analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

a formula for the derivative of the p adic l function of the symmetric square of a finite slope modular form let f be a modular form of weight k and nebentypus psi by generalizing a construction of dabrowski and delbourgo we construct a p adic l function interpolating the special values of the l function l s rm sym 2 f otimes xi where xi is a dirichlet character when s k 1 and xi psi 1 this p adic l function vanishes due to the presence of a so called trivial zero we give a formula for the derivative at s k 1 of this p adic l function when the form f is steinberg at p if the weight of f is even the conductor is even and squarefree and the nebentypus is trivial this formula implies a conjecture of benois

steady periodic water waves with constant vorticity regularity and local bifurcation this paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed using conformal mappings the free boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable the new formulation leads to a regularity result and by use of bifurcation theory to the existence of waves of small amplitude even in the presence of stagnation points in the flow

on certain zeta functions attached to two hilbert modular forms ii the case of automorphic forms on a quaternion algebra as to the meaning of the symbols the reader is referred to the introduction of part i we recall here only that f is a totally real algebraic number field of degree n and w denotes the fourier coefficients of an elliptic modular form 52 z e wq a e2q iaz in part i we investigated d assuming that ex a n q s is an l function of a cm field with an algebraic valued hecke character in the present part ii we treat the case where x a can be obtained as eigenvalues of hecke operators t a on a quaternion algebra b over f thus we begin our study by recalling the theory of automorphic forms on the product o r of r copies of the upper half plane with respect to congruence subgroups of b x here r is the number of archimedean primes of f unramified in b then we introduce in section 2 the notion of arithmeticity or q rationality of such automorphic forms and show that the space of automorphic forms can be spanned by the arithmetic ones our main theorems will be stated in section 3 and proved in section 6 sections 4 and 5 are preliminaries to the proof the central result specialized to the case r 1 asserts that if g is a q rational eigenform such that gj t a x a g for all integral ideals a of f then d s0 for certain integers so are algebraic numbers times 7tdg g where d is an integer determined by the weight of g and is the petersson inner product

finite volume flows and morse theory in this paper we present a new approach to morse theory based on the de rham federer theory of currents the full classical theory is derived in a transparent way the methods carry over uniformly to the equivariant and the holomorphic settings moreover the methods are substantially stronger than the classical ones and have interesting applications to geometry they lead for example to formulas relating characteristic forms and singularities of bundle maps

stability properties of steady water waves with vorticity we present two stability analyses for exact periodic traveling water waves with vorticity the first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth the second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small amplitude or long wave approximation 2006 wiley periodicals inc

symmetry of steady periodic gravity water waves with vorticity we prove that steady periodic two dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest irrespective of the vorticity distribution within the fluid

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

on a class of nearly holomorphic automorphic forms with a b 1 e c and 0 n e z the point that attracts our attention is its nonholomorphic nature it is noteworthy that f is holomorphic if k 2 and also such nonholomorphy doesn t appear in the hilbert modular case therefore one may naturally ask whether this is an isolated oddity in our previous papers 10 and 13 we showed that it was not so in fact a similar eisenstein series on sp n q or on su n n of a particular weight can have the form

arithmetic properties of fourier jacobi expansions of automorphic forms in several variables in this paper we establish some basic arithmetic results concerning automorphic forms in several variables our results are in part analogues of the classical facts for elliptic modular forms and results of shimura on hilbert and siegel modular forms 9 and 10 in essence this involves relating rationality properties of fourier jacobi coefficients to rationality properties in algebraic geometry our automorphic forms are c valued holomorphic functions on certain cartesian powers hnd of siegel upper half spaces hn we consider appropriate arithmetic groups r acting on hnd such that r h d is noncompact in particular we reduce our study to that of congruence subgroups r of reductive linear algebraic groups g such that the derived group g is almost simple over q further let k be a maximal compact subgroup of g r the connected component of the identity in the real

