal compact subgroup of the real lie group g mathbb r where x g mathbb r is the associated symmetric space this investigation has various applications one of these is a vanishing result for the cohomology in the generic case i e where the representation determining the coefficient system e has regular highest weight in degrees below q 0 g mathbb r this is a sharp bound depending only on the underlying real lie group g mathbb r corollary 5 6 proposition 5 8 this result is supplemented by a qualitative structural result in the description of the cohomology in higher degrees by means of regular eisenstein cohomology classes

ghost classes in mathbb q q rank two orthogonal shimura varieties in this article the existence of ghost classes for the shimura varieties associated to algebraic groups of orthogonal similitudes of signature 2 n is investigated we make use of the study of the weights in the mixed hodge structures associated to the corresponding cohomology spaces and results on the eisenstein cohomology of the algebraic group of orthogonal similitudes of signature 1 n 1 for the values of n 4 5 we prove the non existence of ghost classes for most of the irreducible representations including most of those with an irregular highest weight for the rest of the cases we prove strong restrictions on the possible weights in the space of ghost classes and in particular we show that they satisfy the weak middle weight property

torsion homology growth and cycle complexity of arithmetic manifolds let m be an arithmetic hyperbolic 3 manifold such as a bianchi manifold we conjecture that there is a basis for the second homology of m where each basis element is represented by a surface of low genus and give evidence for this we explain the relationship between this conjecture and the study of torsion homology growth

boundary and eisenstein cohomology of mathrm sl _3 mathbb z sl3 z in this article several cohomology spaces associated to the arithmetic groups mathrm sl 3 mathbb z and mathrm gl 3 mathbb z with coefficients in any highest weight representation mathcal m lambda have been computed where lambda denotes their highest weight consequently we obtain detailed information of their eisenstein cohomology with coefficients in mathcal m lambda when mathcal m lambda is not self dual the eisenstein cohomology coincides with the cohomology of the underlying arithmetic group with coefficients in mathcal m lambda in particular for such a large class of representations we can explicitly describe the cohomology of these two arithmetic groups we accomplish this by studying the cohomology of the boundary of the borel serre compactification and their euler characteristic with coefficients in mathcal m lambda at the end we employ our study to discuss the existence of ghost classes

bounds for the multiplicities of cohomological automorphic forms on gl2 we prove a power saving for the dimension of the space of cohomological automorphic forms of xed level and growing weight on gl 2 over any number eld that is not totally real our proof involves the theory of p adically completed cohomology developed by calegari and emerton and a bound for the growth of coinvariants in certain nitely generated noncommutative iwasawa modules

the phylogenetic kantorovich rubinstein metric for environmental sequence samples it is now common to survey microbial communities by sequencing nucleic acid material extracted in bulk from a given environment comparative methods are needed that indicate the extent to which two communities differ given data sets of this type unifrac which gives a somewhat ad hoc phylogenetics based distance between two communities is one of the most commonly used tools for these analyses we provide a foundation for such methods by establishing that if we equate a metagenomic sample with its empirical distribution on a reference phylogenetic tree then the weighted unifrac distance between two samples is just the classical kantorovich rubinstein or earth mover s distance between the corresponding empirical distributions we demonstrate that this kantorovich rubinstein distance and extensions incorporating uncertainty in the sample locations can be written as a readily computable integral over the tree we develop l p zolotarev type generalizations of the metric and we show how the p value of the resulting natural permutation test of the null hypothesis no difference between two communities can be approximated by using a gaussian process functional we relate the l 2 case to an analysis of variance type of decomposition finding that the distribution of its associated gaussian functional is that of a computable linear combination of independent formula see text random variables

on special zeros of p adic l functions of hilbert modular forms let e be a modular elliptic curve over a totally real number field f we prove the weak exceptional zero conjecture which links a higher derivative of the p adic l function attached to e to certain p adic periods attached to the corresponding hilbert modular form at the places above p where e has split multiplicative reduction under some mild restrictions on p and the conductor of e we deduce the exceptional zero conjecture in the strong form i e where the automorphic p adic periods are replaced by the mathcal l invariants of e defined in terms of tate periods from a special case proved earlier by mok crucial for our method is a new construction of the p adic l function of e in terms of local data

inference for empirical wasserstein distances on finite spaces summary the wasserstein distance is an attractive tool for data analysis but statistical inference is hindered by the lack of distributional limits to overcome this obstacle for probability measures supported on finitely many points we derive the asymptotic distribution of empirical wasserstein distances as the optimal value of a linear programme with random objective function this facilitates statistical inference e g confidence intervals for sample based wasserstein distances in large generality our proof is based on directional hadamard differentiability failure of the classical bootstrap and alternatives are discussed the utility of the distributional results is illustrated on two data sets

