heights of heegner cycles and derivatives of l series 0 introduction 99 1 intersections and heights 103 2 kuga sato varieties and cm cycles 117 3 heights of cm cycles 124 4 heights of heegner cycles 137 5 proof of the main identity and the consequences 147 references 151

analytic torsion and holomorphic determinant bundles in this paper we prove that in the case of holomorphic locally kahler fibrations the analytic and algebraic geometry constructions of determinant bundles for direct images coincide we calculate the curvature of the holomorphic hermitian connection for the quillen metric on the determinant bundle we study the behavior of the quillen metric under change of metrics in the fibers and also on the twisting vector bundles we thus generalize the conformal anomaly formula to kahler manifolds of arbitrary dimension we also study the quillen metrics on determinants associated with exact sequences of vector bundles we prove that the quillen metric is smooth on the grothendieck knudsen mumford determinant for arbitrary holomorphic fibrations

analytic torsion and holomorphic determinant bundles i bott chern forms and analytic torsion we attach secondary invariants to any acyclic complex of holomorphic hermitian vector bundles on a complex manifold these were first introduced by bott and chern bot c our new definition uses quillen s superconnections we also give an axiomatic characterization of these classes these results will be used in bgs2 and bgs3 to study the determinant of the cohomology of a holomorphic vector bundle

k polystability of q fano varieties admitting kahler einstein metrics it is shown that any possibly singular fano variety x admitting a kahler einstein metric is k polystable thus confirming one direction of the yau tian donaldson conjecture in the setting of mathbb q fano varieties equipped with their anti canonical polarization the proof is based on a new formula expressing the donaldson futaki invariants in terms of the slope of the ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti canonical line bundle of x one consequence is that a toric fano variety x is k polystable iff it is k polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope p associated to x the results also extend to the logarithmic setting and in particular to the setting of kahler einstein metrics with edge cone singularities applications to geodesic stability bounds on the ricci potential and perelman s lambda entropy functional on k unstable fano manifolds are also given

on the turan number for the hexagon abstract a long standing conjecture of erd s and simonovits is that ex n c 2 k the maximum number of edges in an n vertex graph without a 2 k gon is asymptotically 1 2 n 1 1 k as n tends to infinity this was known almost 40 years ago in the case of quadrilaterals in this paper we construct a counterexample to the conjecture in the case of hexagons for infinitely many n we prove that ex n c 6 3 5 2 5 1 4 3 n 4 3 o n 0 5338 n 4 3 we also show that ex n c 6 n 4 3 o n 0 6272 n 4 3 if n is sufficiently large where is the real root of 16 3 4 2 3 0 this yields the best known upper bound for the number of edges in a hexagon free graph the same methods are applied to find a tight bound for the maximum size of a hexagon free 2 n n bipartite graph

a criterion for the properness of the k energy in a general kahler class in this paper we give a criterion for the properness of the k energy in a general kahler class of a compact kahler manifold by using song weinkove s result in commun pure appl math 61 2 210 229 2008 as applications we give some kahler classes on mathbb c mathbb p 2 3 overline mathbb c mathbb p 2 and mathbb c mathbb p 2 8 overline mathbb c mathbb p 2 in which the k energy is proper finally we prove song weinkove s result on the existence of critical points of hat j functional by the continuity method

conditional probability logic lifted bayesian networks and almost sure quantifier elimination abstract we introduce a formal logical language called conditional probability logic cpl which extends first order logic and which can express probabilities conditional probabilities and which can compare conditional probabilities intuitively speaking although formal details are different cpl can express the same kind of statements as some languages which have been considered in the artificial intelligence community we also consider a way of making precise the notion of lifted bayesian network where this notion is a type of lifted probabilistic graphical model used in machine learning data mining and artificial intelligence a lifted bayesian network in the sense defined here determines in a natural way a probability distribution on the set of all structures in the sense of first order logic with a common finite domain d our main result theorem 3 14 is that for every noncritical cpl formula x there is a quantifier free formula x which is almost surely equivalent to x as the cardinality of d tends towards infinity this is relevant for the problem of making probabilistic inferences on large domains d because a the problem of evaluating by brute force the probability of x being true for some sequence d of elements from d has in general highly exponential time complexity in the cardinality of d and b the corresponding probability for the quantifier free x depends only on the lifted bayesian network and not on d some conclusions regarding the computational complexity of finding are given in remark 3 17 the main result has two corollaries one of which is a convergence law and zero one law for noncritial cpl formulas

