rank 2 symmetric hyperbolic kac moody algebras and hilbert modular forms in this paper we study rank 2 symmetric hyperbolic kac moody algebras h a with the cartan matrices 2 a a2 a 3 and their automorphic correction in terms of hilbert modular forms we associate a family of h a s to the quadratic field q p for each odd prime p and show that there exists a chain of embeddings in each family when p 5 13 17 we show that the first h a in each family i e h 3 h 11 h 66 is contained in a generalized kac moody superalgebra whose denominator function is a hilbert modular form given by a borcherds product hence our results provide automorphic correction for those h a s we also compute asymptotic formulas for the root multiplicities of the generalized kac moody superalgebras using the fact that the exponents in the borcherds products are fourier coefficients of weakly holomorphic modular forms of weight 0

weakly holomorphic modular forms and rank two hyperbolic kac moody algebras in this paper we compute basis elements of certain spaces of weight 0 weakly holomorphic modular forms and consider the integrality of fourier coefficients of the modular forms we use the results to construct automorphic correction of the rank 2 hyperbolic kac moody algebras h a a 4 5 6 through hilbert modular forms explicitly given by borcherds lifts of the weakly holomorphic modular forms we also compute asymptotics of the fourier coefficients as they are related to root multiplicities of the rank 2 hyperbolic kac moody algebras this work is a continuation of an earlier work of the first and second authors where automorphic correction was constructed for h a a 3 11 66

missing modules the gnome lie algebra and e10 we study the embedding of kac moody algebras into borcherds or gener alized kac moody algebras which can be explicitly realized as lie algebras of physical states of some completely compactified bosonic string the extra missing states can be decomposed into irreducible highest or lowest weight missing modules w r t the relevant kac moody subalgebra the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of cer tain polarization states of the associated string we analyse in detail two examples where the momentum lattice of the string is given by the unique even unimodular lorentzian latticeii 1 1 orii 9 1 respectively the former leads to the borcherds algebra gii 1 1 which we call gnome lie algebra with maximal kac moody subalgebraa1 by the use of the denominator formula a complete set of imaginary simple roots can be exhibited while the ddf construction provides an explicit lie algebra basis in terms of purely longitudinal states of the compactified string in two dimensions the second example is the borcherds algebra gii 9 1 whose maximal kac moody subalgebra is the hyperbolic algebra e10 the imaginary simple roots at level 1 which give rise to irreducible low est weight modules fore10 can be completely characterized furthermore our explicit analysis of two non trivial level 2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for gii 9 1

holomorphic anomaly equations and the igusa cusp form conjecture let s be a k3 surface and let e be an elliptic curve we solve the reduced gromov witten theory of the calabi yau threefold s times e for all curve classes which are primitive in the k3 factor in particular we deduce the igusa cusp form conjecture the proof relies on new results in the gromov witten theory of elliptic curves and k3 surfaces we show the generating series of gromov witten classes of an elliptic curve are cycle valued quasimodular forms and satisfy a holomorphic anomaly equation the quasimodularity generalizes a result by okounkov and pandharipande and the holomorphic anomaly equation proves a conjecture of milanov ruan and shen we further conjecture quasimodularity and holomorphic anomaly equations for the cycle valued gromov witten theory of every elliptic fibration with section the conjecture generalizes the holomorphic anomaly equations for elliptic calabi yau threefolds predicted by bershadsky cecotti ooguri and vafa we show a modified conjecture holds numerically for the reduced gromov witten theory of k3 surfaces in primitive classes

elliptic genera of symmetric products and second quantized strings in this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the n fold symmetric product m n sn of a manifold m to the partition function of a second quantized string theory on the space m s 1 the generating function of these elliptic genera is shown to be almost an automorphic form for o 3 2 z in the context of d brane dynamics this result gives a precise computation of the free energy of a gas of d strings inside a higher dimensional brane

siegel modular forms of genus 2 with the simplest divisor we prove that there exist exactly eight siegel modular forms with respect to the congruence subgroups of hecke type of the paramodular groups of genus 2 vanishing precisely along the diagonal of the siegel upper half plane this is a solution of a question formulated during the conference black holes black rings and modular forms ens paris august 2007 these modular forms generalize the classical igusa form and the forms constructed by gritsenko and nikulin in 1998