on the non vanishing of the first betti number of hyperbolic three manifolds we show the non vanishing of cohomology groups of sufficiently small congruence lattices in sl 1 d where d is a quaternion division algebra defined over a number field e contained inside a solvable extension of a totally real number field as a corollary we obtain new examples of compact arithmetic hyperbolic three manifolds with non torsion first homology group confirming a conjecture of waldhausen the proof uses the characterisation of the image of solvable base change by the author and the construction of cusp forms with non zero cusp cohomology by labesse and schwermer

optimal estimation of wasserstein distance on a tree with an application to microbiome studies the weighted unifrac distance a plug in estimator of the wasserstein distance of read counts on a tree has been widely used to measure the microbial community difference in microbiome studies ou

critical values of the twisted tensor l function in the imaginary quadratic case the twisted tensor l function of f which we denote by g s f is a certain dirichlet series associated to a quadratic extension of number fields k f and a cuspidal automorphic function f over k it was introduced in 1 by asai following previous work of shimura in the case when f is a hilbert modular cusp form over a real quadratic extension k of q in the past twenty odd years this l function has been considered more generally for instance 11 and 12 deal with quadratic extensions of totally real fields 17 with imaginary quadratic extensions of q and 3 4 and 14 with general quadratic extensions of number fields all these papers have been primarily concerned with establishing analytic properties of g s f analogous to those in 1 such as meromorphic continuation to the entire complex plane location and finiteness of the number of poles and functional equation the aim of this paper is to prove a rationality result for g s f in the imaginary quadratic setting if k is an imaginary quadratic field and f a cusp form associated to k we establish that there is a period j f such that

module de congruences pour gl 2 d un corps imaginaire quadratique et theorie d iwasawa d un corps cm biquadratique

on the method of thue siegel the methods of thue and siegel based on explicit pade approximations to algebraic functions are used to examine diophantine approximations to roots of

a model for paired multinomial data and its application to analysis of data on a taxonomic tree in human microbiome studies sequencing reads data are often summarized as counts of bacterial taxa at various taxonomic levels specified by a taxonomic tree this article considers the problem of analyzing two repeated measurements of microbiome data from the same subjects such data are often collected to assess the change of microbial composition after certain treatment or the difference in microbial compositions across body sites existing models for such count data are limited in modeling the covariance structure of the counts and in handling paired multinomial count data a new probability distribution is proposed for paired multinomial count data which allows flexible covariance structure and can be used to model repeatedly measured multivariate count data based on this distribution a test statistic is developed for testing the difference in compositions based on paired multinomial count data the proposed test can be applied to the count data observed on a taxonomic tree in order to test difference in microbiome compositions and to identify the subtrees with different subcompositions simulation results indicate that proposed test has correct type 1 errors and increased power compared to some commonly used methods an analysis of an upper respiratory tract microbiome data set is used to illustrate the proposed methods

dyson s lemma with moving parts in its original form d dyson s lemma bounds the order of vanishing of a polynomial in two variables at a finite set of points and was used in studying rational approximations to a fixed algebraic irrational number bombieri b revived interest in dyson s lemma and subsequently esnault and viehweg ev and vojta v1 vastly extended the scope of the result to polynomials in several variables and to a section of a line bundle on a product of two algebraic curves respectively the theorems of vojta and of esnault and viehweg are highly refined geometric results which are used to prove the mordell conjecture and roth s theorem respectively the main theorems of both ev and up to a factor of two v1 were recovered in n as special cases of a more general result before n had been written vojta v3 formulated a result which would vastly improve the main result of n the goal of this note is to prove vojta s statement on a product of an arbitrary number of projective lines and to give a counterexample on a product of three or more curves of genus 1 in order to state our main result we need to fix some notation let

p adic l functions of bianchi modular forms the theory of overconvergent modular symbols developed by rob pollack and glenn stevens gives a beautiful and effective construction of the p adic l function of a modular form in this paper we give an analogue of their results for bianchi modular forms that is modular forms over imaginary quadratic fields in particular we prove control theorems that say that the canonical specialisation map from overconvergent to classical bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable hecke operators we also give an explicit link between the classical modular symbol attached to a bianchi modular form and critical values of its l function which then allows us to construct p adic l functions of bianchi modular forms