_ 1 free graphs asymptotic structure and a 0 1 law etude asymptotique de la structure des graphes etiquetes sans k l 1 etablissement de la l colorabilite et de la loi 0 1

independent sets in hypergraphs many important theorems and conjectures in combinatorics such as the the orem of szemeredi on arithmetic progressions and the erd os stone theorem in extremal graph theory can be phrased as statements about families of independent sets in certain uniform hypergraphs in recent years an important trend in the area has been to extend such classical results to the so called sparse random setting this line of research has recently culminated in the breakthroughs of conlon and gowers and of schacht who de veloped general tools for solving problems of this type although these two papers solved very similar sets of longstanding open problems the methods used are very different from one another and have different strengths and weaknesses in this paper we provide a third completely different approach to proving extremal and structural results in sparse random sets that also yields their natural counting coun terparts we give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets we then derive many interesting results as fairly straightforward consequences of this abstract theorem in particular we prove the well known conjecture of kohayakawa luczak and rodl a probabilistic embedding lemma for sparse graphs for all 2 balanced graphs we also give alternative proofs of many of the results of conlon and gowers and schacht such as sparse random versions of szemeredi s theorem the erd os stone theorem and the erd os simonovits stability theorem and ob tain their natural counting versions which in some cases are considerably stronger we also obtain new results such as a sparse version of the erd os frankl rodl theorem on the number of h free graphs and as a consequence of the k lr conjecture we extend a re sult of rodl and rucinski on ramsey properties in sparse random graphs to the general non symmetric setting similar results have been discovered independently by saxton and thomason

hyperdiscriminant polytopes chow polytopes and mabuchi energy asymptotics letx n p n be a smooth linearly normal algebraic variety it is shown that the mabuchi energy of x fsjx restricted to the bergman metrics is completely determined by the x hyperdiscriminant of format n 1 and the chow form of x as a corollary it is shown that the mabuchi energy is bounded from below for all degenerations in g if and only if the hyperdiscriminant polytope dominates the chow polytope for all maximal algebraic tori h of g

a logical approach to asymptotic combinatorics i first order properties we shall present a general framework for dealing with an extensive set of problems from asymptotic combinatorics this framework provides methods for determining the probability that a large finite structure ran domly chosen from a given class will have a given property our main concern is the asymptotic probability the limiting value as the size of the structure increases for example a common problem in elementary probability texts is to show that the asymptotic probabity that a per mutation will have no fixed point is l e we shall say nothing about the closely related problem of determining rates of convergence although the methods presented here may extend to such problems to develop a general approach we must fix a language for specifying properties of structures thus our approach is logical logic is the branch of mathematics that deals with problems of language in this paper we con sider properties expressible in the language of first order logic and speak of probabilities of first order sentences rather than properties in the sequel to this paper we consider properties expressible in the more general language of monadic second order logic clearly we must restrict the classes of structures we consider in order for questions about asymptotic probabilities to be meaningful and significant therefore we choose to consider only classes closed under disjoint unions and components see sect 1 for definitions this condition is general enough to include the classes of graphs permutations unary functions and many other important examples the main theorems of the paper are about classes satisfying this condition theorems 5 7 and 5 8 state that the existence of asymptotic probabilities for a special set of sentences called component bounded sentences defined in sect 5 is equivalent to a con dition on the growth of such a class theorem 5 9 gives necessary and suf ficient conditions for every first order sentence to have an asymptotic probability of 0 or 1 in nonfast growing classes in section 8 we present a long list of examples from the literature

