discontinuous galerkin methods for friedrichs systems i general theory this paper presents a unified analysis of discontinuous galerkin methods to approximate friedrichs systems an abstract set of conditions is identified at the continuous level to guarantee existence and uniqueness of the solution in a subspace of the graph of the differential operator then a general discontinuous galerkin method that weakly enforces boundary conditions and mildly penalizes interface jumps is proposed all the design constraints of the method are fully stated and an abstract error analysis in the spirit of strang s second lemma is presented finally the method is formulated locally using element fluxes and links with other formulations are discussed details are given for three examples namely advection reaction equations advection diffusion reaction equations and the maxwell equations in the so called elliptic regime

a posteriori error analysis for stabilised finite element approximations of transport problems we develop the a posteriori error analysis of stabilised finite element approximations to linear transport problems via duality arguments two alternative dual problems are considered one is based on the formal adjoint of the hyperbolic differential operator the other on the transposition of the bilinear form for the stabilised finite element method we show both analytically and through numerical experiments that the second approach is superior in the sense that it leads to sharper a posteriori error bounds and more economical adaptively refined meshes

normal bases of ray class fields over imaginary quadratic fields we develop a criterion for a normal basis theorem 2 4 and prove that the singular values of certain siegel functions form normal bases of ray class fields over imaginary quadratic fields other than mathbb q sqrt 1 and mathbb q sqrt 3 theorem 4 2 this result would be an answer for the lang schertz conjecture on a ray class field with modulus generated by an integer 2 remark 4 3

the f q mock theta function conjecture and partition ranks in 1944 freeman dyson initiated the study of ranks of integer partitions here we solve the classical problem of obtaining formulas for ne n resp no n the number of partitions of n with even resp odd rank thanks to rademacher s celebrated formula for the partition function this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f q a problem with its own long history dating to ramanujan s last letter to hardy little was known about this problem until dragonette in 1952 obtained asymptotic results in 1966 g e andrews refined dragonette s results and conjectured an exact formula for the coefficients of f q by constructing a weak maass poincare series whose holomorphic part is q 1f q24 we prove the andrews dragonette conjecture and as a consequence obtain the desired formulas for ne n and no n

robustness in a posteriori error analysis for fem flow models we derive a residual based a posteriori error estimator for a stabilized finite element discretization of certain incompressible oseen like equations we focus our attention on the behaviour of the effectivity index and we carry on a numerical study of its sensitiveness to the problem and mesh parameters we also consider a scalar reaction convection diffusion problem and a divergence free projection problem in order to investigate the effects on the robustness of our a posteriori error estimator of the reaction convection diffusion phenomena and separately of the incompressibility constraint

modular units and cuspidal divisor classes on x0 n2m with n 24 and m squarefree abstract for a positive integer n let c n be the subgroup of j 0 n generated by the equivalence classes of cuspidal divisors of degree 0 and c n q c n j 0 n q be its q rational subgroup let also c q n be the subgroup of c n q generated by q rational cuspidal divisors we prove that when n n 2 m for some integer n dividing 24 and some squarefree integer m the two groups c n q and c q n are equal to achieve this we show that all modular units on x 0 n on such n are products of functions of the form m k h m h 2 n and k z and determine the necessary and sufficient conditions for products of such functions to be modular units on x 0 n

robust a posteriori estimator for advection diffusion reaction problems we propose an almost robust residual based a posteriori estimator for the advection diffusion reaction model problem the theory is developed in the one dimensional setting the numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators in particular the mentioned norm possesses features that allow us to obtain a meaningful a posteriori estimator robust up to a log pe factor where pe is the global peclet number of the problem various numerical tests are performed in one dimension to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature we also consider a possible two dimensional extension of our result and only present a few basic numerical tests indicating that the estimator seems to preserve the good features of the one dimensional setting