the monster lie algebra abstract we calculate the multiplicities of all the roots of the monster lie algebra this gives an example of a lie algebra all of whose simple roots and root multiplicities are known and which is not finite dimensional or an affine kac moody algebra there seem to be several similar infinite dimensional lie algebras which to a sympathetic eye appear to correspond to some of the sporadic simple groups

lattice polarized k3 surfaces and siegel modular forms abstract the goal of the present paper is two fold first we present a classification of algebraic k3 surfaces polarized by the lattice h e 8 e 7 key ingredients for this classification are as follows a normal form for these lattice polarized k3 surfaces a coarse moduli space and an explicit description of the inverse period map in terms of siegel modular forms second we give explicit formulas for a hodge correspondence that relates these k3 surfaces to principally polarized abelian surfaces the hodge correspondence in question underlies a geometric two isogeny of k3 surfaces the details of which are described by the authors in clingher and doran 2011 7

graded lie superalgebras and the superdimension formula in this paper we investigate the structure of graded lie superalgebras l a a l a where is a countable abelian semigroup and a is a countable abelian group with a coloring map satisfying a certain finiteness condition given a denominator identity for the graded lie superalgebra l we derive a superdimension formula for the homogeneous subspaces l a a a which enables us to study the structure of graded lie superalgebras in a unified way we discuss the applications of our superdimension formula to free lie superalgebras generalized kac moody superalgebras and monstrous lie superalgebras in particular the product identities for normalized formal power series are interpreted as the denominator identities for free lie superalgebras we also give a characterization of replicable functions in terms of product identities and determine the root multiplicities of monstrous lie superalgebras

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

applications of a computer implementation of poincare s theorem on fundamental polyhedra poincarcs theorem asserts that a group f of isometries of hyperbolic space h is discrete if its generators act suitably on the boundary of some polyhedron in h and when this happens a presentation of f can be derived from this action we explain methods for deducing the precise hypotheses of the theorem from calculation in f when f is algorithmi cally defined and we describe a file of fortran programs that use these methods for groups f acting on the upper half space model of hyperbolic 3 space h we exhibit one modest example of the application of these programs and we summarize computations of repesenta tions of groups psl 2 0 where 6 is an order in a complex quadratic number field in the early 1880 s h poincare discovered a general theorem allowing one to deduce the discreteness of and a presentation for a group g of isometries of hyperbolic space from its action on a hyperbolic polyhedron under certain condi tions theorems of this sort are part of the foundations of his theories of fuchsian and kleinian groups that have become very popular again and h seifert has recently given us a modern proof of a fairly general version of poincare s theorem in 12 see also 7 this theorem has been little used over the past century perhaps partly because its hypotheses have seemed very difficult to verify for a given group g except in very special circumstances one reason for doubting that poincare s theorem is unreasonably difficult to apply to fairly general discrete groups is that no general alternative method for accomplishing its tasks has been proposed the present paper is devoted to demonstrating that poincare s theorem can indeed be applied to given groups in apparently difficult cases and that much of the work can be done by a computer our experience suggests that the theorem is really very helpful in guiding the user to an understanding of the details of the action of g starting from a state of near ignorance an outline of this paper is as follows in section 1 we begin by stating seifert s version of poincare s theorem and explaining how we would apply it to an algorithmically defined group g of isometries of hyperbolic space h o in section 2 we specialize to the situation 00 for which we wrote our file poincare of fortran

a character formula for generalized kac moody superalgebras abstract in 1 borcherds defined generalized kac moody denoted by gkm for short algebras and gave a character formula for lowest weight modules of these algebras in this paper we give a character formula for lowest weight modules of gkm superalgebras for any standard result about contragredient superalgebras that have an immediate generalization to gkm superalgebras see 3 and 4 all vector spaces are assumed to be over the real field