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

polynomial extensions of van der waerden s and szemeredi s theorems an extension of the classical van der waerden and szemeredi the orems is proved for commuting operators whose exponents are polynomials as a consequence for example one obtains the following result let s zl be a set of positive upper banach density let p1 n pk n be polynomials with rational coefficients taking integer values on the integers and satisfying pi 0 0 i 1 k then for any v1 vk zl there exist an integer n and a vector u zl such that u pi n vi s for each i k department of mathematics ohio state university columbus ohio 43210 e mail address vitaly math ohio state edu department of mathematics technion haifa 23000 israel e mail address sashal techunix technion ac il current address department of mathematics stanford university stanford california 94305 e mail address leibman math stanford edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

heights of kudla rapoport divisors and derivatives of l functions we study special cycles on integral models of shimura varieties associated with unitary similitude groups of signature n 1 1 we construct an arithmetic theta lift from harmonic maass forms of weight 2 n to the arithmetic chow group of the integral model of a unitary shimura variety by associating to a harmonic maass form f a linear combination of kudla rapoport divisors equipped with the green function given by the regularized theta lift of f our main result is an equality of two complex numbers 1 the height pairing of the arithmetic theta lift of f against a cm cycle and 2 the central derivative of the convolution l function of a weight n cusp form depending on f and the theta function of a positive definite hermitian lattice of rank n 1 when specialized to the case n 2 this result can be viewed as a variant of the gross zagier formula for shimura curves associated to unitary groups of signature 1 1 the proof relies on among other things a new method for computing improper arithmetic intersections

the difference between consecutive primes ii with enough effort the value of x0 could be determined effectively the paper has much in common with 1 in particular we use the sieve method of harman 4 5 we no longer use zero density estimates however but rather mean value results on dirichlet polynomials similar to those that give rise to such estimates compare for example work of iwaniec and pintz 9 and baker harman and pintz 2 much of the improvement over 1 arises from the use of watt s theorem 11 on a particular kind of mean value more accurate estimates for six dimensional integrals are also used to good effect there is in addition a device which uses a two dimensional sieve to get an asymptotic formula for a one dimensionally sieved set see lemmas 16 17 unfortunately these lemmas which would be of great signi cance for v 0 53 are not very numerically signi cant when v drops to 0 525 the same applies to the roale reversals discussed below let us introduce enough notation to permit an outline of the proof when e is a nite sequence of positive integers counted with multiplicity we write jej for the number of terms of e and ed fm dm 2 eg let p z y

geometric analysis of chow mumford stability abstract in this paper we relate the chow mumford stability of a projective manifold x to the properness of a certain energy functional on the subspace of bergman metrics in the kahler cone of x we use this relationship to rederive the stability of smooth curves over c this is accomplished by analyzing the asymptotics of some rational integrals

tools for template dependencies template dependencies td s are a class of data dependencies that include multivalued and join dependencies and embedded versions of these a collection of techniques examples and results about td s are presented the principal results are 1 finite implication implication over relations with a finite number of tuples is distinct from unrestricted implication for td s 2 there are for td s over three or more attributes infinite chains of increasingly weaker and increasingly stronger full td s 3 however there are weakest nontrivial and strongest full td s over any given set of attributes 4 over two attributes there are only three distinct td s 5 there is no weakest not necessarily full td over any set of three or more attributes 6 there is a finite relation that obeys every strictly partial td but no full td 7 the conjunction of each finite set of full td s is equivalent to a single full td however the conjunction of a finite set of not necessarily full td s is not necessarily e