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

a flow force reformulation for steady irrotational water waves abstract the classical irrotational water wave problem described by euler equations with a nonlinear free surface boundary condition and influenced by gravity over a flat bed is considered exploiting the monotonicity property of the flow force lines with depth the unknown boundary problem is transformed into a problem with fixed domain a flow force function formulation of the irrotational water wave problem is developed variational approach is used to prove the existence of small amplitude irrotational traveling wave solutions local bifurcation results have been proved relying on crandall rabinowitz bifurcation theorem

on the transformation formulas of theta series here s is a nondegenerate rational symmetric matrix of signature q s s 3 x iy is a variable in the siegel half space hn of degree n u is a variable in the space cn of all complex n x q matrices runs over the set qq of all rational q x n matrices a is a c valued function on qn that is a finite linear combination of characteristic functions of cosets of qq modulo a z lattice a is a real matrix such that s ta diag lq 1 a and e c exp 2iric for c e c if q 1 s a 1 and a is suitably chosen the series g becomes a classical theta function of jacobi riemann if s is positive definite n 1 u 0 and q is even g is an elliptic modular form of weight q 2 studied by hecke also siegel introduced the series with u 0 in his theory of quadratic forms in si now g satisfies a transformation formula of the following type for every

on the lifting of elliptic cusp forms to siegel cusp forms of degree 2n i 1 cf 8 in this paper we will prove that their conjecture is true moreover we obtain a simple fourier coefficient formula for f z ibukiyama 23 also made a similar conjecture which was formulated in terms of the koecher maass series which we will not consider here let us describe our results throughout this paper we fix positive integers k and n such that k n mod 2 and put e 1 k 1 n for any positive rational number n we denote the absolute value of the discriminant of q vn q by dn the primitive dirichlet character corresponding to q v n q is denoted by xn let fn be the positive rational number such that n nf2nlet b e m2n q be a positive definite half integral symmetric matrix put db det 2b db d db fb f db and xb x db note that edb 0 1 mod 4 and that fb is an integer for each prime p let p qp cx be the unique additive character such that bp x exp 27ir ltx for x e z p 1 recall that the siegel series for b is defined by

chern gauss bonnet and lefschetz duality from a currential point of view abstract we use the mapping cone for the relative derham cohomology of a manifold with boundary in order to show that the chern gauss bonnet theorem for oriented riemannian vector bundles over such manifolds is a manifestation of lefschetz duality in any of the two embodiments of the latter we explain how thom isomorphism fits into this picture complementing thus the classical results about thom forms with compact support when the rank is odd we construct by using secondary transgression forms introduced here a new closed pair of forms on the disk bundle associated to a vector bundle pair which is lefschetz dual to the zero section

a spectral proof of class number one we continue our previous work on the subject of re proving the heegner baker stark theorem giving another effective resolution of this conjecture of gauss namely there are exactly 9 imaginary quadratic fields mathbf q sqrt q with class number 1 specifically the list is q in 3 4 7 8 11 19 43 67 163 we again follow an ansatz of goldfeld considering a modular form l function of analytic rank 2 and reducing the situation to showing sufficient cancellation of the dirichlet series coefficients when restricted to the principal form previously we then chose a specific rank 2 elliptic curve with complex multiplication by mathbf q sqrt 1 and deduced the desired cancellation from an equi distribution result of hooley s for roots of a quadratic polynomial to varying moduli herein we instead use spectral techniques filling in details of work of templier and tsimerman to complete the proof relying on the duke iwaniec bound for fourier coefficients of half integral weight maass forms unlike the previous method we do not require complex multiplication and any analytic rank 2 modular form whose level has an even number of prime factors suffices

on a conjecture of shimura concerning periods of hilbert modular forms introduction in this paper we shall give an affirmative answer to an essential part of the conjecture of shimura on p invariants of hilbert modular forms let f be a totally real algebraic number field of degree n and jf be the set of all isomorphisms of f into c let fa resp fa be the adele ring resp the idele group of f and fx be the archimedean part of fa let x be a primitive system of eigenvalues of hecke operators which occurs in the space of holomorphic hilbert modular cusp forms on gl 2 fa of weight k and f be the primitive form which belongs to x in s 1 shimura introduced an invariant u e f e cx for every 6 e z 2z jf such that