siegel automorphic form corrections of some lorentzian kac moody lie algebras we find automorphic form corrections which are generalized lorentzian kac moody superalgebras without odd real simple roots for two elliptic lorentzian kac moody algebras of rank 3 with a lattice weyl vector and calculate multiplicities of their simple and arbitrary roots these kac moody algebras are defined by hyperbolic symmetrized generalized cartan matrices inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of rank 3 both these algebras have elliptic type i e their weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces and have a lattice weyl vector the correcting automorphic forms are siegel modular forms the form corresponding to g 1 is the classical siegel cusp form of weight 5 which is the product of ten even theta constants in particular we find an infinite product formula for this modular form

stability problem of ulam for generalized forms of cauchy functional equation abstract let g 1 be a vector space and g 2 a banach space in this paper we solve the generalized hyers ulam rassias stability problem for a generalized form g i 1 n a i x i c i 1 n a i g x i c x i g 1 of cauchy functional equation f x y f x f y for a mapping f g 1 g 2 where a i a i are scalars for all i 1 n and c g 1 c g 2 are vectors

automorphic correction of the hyperbolic kac moody algebra e10 in this paper we study automorphic correction of the hyperbolic kac moody algebra e10 using the borcherds product for o 10 2 attached to a weakly holomorphic modular form of weight 4 for sl2 z we also clarify some aspects of automorphic correction for lorentzian kac moody algebras and give heuristic reasons for the expectation that every lorentzian kac moody algebra has an automorphic correction

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

computing arithmetic kleinian groups arithmetic kleinian groups are arithmetic lattices in psl 2 c we present an algorithm which given such a group gamma returns a fundamental domain and a finite presentation for gamma with a computable isomorphism

presentations of the unit group of an order in a non split quaternion algebra abstract we give an algorithm to determine a finite set of generators of the unit group of an order in a non split classical quaternion algebra h k over an imaginary quadratic extension k of the rationals we then apply this method to obtain a presentation for the unit group of h z 1 7 2 as a consequence a presentation is discovered for the orthogonal group so 3 z 1 7 2 these results provide the first examples of a characterization of the unit group of some group rings that have an epimorphic image that is an order in a non commutative division algebra that is not a totally definite quaternion algebra

the arithmetic mirror symmetry and calabi yau manifolds we extend our variant of mirror symmetry for k3 surfaces gn3 and clarify its relation with mirror symmetry for calabi yau manifolds we introduce two classes for the models a and b of calabi yau manifolds fibrated by k3 surfaces with some special picard lattices these two classes are related with automorphic forms on iv type domains which we studied in our papers gn1 gn6 conjecturally these automorphic forms take part in the quantum intersection pairing for model a yukawa coupling for model b and mirror symmetry between these two classes of calabi yau manifolds recently there were several papers by physicists where it was shown on some examples we propose a problem of classification of introduced calabi yau manifolds our papers gn1 gn6 and n3 n14 give hope that this is possible they describe possible picard or transcendental lattices of general k3 fibers of the calabi yau manifolds

the fake monster superalgebra abstract we show that the physical states of a 10 dimensional superstring moving on a torus form a generalized kac moody superalgebra this gives the first explicit realizations of these algebras for a special torus the denominator function of this algebra is an automorphic form so that we can determine the simple roots we call this algebra the fake monster superalgebra

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

sturm bounds for siegel modular forms of degree 2 and odd weights we correct the proof of the theorem in the previous paper presented by kikuta which concerns sturm bounds for siegel modular forms of degree 2 and of even weights modulo a prime number dividing 2 cdot 3 we give also sturm bounds for them of odd weights for any prime numbers and we prove their sharpness the results cover the case where fourier coefficients are algebraic numbers

generalized moonshine ii borcherds products the goal of this paper is to construct infinite dimensional lie algebras by using infinite product identities and to use these lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and lie algebras we expect the lie algebras that we construct to manifest as algebras of physical states in an orbifold conformal field theory yet to be fully constructed with symmetries given by the monster simple group we introduce vector valued modular functions attached to families of modular functions of different levels and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half planes we recast this result using the borcherds harvey moore singular theta lift and show that the vector valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps for each element of the monster simple group we construct an infinite dimensional lie algebra such that its denominator formula is an infinite product expansion of the automorphic function arising from that element s mckay thompson series these lie algebras have the unusual property that their simple roots and all root multiplicities are known we show that under certain hypotheses characters of groups acting on these lie algebras form functions on the upper half plane that are either constant or invariant under a genus zero congruence group