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

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

integrality of a ratio of petersson norms and level lowering congruences we prove integrality of the ratiof f g g outside an explicit finite set of primes where g is an arithmetically normalized holomorphic newform on a shimura curve f is a normalized hecke eigenform on gl 2 with the same hecke eigenvalues as g and denotes the petersson inner product the primes dividing this ratio are shown to be closely related to certain level lowering con gruences satisfied by f and to the central values of a family of rankin selberg l functions finally we give two applications the first to proving the integral ity of a certain triple product l value and the second to the computation of the faltings height of jacobians of shimura curves

cuspidal class number of the tower of modular curves x1 npn we consider a cuspidal class number which is the order of a subgroup of the full cuspidal divisor class group of x1 npn with p nmid n and n 1 by studying the second generalized bernoulli numbers we obtain results similar to ones ferrero and washington in ann math 2 109 2 377 395 1979 washington in invent math 49 87 97 1978 about the relative class numbers of cyclotomic mathbb z p extension of an abelian number field

heegner points on mumford tate curves 1 shimura curves 417 2 heegner points 429 3 the regulator term 441 4 the conjecture 445 5 applications and re nements 447

the cuspidal class number formula for the modular curves x1 pm abstract the purpose of the paper is to determine the cuspidal class number of the modular curve x 1 2 2 n 1 it is a continuation of the previous work of the determination of the cuspidal class number for the modular curves x 1 p m for the case p 2

hierarchical a posteriori error estimators for mortar finite element methods with lagrange multipliers hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods a weak continuity condition at the interfaces is enforced by means of lagrange multipliers the two proposed error estimators are based on a defect correction in higher order finite element spaces and an adequate hierarchical two level splitting the first provides upper and lower bounds for the discrete energy norm of the mortar finite element solution whereas the second also estimates the error for the lagrange multiplier it is shown that an appropriate measure for the nonconformity of the mortar finite element solution is the weighted l2 norm of the jumps across the interfaces

a unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty we discuss stabilized galerkin approximations in a new framework widening the scope from the usual dichotomy of the discontinuous galerkin method on the one hand and petrov galerkin methods such as the supg method on the other the idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation thus circumventing the need of a petrov galerkin type choice of spaces this is made possible by adding a higher order penalty term giving l2 control of the jumps in the gradients between adjacent elements we consider convection diffusion reaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces the standard h1 conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides the crouzeix raviart element moreover we show how the formulation extends to discontinuous galerkin interior penalty methods in a natural way by domain decomposition using nitsche s method

multiplicative lattices in global fields consider the classical problem of packin9 equal non overlappin9 spheres in ir n let l be the set of centers and let a l be the density set 2 l log 2 a l n and call it the density exponent let us restrict ourselves to the case when l is a lattice in r n it corresponds to a quadratic formf x on a free z module of rank n and the problem of finding the least possible 2 is equivalent to the other classical problem of finding a form of discriminant 1 with the maximum value of 7 l min f x cf 9 x n 0 consider a family of lattices l n c u with n and let 2 ln lira sup 2 ln it is known that 2 ln 0 599 the kabatiansky levenshtein

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

adaptive streamline diffusion finite element methods for stationary convection diffusion problems adaptive finite element methods for stationary convectiondiffusion problems are designed and analyzed the underlying discretization scheme is the shock capturing streamline diffusion method the adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed a priori error estimates are used to show that the algorithms are efficient in a certain sense

small data oscillation implies the saturation assumption the saturation assumption asserts that the best approximation error in h 1 0 with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements we establish a link between this assumption and the oscillation of f delta u and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption we also show that this condition is necessary and asymptotically valid provided f in l 2

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

hauteur de faltings de quotients de j 0 n discriminants d algebres de hecke et congruences entre formes modulaires in this paper we compute the faltings height of the jacobian of the modular curve j 0 n we explain how it is related to the discriminant of the hecke algebra this discriminant measures congruences between modular forms as a corollary we obtain a lower bound for this discriminant the faltings height of a quotient a of j 0 n is also computed

modular units and cuspidal divisor class groups of x 1 n abstract in this article we consider the group f 1 n of modular units on x 1 n that have divisors supported on the cusps lying over of x 0 n called the cusps for each positive integer n we will give an explicit basis for the group f 1 n this enables us to compute the group structure of the rational torsion subgroup c 1 n of the jacobian j 1 n of x 1 n generated by the differences of the cusps in addition based on our numerical computation we make a conjecture on the structure of the p primary part of c 1 p n for a regular prime p

