diameters and eigenvalues we derive a new upper bound for the diameter of a k regular graph g as a function of the eigenvalues of the adjacency matrix namely suppose the adjacency matrix of g has eigenvalues ai a2 a with ia i i vi21 ia i where ai k a ia21 then the diameter d g must satisfy d g we will consider families of graphs whose eigenvalues can be explicitly de termined these graphs are determined by sums or differences of vertex labels namely the pair i j being an edge depends only on the value i j or i j for directed graphs we will show that these graphs are expander graphs with small diameters by using an inequality on character sums which was recently proved by n m katz bell communications research 445 south street morristown new jersey 07960 this content downloaded from 207 46 13 111 on sat 21 may 2016 07 00 04 utc all use subject to http about jstor org terms

counterexamples on l_p maximal regularity we establish a transference result for l p maximal regularity for the abstract cauchy problem on banach space from this result we deduce counterexamples to l p maximal regularity 1 p infty in particular we obtain an operator b without any l p maximal regularity although it admits bounded imaginary powers with vert b is vert 1 for all s in mathbb r we also derive an operator which satisfies l p maximal regularity on bounded intervals 0 t but not on the half line mathbb r

on the inverse of the sum of two sectorial operators abstract we study an abstract linear operator equation on a banach space by using the inverse of the sum of two sectorial operators we prove that the boundedness of a special type of operator valued h calculus is sufficient for maximal regularity of the solution we apply the result to the abstract parabolic problem to give a maximal l p regularity condition we also study the abstract hyperbolic problem and give a sufficient condition for the existence of solution

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

boundedness of riesz transforms for elliptic operators on abstract wiener spaces let e h be an abstract wiener space and let dv vd where d denotes the malliavin derivative and v is a closed and densely defined operator from h into another hilbert space h given a bounded operator b on h coercive on the range r v we consider the operators a v bv in h and a vv b in h as well as the realisations of the operators l dv bdv and l dvdv b in lp e and lp e h respectively where 1 p our main result asserts that the following four assertions are equivalent 1 d l d dv with lf p dvf p for f d l 2 l admits a bounded h functional calculus on r dv 3 d a d v with ah vh for h d a 4 a admits a bounded h functional calculus on r v moreover if these conditions are satisfied then d l d dv2 d da the equivalence 1 4 is a non symmetric generalisation of the classical meyer inequalities of malliavin calculus where h h v i b 12i a one sided version of 1 4 giving lp boundedness of the riesz transform dv l in terms of a square function estimate is also obtained as an application let a generate an analytic c0 contraction semigroup on a hilbert space h and let l be the lp realisation of the generator of its second quantisation our results imply that two sided bounds for the riesz transform of l are equivalent with the kato square root property for a the boundedness of the riesz transform is used to obtain an lp domain characterisation for the operator l

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

product formulas in functional calculi for sectorial operators we study the product formula fg a f a g a in the framework of unbounded functional calculus of sectorial operators a we give an abstract result and as corollaries we obtain new product formulas for the holomorphic functional calculus an extended stieltjes functional calculus and an extended hille phillips functional calculus our results generalise previous work of hirsch martinez and sanz and schilling

martingale transforms and complex uniform convexity les transformees de martingale et les operateurs integraux singuliers de calderon zygmund sont bornees comme des operateurs de l 2 l 1 dans l 2 l q quand 0 q 1 si y est un sous espace reflexif de l 1 alors l 1 y peut se renormer pour etre 2 complexe uniformement convexe on donne une nouvelle demonstration de la propriete de cotype 2 de l 1 h

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

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

some results about complex powers of closed operators the aim of this paper is to give some further results about complex powers of operators a tool that the authors already employed in 2 and s to study some questions of operator theory our first result is essentially a completion of corollary 2 9 of 2 since it gives an estimate for the rate of decrease of a a at infinity when a is a positive operator with bounded imaginary powers we remark that the converse question i e the estimate of the imaginary powers in terms of the amplitude of the spectrum remains open next we prove a result of pertubation if we add to a positive operator