reasoning about functional and full hierarchical dependencies over partial relations we study the implication problem for the combined class of functional and full hierarchical dependencies in the presence of sql s not null constraints two different notions of implication are addressed one where a dependency is implied by the given set of dependencies plus the underlying schema and one where a dependency is implied by the given set of dependencies alone we establish axiomatizations for both notions of implication and reveal deep relationships between them

a new proof of the graph removal lemma let h be a xed graph with h vertices the graph removal lemma states that every graph on n vertices with o n h copies of h can be made h free by removing o n 2 edges we give a new proof which avoids szemer edi s regularity lemma and gives a better bound this approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma this answers questions of alon and gowers

membership problem for embedded multivalued dependencies under some restricted conditions abstract it is known that there is no finite set of inference rules that is complete for embedded multivalued dependencies this paper considers the problem called membership problem of deciding whether a given embedded multivalued dependency x v on u or functional dependency x v is implied by a given set d of functional and embedded multivalued dependencies a new inference rule for embedded multivalued dependencies is presented and the following results about membership problems are shown let u be a set of attributes and let w n w 0 u let d f e e 1 e n where 1 i f is a set of functional dependencies y z satisfying y w 0 or z w 0 2 ii e is a set of embedded multivalued dependencies y z on t satisfying at least one of y w 0 t z w 0 and t y z w 0 and 3 iii each e 1 i n is a set of embedded multivalued dependencies on w 1 1 if x v u w 0 then the membership problem is solaable that is it is decidable whether x v on u or x v is implied by d 2 2 if x v u w n then the membership problem cah be solved in o d v x time where d is the size of the description of d 3 3 if x w n and x v u w 0 then the membership problem can be solved in o d 2 w n x n t 1 w 1 w 1 l 1 time

extremal results for random discrete structures we study thresholds for extremal properties of random discrete structures we determine the threshold for szemer edi s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we determine the threshold for tur an type problems for random graphs and hypergraphs in particular we verify a conjecture of

kahler einstein metrics and the generalized futaki invariant in 1983 futaki introduced his famous invariant this invariant generalizes the obstruction of kazdan warner to prescribing gauss curvature on s 2 cf ful l the futaki invariant is defined for any compact k ihler manifold with positive first chern class that has nontrivial holomorphic vector fields it is a lie algebraic character from the lie algebra of holomorphic vector fields into and its vanishing is a necessary condition for the existence of a k ihler einstein metric on the underlying manifold therefore it can be used to test the existence of k ihlereinstein metrics on a given compact k ihler manifold with positive first chern class an excellent reference on the futaki invariant is futaki s book fu2 until now all known nontrivial obstructions to k ihler einstein metrics come from holomorphic vector fields this suggests the following conjecture

regularity of kahler ricci flows on fano manifolds in this paper we will establish a regularity theory for the kahler ricci flow on fano n manifolds with ricci curvature bounded in l p norm for some p n using this regularity theory we will also solve a long standing conjecture for dimension 3 as an application we give a new proof of the yau tian donaldson conjecture for fano 3 manifolds the results have been announced in 45

the number of c2l free graphs abstract one of the most basic questions one can ask about a graph h is how many h free graphs on n vertices are there for non bipartite h the answer to this question has been well understood since 1986 when erd s frankl and rodl proved that there are 2 1 o 1 ex n h such graphs for bipartite graphs however much less is known even the weaker bound 2 o ex n h has been proven in only a few special cases for cycles of length four and six and for some complete bipartite graphs for even cycles bondy and simonovits proved in the 1970s that ex n c 2 l o n 1 1 l and this bound is conjectured to be sharp up to the implicit constant in this paper we prove that the number of c 2 l free graphs on n vertices is at most 2 o n 1 1 l confirming a conjecture of erd s our proof uses the hypergraph container method which was developed recently and independently by balogh morris and samotij and by saxton and thomason together with a new balanced supersaturation theorem for even cycles we moreover show that there are at least 2 1 c ex n c 6 c 6 free graphs with n vertices for some c 0 and infinitely many values of n n disproving a well known and natural conjecture as a further application of our method we essentially resolve the so called turan problem on the erd s renyi random graph g n p for both even cycles and complete bipartite graphs