dyson s lemma and a theorem of esnault and viehweg in the classical theory of diophantine approximation as encountered in the work of thue siegel dyson roth and schmidt finiteness results are obtained by constructing an auxilliary polynomial one knows a priori that the polynomial vanishes to high order at certain approximating points and this contradicts an upper bound on the order of vanishing obtained by other techniques the difficult part of the argument is always bounding the order of vanishing from above this technique of diophantine approximation has recently been extended with great success v3 v4 v5 f1 f2 in order to obtain finiteness results for rational points on certain subvarieties of abelian varieties an intermediate stage in this development occurred when esnault and viehweg ev1 proved a generalized version of dyson s lemma strong enough to imply roth s theorem a few years later vojta v1 built upon this work to prove a variant of dyson s lemma valid on a product of curves of arbitrary genus this played an essential role in his proof v3 of the mordell conjecture faltings f1 further extending the work of vojta was able to prove finiteness results for certain higher dimensional subvarieties of abelian varieties dyson s lemma however plays no immediate role in faltings proof instead he uses a geometrical result called the product theorem though dyson s lemma is not used in f1 there is nonetheless a close relationship between the arguments used in section 4 of f1 and those occurring in section 5 of ev1 in particular in both instances the proof is by induction on the dimension of product subvarieties with lemma 2 9 of ev1 replacing corollary 4 3 of f1 and with the main lemma 5 3 in ev1 taking the place of the product theorem in f1 in this note we propose to generalize the main theorems of ev1 and v1 in order to state our results we need to fix some notation let

the torsion in symmetric powers on congruence subgroups of bianchi groups in this paper we prove that for a fixed neat principal congruence subgroup of a bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated to the m th symmetric power of the standard representation of sl 2 c grows exponentially in m 2 we give upper and lower bounds for the growth rate our result extends a a result of w muller and s marshall who proved the corresponding statement for closed arithmetic 3 manifolds to the finite volume case we also prove a limit multiplicity formula for combinatorial reidemeister torsions on higher dimensional hyperbolic manifolds

on the number of solutions of polynomial congruences and thue equations has only finitely many solutions in integers x and yv in the first part of this paper we shall establish upper bounds for the number of solutions of 1 in coprime integers x and y under the assumption that the discriminant d f of f is nonzero for most integers h these bounds improve upon those obtained by bombieri and schmidt in 5 in the course of proving these bounds we shall establish a result on polynomial congruences that extends earlier work of nagell 30 ore 32 sandor 33 and huxley 19 in fact we shall establish an upper bound for the number of solutions of a polynomial congruence that is in general best possible in the second part we shall address the problem of finding forms f for which 1 has many solutions for arbitrarily large integers h finally we shall obtain upper bounds for the number of solutions of certain thue mahler and ramanujan nagell equations by appealing to estimates of evertse gy6ry stewart and tijdeman 17 for the number of solutions of s unit equations

the residual eisenstein cohomology of sp_ 4 over a totally real number field let g sp4 k be the k split symplectic group of k rank 2 where k is a totally real numbereld in this paper we compute the eisenstein cohomology of g with respect to anynite dimensional irreducible k rational representation e of g1 rk qg r where rk q denotes the restriction of scalars from k to q the approach is based on the work of schwermer regarding the eisenstein cohomology for sp4 q kim s description of the residual spectrum of sp4 and the frankeltration of the space of automorphic forms in fact taking the representation theoretic point of view we write for the group g the frankeltration with respect to the cuspidal support and give a precise description of theltration quotients in terms of induced representations this is then used as a prerequisite for the explicit computation of the eisenstein cohomology the special focus is on the residual eisenstein cohomology under a certain compatibility condition for the coefficient system e and the cuspidal support we prove the existence of non trivial residual eisenstein cohomology classes which are not square integrabl e that is represented by a non square integrabl residue of an eisenstein series

the gorenstein conjecture fails for the tautological ring of mathcal overline m _ 2 n we prove that for n equal to at least one of the integers 8 12 16 20 the tautological ring r bullet overline mathcal m 2 n is not gorenstein in fact our n equals the smallest integer such that there is a non tautological cohomology class of even degree on overline mathcal m 2 n by work of graber and pandharipande such a class exists on overline mathcal m 2 20 and we present some evidence indicating that n is in fact 20

on the fundamental periods of hilbert modular forms the main purpose of this paper is to establish the existence of fundamental periods of primitive cusp forms of hilbert modular type of several variables as well as the relationship between those fundamental periods and the special values of the associated l functions these results together with some recent results of shimura give us the means of translating with ease results concerning periods of automorphic forms derived from various points of view we also verify several conjectures of shimura on the properties of such fundamental periods

