congruences involving arithmetic progressions for weakly holomorphic modular forms abstract in this paper we give a classification of weights k such that there is a nonzero weakly holomorphic modular form f a n q n of weight k on 1 n having infinitely many congruences of the form a l n 0 mod l where l is a prime and is an integer in 0 1 l 1 these are similar to congruences for the partition function investigated by ramanujan furthermore we characterize linear combinations of shimura theta functions with odd characters in terms of these congruences as an application of our main theorem we consider a generalization of newman s conjecture for weakly holomorphic modular forms on 0 n with real dirichlet characters finally we use these results to study arithmetic properties of colored partitions and generalized frobenius partitions

traces of cm values of modular functions zagier proved that the traces of singular moduli i e the sums of the values of the classical j invariant over quadratic irrationalities are the fourier coefficients of a modular form of weight 3 2 with poles at the cusps using the theta correspondence we generalize this result to traces of cm values of weakly holomorphic modular functions on modular curves of arbitrary genus we also study the theta lift for the weight 0 eisenstein series for sl2 and realize a certain generating series of arithmetic intersection numbers as the derivative of zagier s eisenstein series of weight 3 2 this recovers a result of kudla rapoport and yang

congruences for the coefficients of weakly holomorphic modular forms recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli we show that this type of phenomenon is completely general by finding similar congruences for the coefficients of any weakly holomorphic modular form on any congruence subgroup gamma 0 n in particular we give congruences for a wide class of partition functions and for traces of cm values of arbitrary modular functions on certain congruence subgroups of prime level

partition congruences and the andrews garvan dyson crank in 1944 freeman dyson conjectured the existence of a crank function for partitions that would provide a combinatorial proof of ramanujan s congruence modulo 11 forty years later andrews and garvan successfully found such a function and proved the celebrated result that the crank simultaneously explains the three ramanujan congruences modulo 5 7 and 11 this note announces the proof of a conjecture of ono which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function

a geometric effective nullstellensatz the purpose of this paper is to present a geometric theorem which clarifies and extends in several directions work of brownawell kollar and others on the effective nullstellensatz specifically we work on an arbitrary smooth complex projective variety x with the previous classical results corresponding to the case when x is projective space in this setting we prove a local effective nullstellensatz for ideal sheaves and a corresponding global division theorem for adjoint type bundles we also make explicit the connection with the intersection theory of fulton and macpherson finally constructions involving products of prime ideals that appear in earlier work are replaced by geometrically more natural conditions involving orders of vanishing along subvarieties much of the previous activity in this area has been algebraic in nature and seems perhaps not well known in detail among geometers therefore we have felt it worthwhile to include here a rather extended introduction we start with an overview of the questions and earlier work on them then we present the set up and statement of our main theorem we conclude with a series of examples which can be read before the general result of what it yields in special cases

deformations of canonical singularities m reid defined the concept of canonical singularities as those which appear on canonical models of algebraic varieties whose canonical rings are finitely generated from the view point of the classification theory of algebraic varieties it is expected that small deformations of a variety which has only canonical singularities have only canonical singularities the main result of this paper states that this is the case

dyson s ranks and maass forms which were defined by ramanujan and watson decades ago in his last letter to hardy dated january 1920 see pages 127 131 of 27 ramanujan lists 17 such functions and he gives 2 more in his lost notebook 27 in his paper the final problem an account of the mock theta functions 32 watson defines 3 further functions surprisingly much remains unknown about these enigmatic series ramanujan s claims about their analytic properties remain open and there is even debate concerning the rigorous definition of such a function despite these seemingly problematic issues ramanujan s mock theta functions indeed possess many striking properties and they have been the subject of an astonishing number of important works for example see 5 6 7 8 12 13 14 18 19 20 23 27 28 32 33 35 36 to name a few watson predicted this high level of activity in his 1936 presidential address to the london mathematical society with his prophetic words see page 80 of 32

invariance of plurigenera in this paper we give a proof of the following long conjectured result on the invariance of the plurigenera main theorem let p x d be a smooth projective family of compact complex manifolds parametrized by the open unit 1 disk d assume that the bers xt py1 t t 2 d are of general type then for every positive integer m the plurigenus dimc c xt mkxt is independent of t 2 d where kxt is the canonical line bundle of xt

uniform bounds and symbolic powers on smooth varieties the purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety in particular weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety let

reductions of pdes to second order odes and symbolic computation a new method to obtain second order reductions for ordinary differential equations which are polynomial in the derivatives of the dependent variable is presented the method is applied to obtain reductions and new solutions to several well known equations of mathematical physics a lubrication equation a thin film equation the zoomeron equation and a family of 5 th order partial differential equations which includes the caudrey dodd gibbon sawada kotera kaup kupershmidt ito and lax equations some pieces of computer algebra code to derive the reductions are also included