an equivalence between relational database dependencies and a fragment of propositional logic it is known that there is an eqmvalence between functional dependencies m a relatmonal database and a certain fragment of proposmonal logic thins eqmvalence is extended to include both functional and multivalued dependencmes thus for each dependency there is a corresponding statement m proposmonal logic it ms then shown that a dependency funcuonal or multivalued is a consequence of a set of dependencies ff and only ff the corresponding proposiuonal statement s a consequence of the corresponding set of proposmonal statements examples are given to show that these techniques are valuable mn provmdmg much shorter proofs of theorems about dependencies than have been obtained by more tradmonal means it is shown that this eqmvalence cannot be extended to include either join dependencies or embedded multmvalued dependencies

a proof procedure for data dependencies definition d une classe de dependances dependances generant l equalite et les triples et generalisation du processus de poursuite pour traiter ces dependances on montre quelques cas pour lesquels il y a une procedure de decision redondance de l egalite pour decider de l implication des dependances generant les triples et presque redondance pour decider de l implication des dependances generant l egalite

space of ricci flows i based on the compactness of the moduli of non collapsed calabi yau spaces with mild singularities we set up a structure theory for polarized kahler ricci flows with proper geo metric bounds our theory is a generalization of the structure theory of non collapsed kahler einstein manifolds as applications we prove the hamilton tian conjecture and the partial c 0 conjecture of tian

sort sets in the relational model the notion of sort set is introduced here to formalize the fact that certain database relations can be sorted so that two or more columns are simultaneously listed in order this notion is shown to be applicable in several ways to enhance the efficiency of an implemented database a characterization of when order dependency implies the existence of sort sets in a database is presented along with several corollaries concerning complexity armstrong relations and cliques of certain graphs sort set dependencies are then introduced a finite sound and complete set of inference rules for sort set dependencies is presented as well as a proof that there is no such set for functional and sort set dependencies taken together deciding logical implication for sort set dependencies is proved to be polynomial but if functional dependencies are included the problem is co np complete each set of sort set and functional dependencies is shown to have an armstrong relation a natural generalization of armstrong relation here called separator is given and then used to study the relationship between order and sort set dependencies

functional dependencies and constraints on null values in database relations database relations with incomplete information are considered the no information interpretation of null values is adopted due to its characteristics of generality and naturalness coherently with the framework and its motivation two meaningful classes of integrity constraints are studied a functional dependencies which have been widely investigated in the classical relational theory and b constraints on null values which control the presence of nulls in the relations specifically three types of constraints on null values are taken into account nullfree subschemes existence constraints disjunctive existence constraints and the interaction of each of them with functional dependencies is studied in each of the three cases the inference problem is solved the complexity of the algorithms for its solution analyzed and the existence of a complete axiomatization discussed

on the undecidability of implications between embedded multivalued database dependencies the implication and finite implication problems for embedded multivalued database dependencies are both shown to be algorithmically undecidable the proof is by an interpretation of semigroup word problems via systems of permuting equivalence relations into database dependencies in contrast it is shown that for each fixed premise h one has a decision procedure for implications h f

inferences for numerical dependencies abstract we introduce and motivate the study of numerical dependencies which are a generalization of functional dependencies we prove that there does not exist a finite set of sound and complete inference rules for numerical dependencies in contrast to the case of functional dependencies we also prove that nontrivial numerical dependencies which are not functional dependencies cannot be expressed by horn formulas in first order logic and show some applications of numerical dependencies