inhomogeneous quadratic forms and triangular numbers we prove an explicit formula for the number of representations of an integer as the sum of n triangular numbers for each n in the range 2 n 8 as special cases of a more general formula applicable to an inhomogeneous quadratic form over a totally real number field the formula can be derived by calculating explicitly the fourier coefficients of a certain eisenstein series that appears in the siegel weil formula for an inhomogeneous quadratic form

zeta functions and eisenstein series on classical groups we construct an euler product from the hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a cm field and the eigenform is holomorphic we also prove analytic continuation of an eisenstein series on another unitary group containing the group just mentioned defined with such an eigenform as an application of our methods we prove an explicit class number formula for a totally definite hermitian form over a cm field

zeta functions and eisenstein series on metaplectic groups the main objectives of this paper are 1 to construct an euler product from a hecke eigenform of half integral weight 2 to prove its meromorphic continuation to the whole plane 3 to examine its relationship with the fourier coefficients of the form and 4 to apply these results to the investigations of eisenstein series of half integral weight the automorphic forms are holomorphic ones considered on the metaplectic cover m of g where g n sp n f with a totally real algebraic number field f the first three are generalizations of the corresponding results in the elliptic and hilbert modular cases obtained in 2 and 6 at the same time all four closely parallel similar results in the case of integral weight in 9 and s10 to describe our results let us take for simplicity f to be q in which case the weight k is half an odd positive integer we consider a holomorphic form f of weight k with respect to a principal congruence subgroup of sp n z of level n which is an eigenform in the sense that f it m 2 m f with a complex number 2 m where n is a multiple of 4 and t m is a hecke operator defined for each positive integer m in a natural way as our first main result we shall prove theorem 4 4 that

capillary gravity water waves modified flow force formulation abstract the classical irrotational capillary gravity water wave problem described by the euler equations with a nonlinear free surface boundary condition over a flat bed is considered a modified flow force has been defined and a new formulation of capillary gravity waves in the framework of the modified flow force function has been developed using bifurcation theory the local existence of waves of small amplitude is proved

on the bloch kato conjecture for elliptic modular forms of square free level let kappa ge 6 be an even integer m an odd square free integer and f in s 2 kappa 2 gamma 0 m a newform we prove that under some reasonable assumptions that half of the lambda part of the bloch kato conjecture for the near central critical value l kappa f is true we do this by bounding the ell valuation of the order of the appropriate bloch kato selmer group below by the ell valuation of algebraic part of l kappa f we prove this by constructing a congruence between the saito kurokawa lift of f and a cuspidal siegel modular form

on certain zeta functions attached to two hilbert modular forms i the case of hecke characters to make our exposition smooth let us first introduce some notational conventions for an algebraic number field k of finite degree we denote by jk the set of all embeddings of k into c and by ik the free z module generated by the elements of jk we then put rik ik0 r and cik ik0 c if p eapau e ik with a e jk and po e z we put xp fl xa pfor 0 x e k this is meaningful for p e cik if xa are all real and positive throughout the paper we denote by d the complex upper half plane and by f a totally real algebraic number field of degree n now the zeta function to be studied in this paper when suitably specialized has the form

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

exact steady periodic water waves with vorticity we consider the classical water wave problem described by the euler equations with a free surface under the influence of gravity over a flat bottom we construct two dimensional inviscid periodic traveling waves with vorticity they are symmetric waves whose profiles are monotone between each crest and trough we use bifurcation and degree theory to construct a global connected set of such solutions c 2004 wiley periodicals inc

on the fundamental periods of automorphic forms of arithmetic type the automorphic forms to be investigated are defined with respect to a congruence subgroup f of an algebraic group g which consists of the invertible elements of a quaternion algebra b over a totally real algebraic number field e we have g glz r i where 6 resp 6 is the set of archimedean primes of e unramified resp ramified in b and n denotes the hamilton quaternions such a b is said to be of signature 5 6 we consider a holomorphic map h of h a into a complex vector space v such that

periodic traveling gravity water waves with discontinuous vorticity we use elliptic theory to prove the existence of steady two dimensional periodic waterwaves of large amplitude in a flowwith an arbitrary bounded but discontinuous vorticity this is achieved by developing a local and global bifurcation construction of weak solutions of the elliptic partial differential equations that are relevant to this hydrodynamical context