modularity lifting beyond the taylor wiles method we prove new modularity lifting theorems for p adic galois representations in situations where the methods of wiles and taylor wiles do not apply previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology in practice this imposes several restrictions one must be in a shimura variety setting and the automorphic forms must be of regular weight at infinity in this paper we essentially show how to remove these restrictions our most general result is a modularity lifting theorem which on the automorphic side applies to automorphic forms on the group mathrm gl n over a general number field it is contingent on a conjecture which in particular predicts the existence of galois representations associated to torsion classes in the cohomology of the associated locally symmetric space we show that if this conjecture holds then our main theorem implies the following if e is an elliptic curve over an arbitrary number field then e is potentially automorphic and satisfies the sato tate conjecture in addition we also prove some unconditional results for example in the setting of mathrm gl 2 over mathbf q we identify certain minimal global deformation rings with the hecke algebras acting on spaces of p adic katz modular forms of weight 1 such algebras may well contain p torsion moreover we also completely solve the problem for p odd of determining the multiplicity of an irreducible modular representation overline rho in the jacobian j 1 n where n is the minimal level such that overline rho arises in weight two

eigenvarieties for reductive groups we develop the theory of overconvergent cohomology introduced by g stevens and we use it to give a construction of eigenvarieties associated to any reductive group g over q such that g r has discrete series we prove that the so called eigenvarieties are equidimensional and generically at over the weight space

uniformization of modular elliptic curves via p adic periods abstract the langlands programme predicts that a weight 2 newform f over a number field k with integer hecke eigenvalues generally should have an associated elliptic curve e f over k in 19 we associated building on works of darmon 8 and greenberg 20 a p adic lattice to f under certain hypothesis and implicitly conjectured that is commensurable with the p adic tate lattice of e f in this paper we present this conjecture in detail and discuss how it can be used to compute directly from f a weierstrass equation for the conjectural e f we develop algorithms to this end and implement them in order to carry out extensive systematic computations in which we compute weierstrass equations of hundreds of elliptic curves some with huge heights over dozens of number fields the data we obtain give extensive support for the conjecture and furthermore demonstrate that the conjecture provides an efficient tool to building databases of elliptic curves over number fields

on the arithmetic of shalika models and the critical values of l functions for gl2n let pi be a cohomological cuspidal automorphic representation of rm gl 2n bbb a over a totally real number field f suppose that pi has a shalika model we define a rational structure on the shalika model of pi f comparing it with a rational structure on a realization of pi f in cuspidal cohomology in top degree we define certain periods omega epsilon pi f we describe the behavior of such top degree periods upon twisting pi by algebraic hecke characters chi of f then we prove an algebraicity result for all the critical values of the standard l functions l s pi otimes chi here we use the recent work of b sun on the non vanishing of a certain quantity attached to pi infty as applications we obtain algebraicity results in the following cases firstly for the symmetric cube l functions attached to holomorphic hilbert modular cusp forms we also discuss the situation for higher symmetric powers secondly for certain self dual of symplectic type rankin selberg l functions for rm gl 3 times rm gl 2 assuming langlands functoriality this generalizes to certain rankin selberg l functions of rm gl n times rm gl n 1 thirdly for the degree four l functions attached to siegel modular forms of genus 2 and full level moreover we compare our top degree periods with periods defined by other authors we also show that our main theorem is compatible with conjectures of deligne and gross

eisenstein cohomology for congruence subgroups of so n 2 the automorphic cohomology of a connected reductive algebraic group defined over mathbb q decomposes as a direct algebraic sum of cuspidal and eisenstein cohomology the present paper investigates the eisenstein cohomology for congruence subgroups of a rational form g of mathbb q rank 2 of so n 2 in the generic case the main results provide a description of the internal structure of the summands in the eisenstein cohomology corresponding to the associate classes of maximal parabolic mathbb q subgroups

intersection numbers of hecke cycles on hilbert modular varieties let o be the ring of integers of a totally real number field e and set g res e q gl 2 fix an ideal c 0 for each ideal m 0 let t m denote the mth hecke operator associated to the standard compact open subgroup u 0 c of g a f setting x 0 c g q g a k u 0 c where k is a certain subgroup of g r we use t m to define a hecke cycle z m ih 2 e q x 0 c x 0 c here ih denotes intersection homology we use zucker s conjecture proven by looijenga and independently by saper and stern to obtain a formula relating the intersection number z m z n to the trace of t m o t n considered as an endomorphism of the space of hilbert cusp forms on u 0 c

bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms let goo be a semisimple real lie group with unitary dual goo we produce new upper bounds for the multiplicities with which representations e of cohomological type appear in certain spaces of cusp forms on goo the main new idea is to apply noncommutative iwasawa theory to certain p adic completions of the cohomology of locally symmetric spaces