a posteriori error estimators for convection diffusion equations we derive a posteriori error estimators for convection diffusion equations with dominant convection the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh peclet number the estimators are either based on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

almost universal codes achieving ergodic mimo capacity within a constant gap this paper addresses the question of achieving capacity with lattice codes in multi antenna block fading channels when the number of fading blocks tends to infinity a design criterion based on the normalized minimum determinant is proposed for division algebra multi block space time codes over fading channels this plays a similar role to the hermite invariant for gaussian channels under maximum likelihood decoding it is shown that this criterion is sufficient to guarantee transmission rates within a constant gap from capacity both for deterministic channels and ergodic fading channels moreover if the number of receive antennas is greater than or equal to the number of transmit antennas the same constant gap is achieved under naive lattice decoding as well in the case of independent identically distributed rayleigh fading the error probability vanishes exponentially fast in contrast to the standard approach in the literature which employs random lattice ensembles the existence results in this paper are derived from the number theory first the gap to capacity is shown to depend on the discriminant of the chosen division algebra then class field theory is applied to build families of algebras with small discriminants the key element in the construction is the choice of a sequence of division algebras whose centers are number fields with small root discriminants

on metaplectic forms over function fields we propose a concept of half integral weight in the global function field context and construct natural families of functions with given weight an analogue of shimura correspondence between weight 2 functions and weight frac 3 2 functions via theta series from definite quaternion algebras over function fields is then established from the study of fourier coefficients of these theta series we arrive at a waldspurger type formula this formula is then applied to l series coming from elliptic curves over function fields

a posteriori error estimation for interior penalty finite element approximations of the advection reaction equation in this note we consider residual based a posteriori error estimation for finite element approximations of the transport equation for the discretization we use piecewise affine continuous or discontinuous finite elements and symmetric stabilization of interior penalty type the lowest order discontinuous galerkin method using piecewise constant approximation is included as a special case the key elements in the analysis are a saturation assumption and an approximation result for interpolation between discrete spaces

torsion points on modular curves let n 23 be a prime number in this paper we prove a conjecture of coleman kaskel and ribet about the valued points of the modular curve x 0 n which map to torsion points on j 0 n via the cuspidal embedding we give some generalizations to other modular curves and to noncuspidal embeddings of x 0 n into j 0 n

a posteriori error estimates for lowest order mixed finite element discretizations of convection diffusion reaction equations we establish residual a posteriori error estimates for lowest order raviart thomas mixed finite element discretizations of convection diffusion reaction equations on simplicial meshes in two or three space dimensions the upwind mixed scheme is considered as well and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusion dispersion tensor and on a possible convection dominance global upper bounds for the approximation error in the energy norm are derived where in particular all constants are evaluated explicitly so that the estimators are fully computable our estimators give local lower bounds for the error as well and they hold from the cases where convection or reaction are not present to convection or reaction dominated problems we prove that their local efficiency depends only on local variations in the coefficients and on the local peclet number moreover the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies the main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element known in the mixed finite element method and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

on tensor product l functions for quaternion unitary groups and gl 2 over totally real number fields mixed weight cases abstract we prove an algebraicity of special values of tensor product l functions for quaternion unitary groups and gl 2 in the mixed weight cases over any totally real number field as a consequence we can prove a period relation for automorphic forms of quaternion unitary groups of degree 2 and gsp 4 when these automorphic forms correspond under the conjectural jacquet langlands correspondence and a period relation for endoscopic liftings of gsp 4 in very general situations

twisted hilbert modular surfaces arithmetic intersections and the jacquet langlands correspondence abstract we study arithmetic intersections on quaternionic hilbert modular surfaces and shimura curves over a real quadratic field our first main result is the determination of the degree of the top arithmetic todd class of an arithmetic twisted hilbert modular surface this quantity is then related to the arithmetic volume of a shimura curve via the arithmetic grothendieck riemann roch theorem and the jacquet langlands correspondence

parametrizations of elliptic curves by shimura curves and by classical modular curves abstract fix an isogeny class of semistable elliptic curves over q the elements of have a common conductor n which is a square free positive integer let d be a divisor of n which is the product of an even number of primes i e the discriminant of an indefinite quaternion algebra over q to d we associate a certain shimura curve x0d n d whose jacobian is isogenous to an abelian subvariety of j0 n there is a unique a for which one has a nonconstant map d x0d n d a whose pullback a pic0 x0d n d is injective the degree of d is an integer d which depends only on d and the fixed isogeny class we investigate the behavior of d as d varies