combinatorial theorems in sparse random sets we develop a new technique that allows us to show in a unified way that many well known combinatorial theorems including turan s theorem szemeredi s theorem and ramsey s theorem hold almost surely inside sparse random sets for instance we extend turan s theorem to the random setting by showing that for every 0 and every positive integer t 3 there exists a constant c such that if g is a random graph on n vertices where each edge is chosen independently with probability at least cn 2 t 1 then with probability tending to 1 as n tends to infinity every subgraph of g with at least 1 1 t 1 e g edges contains a copy of kt this is sharp up to the constant c we also show how to prove sparse analogues of structural results giving two main applications a stability version of the random turan theorem stated above and a sparse hypergraph removal lemma many similar results have recently been obtained independently in a different way by schacht and by friedgut rodl and schacht

hypergraph regularity and the multidimensional szemeredi theorem we prove analogues for hypergraphs of szemeredi s regularity lemma and the associated counting lemma for graphs as an application we give the first combinatorial proof of the multidimensional szemeredi theorem of furstenberg and katznelson and the first proof that provides an explicit bound similar results with the same consequences have been obtained independently by nagle rodl schacht and skokan

axioms and algorithms for inferences involving probabilistic independence abstract this paper offers an axiomatic characterization of the probabilistic relation x is independent of y written x y where x and y are two disjoint sets of variables four axioms for x y are presented and shown to be complete based on these axioms a polynomial membership algorithm is developed to decide whether any given independence statement x y logically follows from a set of such statements i e whether x y holds in every probability distribution that satisfies the complexity of the algorithm is o k2 n where is the number of given statements n is the number of variables in x y and k is the number of variables in x y

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

k stability and kahler einstein metrics we prove that if a fano manifold m is k stable then it admits a kahler einstein metric it affirms a longstanding conjecture for fano manifolds 2015 wiley periodicals inc

conical kahler einstein metrics revisited in this paper we introduce the interpolation degeneration strategy to study kahler einstein metrics on a smooth fano manifold with cone singularities along a smooth divisor that is proportional to the anti canonical divisor by interpolation we show the angles in 0 2 that admit a conical kahler einstein metric form a connected interval and by degeneration we determine the boundary of the interval in some important cases as a first application we show that there exists a kahler einstein metric on mathbb p 2 with cone singularity along a smooth conic degree 2 curve if and only if the angle is in 2 2 when the angle is 2 3 this proves the existence of a sasaki einstein metric on the link of a three dimensional a2 singularity and thus answers a question posed by gauntlett martelli sparks yau as a second application we prove a version of donaldson s conjecture about conical kahler einstein metrics in the toric case using song wang s recent existence result of toric invariant conical kahler einstein metrics

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

calabi flow geodesic rays and uniqueness of constant scalar curvature kahler metrics we prove that constant scalar curvature kahler metric adjacent to a fixed kahler class is unique up to isomorphism the proof is based on the study of a fourth order evolution equation namely the calabi flow from a new geometric perspective and on the geometry of the space of kahler metrics

a refinement of the cameron erdős conjecture in this paper we study sum free subsets of the set 1 n that is subsets of the first n positive integers which contain no solution to the equation x y z and erdos conjectured in 1990 that the lumber of such sets is o 2 n 2 this conjecture was confirmed by green and independently by sapozhenko here we prove a refined version of their theorem by showing that the number of sum free subsets of n of size in is 2 o n m n 2 for every 1 to 17 21 for m root t this result is sharp up to the constant implicit in the oh our proof uses a general bound on the number of independent sets of size to in 3 uniform hypergraphs proved recently by the authors and new bounds on the number of integer partitions with small sumset

a new series of dense graphs of high girth let k 1 be an odd integer t b k 2 4 c and q be a prime power we construct a bipartite q regular edge transitive graph cd k q of order v 2qk t 1 and girth g k 5 if e is the the number of edges of cd k q then e v 1 k t 1 these graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g g 5 g 6 11 12 for g 24 this represents a slight improvement on bounds established by margulis and lubotzky phillips sarnak for 5 g 23 g 6 11 12 it improves on or ties existing bounds