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

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

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

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

nilpotent fusion categories abstract in this paper we extend categorically the notion of a finite nilpotent group to fusion categories to this end we first analyze the trivial component of the universal grading of a fusion category c and then introduce the upper central series of c for fusion categories with commutative grothendieck rings e g braided fusion categories we also introduce the lower central series we study arithmetic and structural properties of nilpotent fusion categories and apply our theory to modular categories and to semisimple hopf algebras in particular we show that in the modular case the two central series are centralizers of each other in the sense of m muger

decomposition of products in nilpotent table algebras abstract two theorems on the decomposition of products in nilpotent commutative table algebras are established this generalizes several results of adan bante on the decomposition of products of irreducible characters and of conjugacy classes of finite p groups and extends these results to fusion rule algebras and association schemes

invertible extractors and wiretap protocols a wiretap protocol is a pair of randomized encoding and decoding functions such that knowledge of a bounded fraction of the encoding of a message reveals essentially no information about the message while knowledge of the entire encoding reveals the message using the decoder in this paper the notion of efficiently invertible extractors is studied and it is shown that a wiretap protocol can be constructed from such an extractor then invertible extractors for symbol fixing affine and general sources are constructed and used to create wiretap protocols with asymptotically optimal trade offs between their rate ratio of the length of the message versus its encoding and resilience ratio of the observed positions of the encoding and the length of the encoding the results are further applied to create wiretap protocols for challenging communication problems such as active intruders who change portions of the encoding network coding and intruders observing arbitrary boolean functions of the encoding

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

on thin residues and basis digraphs of nilpotent table algebras and applications to nilpotent groups abstract in this paper we study characterizations of nilpotent table algebras in terms of thin residues thin residue matrices thin residue digraphs and basis digraphs we will also discuss applications to nilpotent groups in particular we give a conceptual explanation of the results in takegahara and yoshida 2008 ty

a joint functional calculus for sectorial operators with commuting resolvents 1 introduction in this paper we are concerned with h 1 functional calculus in the sense of the construction introduced by mcintosh 23 and developed recently by cowling doust mcintosh and yagi 8 given v in 0 p we will use the notation sv fz 2 c n f0g jarg z j vg to denote the open sector of angle 2v around the half line r let x be a complex banach space given a linear operator a on x we denote by d a n a and r a the domain the kernel and the range of a respectively we also denote by j a the spectrum of a and by r a the resolvent set of a

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

erratum to perturbation and interpolation theorems for the h calculus with applications to differential operators we prove comparison theorems for the h calculus that allow to transfer the property of having a bounded h calculus from one sectorial operator to another the basic technical ingredient are suitable square function estimates these comparison results provide a new approach to perturbation theorems for the h calculus in a variety of situations suitable for applications our square function estimates also give rise to a new interpolation method the rademacher interpolation we show that a bounded h calculus is characterized by interpolation of the domains of fractional powers with respect to rademacher interpolation this leads to comparison and perturbation results for operators defined in interpolation scales such as the l p scale we apply the results to give new proofs on the h calculus for elliptic differential operators including schrodinger operators and perturbed boundary conditions as new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded h calculus in certain sobolev spaces and that the stokes operator on bounded domains with c 1 1 has a bounded h calculus in the helmholtz scale l p p 1

square function estimates and the kato problem for second order parabolic operators in rn 1 in 4 the kato square root problem for second order complex uniformly elliptic operators of the form l div a del f with only bounded and measurable coefficients was solved the solution is a

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

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

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

square functions and h calculus on subspaces of l p and on hardy spaces let x be a closed subspace of l p with 1 p and let a be any sectorial operator on x we consider associated square functions on x of the form open image in new window and we show that if a admits a bounded h functional calculus on x then these square functions are equivalent to the original norm of x then we deduce a similar result when x h1 n is the usual hardy space for an appropriate choice of f for example if n 1 the right choice is the sum open image in new window for h h1 where h denotes the hilbert transform

weighted maximal regularity estimates and solvability of non smooth elliptic systems i we develop new solvability methods for divergence form second order real and complex elliptic systems above lipschitz graphs with l 2 boundary data the coefficients a may depend on all

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

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

operators with an absolute functional calculus we study sectorial operators with a special type of functional calculus which we term an absolute functional calculus a typical example of such an operator is an invertible operator a defined on a banach space x considered on the real inter polation space dom a x p where 0 1 and 1 p in general the absolute functional calculus can be characterized in terms of real interpolation spaces we show that operators of this type have a strong form of the h calculus and behave very well with respect to the joint functional calculus we give applications of these results to recent work of arendt batty and bu on the existence of holder continuous solutions for the abstract cauchy problem