on the explicit construction of higher deformations of partition statistics the modularity of the partition generating function has many important consequences for example asymptotics and congruences for p n in a series of papers the author and ono 12 13 connected the rank a partition statistic introduced by dyson to weak maass forms a new class of functions which are related to modular forms and which were rst considered in 11 here we do a further step towards understanding how weak maass forms arise from interesting partition statistics by placing certain 2 marked durfee symbols introduced by andrews 1 into the framework of weak maass forms to do this we construct a new class of functions which we call quasiweak maass forms because they have quasimodular forms as components as an application we prove two conjectures of andrews it seems that this new class of functions will play an important role in better understanding weak maass forms of higher weight themselves and also their derivatives as a side product we introduce a new method which enables us to prove transformation laws for generating functions over incomplete lattices

formulas for multiplier ideals on singular varieties we prove a generalization of demailly ein lazarsfeld s subadditivity formula and musta s summation formula for multiplier ideals to the case of singular varieties using characteristic p methods as an application of our formula we improve hochster huneke s result on the growth of symbolic powers of ideals in singular affine algebras

nonholomorphic ramanujan type congruences for hurwitz class numbers in contrast to all other known ramanujan type congruences we discover that ramanujan type congruences for hurwitz class numbers can be supported on nonholomorphic generating series we establish a divisibility result for such nonholomorphic congruences of hurwitz class numbers the two key tools in our proof are the holomorphic projection of products of theta series with a hurwitz class number generating series and a theorem by serre which allows us to rule out certain congruences

m2 rank of overpartitions and harmonic weak maass forms there has been significant recent interest in constructing many families of harmonic weak maass forms using rank generating functions of various partitions in this paper applying zwegers results on appell lerch sums we provide a new family of harmonic weak maass forms with the m2 rank generating function of overpartitions as applications we prove the modularity of m2 rank differences and ramanujan type congruences for such ranks

a briancon skoda type theorem for graded systems of ideals abstract we establish a generalization of the briancon skoda theorem about integral closures of ideals for graded systems of ideals satisfying a certain geometric condition

divisibility and distribution of partitions into distinct parts abstract we study the generating function for q n the number of partitions of a natural number n into distinct parts using the arithmetic properties of fourier coefficients of integer weight modular forms we prove several theorems on the divisibility and distribution of q n modulo primes p 5

integral closure of ideals in excellent local rings in 1 briancon and skoda proved using analytic methods that if i is an ideal in the convergent power series ring c x1 xn then in the integral closure of i is contained in i extensive work has been done in the direction of proving briancon skoda type theorems that is statements about i t being contained in i t k where k is a constant independent of t and is a closure operation on ideals cf 5 6 8 11 in this paper we study the following related problem given an ideal i of a noetherian local ring r m find a linear integer valued function f n such that i mn i m for all n or for all sufficiently large n an element x of r is said to be in the integral closure j of an ideal j if it satisfies a relation of the form x 1x n 1 2xn 2 n 0 with t j t for all t we first observe that if r m is a noetherian local ring which is complete in the m adic topology then there exists an integer valued function f n with limn f n such that i mn i m to see this we use the fact that i v v 1 iv where the intersection is over all discrete valuation domains v which are r algebras via v and whose maximal ideal contracts to m with this it is easily shown that n i mn i by chevalley s theorem 12 p 270 for r i and the descending sequence of ideals i mn i n then i mn i m for some function f n such that limn f n chevalley s theorem does not help in determining the order of growth of f n and in fact it cannot because it takes into account only the topology determined by a given descending sequence of ideals by also considering the algebraic properties of the sequence we prove a stronger statement

weakly holomorphic modular forms of half integral weight with nonvanishing constant terms modulo l let l be a prime and j 0 be an integer suppose that f z n a n q n is a weakly holomorphic modular form of weight 1 2 and that a 0 0 mod l we prove that if the coefficients of f z are not well distributed modulo l j then 0 or 1 mod l 1 2 this implies that under the additional restriction 0 0 mod l the following conjecture of balog darmon and ono is true if the coefficients of a modular form of weight 1 2 are almost but not all divisible by l then either 0 mod l 1 2 or 1 mod l 1 2 we also prove that if 0 and 1 mod l 1 2 then there does not exist an integer 0 l such that ln 0 mod l for every nonnegative integer n as an application we study congruences for the values of the overpartition function

the multiplier ideals of a sum of ideals we prove that if a b o x are nonzero sheaves of ideals on a complex smooth variety x then for every q we have the following relation between the multiplier ideals of a b and a b i x a b i x a i x b a similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals we use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals this is applied to compare the multiplier ideals associated to a scheme in different embeddings