adaptive least squares finite element methods for linear transport equations based on an h div flux reformulation abstract in this paper we study the least squares finite element methods lsfem for the linear hyperbolic transport equations the linear transport equation naturally allows discontinuous solutions and discontinuous inflow conditions while the normal component of the flux across the mesh faces needs to be continuous traditional lsfems using continuous finite element approximations will introduce unnecessary extra error for discontinuous solutions and boundary conditions in order to separate the continuity requirements a new flux variable is introduced with this reformulation the continuities of the flux and the solution can be handled separately in natural h div l 2 conforming finite element spaces several variants of the methods are developed to handle the inflow boundary condition strongly or weakly with the reformulation the new lsfems can handle discontinuous solutions and boundary conditions much better than the traditional lsfems with continuous polynomial approximations with least squares functionals as a posteriori error estimators the adaptive methods can naturally identify error sources including singularity and non matching discontinuity for discontinuity aligned mesh no extra error is introduced if an r t 0 p 0 pair is used to approximate the flux and the solution the new adaptive lsfems can approximate discontinuous solutions with almost no overshooting even when the mesh is not aligned with discontinuity existence and uniqueness of the solutions and a priori and a posteriori error estimates are established for the proposed methods extensive numerical tests are performed to show the effectiveness of the methods developed in the paper

a posteriori error estimators for the raviart thomas element when error estimators for the rte are developed two difficulties prevent the success of the straightforward application of frequently used arguments the hdiv norm is an anisotropic norm i e it refers to differential operators of different orders moreover the traces of hdiv functions are only in h 1 2 therefore one does not obtain optimal a posteriori error estimates when using natural norms this drawback is overcome by using mesh dependent norms

on some arithmetic properties of siegel functions we deal with several arithmetic properties of the siegel functions which are modular units by modifying the ideas in kubert and lang modular units grundlehren der mathematischen wissenschaften vol 244 spinger heidelberg 1981 we establish certain criterion for determining a product of siegel functions to be integral over mathbb z j we also find generators of the function fields mathcal k x 1 n by examining the orders of siegel functions at the cusps and apply them to evaluate the ramanujan s cubic continued fraction systematically furthermore we construct ray class invariants over imaginary quadratic fields in terms of singular values of j and siegel functions

a robust supg norm a posteriori error estimator for stationary convection diffusion equations a robust residual based a posteriori estimator is proposed for the supg finite element method applied to stationary convection diffusion reaction equations the error in the natural supg norm is estimated the main concern of this paper is the consideration of the convection dominated regime a global upper bound and a local lower bound for the error are derived where the global upper estimate relies on some hypotheses numerical studies demonstrate the robustness of the estimator and the fulfillment of the hypotheses a comparison to other residual based estimators with respect to the adaptive grid refinement is also provided

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

robust a posteriori error estimates for stationary convection diffusion equations we analyze a posteriori error estimators for finite element discretizations of convection dominated stationary convection diffusion equations using locally refined isotropic meshes the estimators are based on either the evaluation of local residuals or the solution of discrete local problems with dirichlet or neumann boundary conditions all estimators yield global upper and lower bounds for the error measured in a norm that incorporates the standard energy norm and a dual norm of the convective derivative they are fully robust in the sense that the ratio of the upper and lower bounds is uniformly bounded with respect to the size of the convection the estimates are also uniform with respect to the size of the zero order reaction term and also hold for the limit case of vanishing reaction

the rational torsion subgroups of drinfeld modular jacobians and eisenstein pseudo harmonic cochains let mathfrak n be a square free ideal of mathbb f q t we study the rational torsion subgroup of the jacobian variety j 0 mathfrak n of the drinfeld modular curve x 0 mathfrak n we prove that for any prime number ell not dividing q q 1 the ell primary part of this group coincides with that of the cuspidal divisor class group we further determine the structure of the ell primary part of the cuspidal divisor class group for any prime ell not dividing q 1