configurations in abelian categories ii ringel hall algebras abstract this is the second in a series on configurations in an abelian category a given a finite poset i an i configuration is a finite collection of objects j and morphisms j k or j k j k in a satisfying some axioms where j k i configurations describe how an object x in a decomposes into subobjects the first paper defined configurations and studied moduli spaces of i configurations in a using the theory of artin stacks it showed well behaved moduli stacks obj a m i a of objects and configurations in a exist when a is the abelian category coh p of coherent sheaves on a projective scheme p or mod k q of representations of a quiver q write cf obj a for the vector space of q valued constructible functions on the stack obj a motivated by the idea of ringel hall algebras we define an associative multiplication on cf obj a using pushforwards and pullbacks along 1 morphisms between configuration moduli stacks so that cf obj a is a q algebra we also study representations of cf obj a the lie subalgebra cf ind obj a of functions supported on indecomposables and other algebraic structures on cf obj a then we generalize all these ideas to stack functions sf obj a a universal generalization of constructible functions containing more information when ext i x y 0 for all x y a and i 1 or when a coh p for p a calabi yau 3 fold we construct lie algebra morphisms from stack algebras to explicit algebras which will be important in the sequels on invariants counting semistable objects in a

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

secondary terms in counting functions for cubic fields we prove the existence of secondary terms of order x5 6 in the davenport heilbronn theorems on cubic fields and 3 torsion in class groups of quadratic fields for cubic fields this confirms a conjecture of datskovsky wright and roberts we also prove a variety of generalizations including to arithmetic progressions where we discover a curious bias in the secondary term roberts s conjecture has also been proved independently by bhargava shankar and tsimerman in contrast to their work our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms developed by shintani and datskovsky wright

the period index problem for fields of transcendence degree 2 using geometric methods we prove the standard period index conjecture for the brauer group of a field of transcendence degree 2 over fp

on the motivic class of the stack of bundles let g be a split connected semisimple group over a field we give a conjectural formula for the motivic class of the stack of g bundles over a curve c in terms of special values of the motivic zeta function of c the formula is true if or g sln if upon applying the poincare or called the serre characteristic by some authors the formula reduces to results of teleman and atiyah bott on the gauge group if upon applying the counting measure it reduces to the fact that the tamagawa number of g over the function field of c is p1 g

nonvanishing of quadratic twists of modular l functions and applications to elliptic curves if d is square free or is a fundamental discriminant then let d d denote the kronecker character for the quadratic field q d whose fundamental discriminant is d throughout d shall denote a fundamental discriminant the d quadratic twist of f denoted f d is the newform corresponding to the twist of f by the character d in particular if gcd m d 1 then f d z n 1 d n a n q n and

localization in equivariant intersection theory and the bott residue formula we prove the localization theorem for torus actions in equivariant intersection theory using the theorem we give another proof of the bott residue formula for chern numbers of bundles on smooth complete varieties in addition our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions this class is rather special but it includes some interesting examples such as complete intersections and schubert varieties 1 introduction the purpose of this paper is to prove the localization theorem for torus actions in equivariant intersection theory using the theorem we give another proof of the bott residue formula for chern numbers of bundles on smooth complete varieties in addition our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions this class is rather special but it includes some interesting examples such as complete intersections cf bfq and schubert varieties let t be a split torus acting on a scheme x the 7 equivariant chow groups of x are a module over rt sym t where f is the character group of t the localization theorem states that up to torsion the equivariant chow groups of the fixed locus xt are isomorphic to those of x such a theorem is a hallmark of any equivariant theory the earliest version for equivariant cohomology is due to borel bo subsequently it theory versions were proved by segal se in topological a theory quart qu for actions of a cyclic group and thomason th2 for algebraic theory th2 for equivariant chow groups the localization isomorphism is given by the equivariant push forward induced by the inclusion of xt to x an interesting aspect of this theory is that the push forward is naturally defined on the level of cycles even in the singular case the closest topological analogue of this is equivariant borel moore homology see e g3 for a definition and a similar proof establishes localization in that theory for smooth spaces the inverse to the equivariant push forward can be written explicitly it was realized independently by several authors i n a b b v

rank one quadratic twists of an infinite family of elliptic curves a conjecture of goldfeld implies that a positive proportion of quadratic twists of an elliptic curve e q has analytic rank 1 this assertion has been confirmed by vatsal v1 and the first author by for only two elliptic curves here we confirm this assertion for infinitely many elliptic curves e q using the heegner divisors the 3 part of the class groups of quadratic fields and a variant of the binary goldbach problem for polynomials

rank one twists of a certain elliptic curve the purpose of this note is to give the first known example for a conjecture of goldfeld on the number of rank one curves appearing in a family of quadratic twists we show unconditionally that the curve x0 19 has the property that a positive proportion of its quadratic twists have analytic rank one this amounts to a strong nonvanishing statement for the derivatives of certain l functions when the sign of the functional equation is 1 we simultaneously obtain the fact that a positive proportion of twists have analytic rank zero when the sign of the functional equation is 1 let e be a modular elliptic curve over q with corresponding newform f z anq n s2 0 n let l s f l s e ann s be the usual l series then l s f satisfies a functional equation under s 2 s with sign 1 e let d be the fundamental discriminant of a quadratic field k k d and write d for the associated quadratic character in this setting we define the twisted l function l s fd d n ann if e is given by the equation y 2 p x then l s fd is the l function attached to the twisted elliptic curve ed given by dy 2 p x therefore l s fd also satisfies a functional equation and if d n 1 then the sign is given by d d n for r 0 1 and a positive real number x define m r f x d d x ords 1l s fd r the following conjecture of goldfeld gol79 is well known

donaldson thomas invariants of 2 dimensional sheaves inside threefolds and modular forms abstract motivated by the s duality conjecture we study the donaldson thomas invariants of the 2 dimensional gieseker stable sheaves on a threefold these sheaves are supported on the fibers of a nonsingular threefold x fibered over a nonsingular curve in the case where x is a k3 fibration we express these invariants in terms of the euler characteristic of the hilbert scheme of points on the k3 fiber and the noether lefschetz numbers of the fibration we prove that a certain generating function of these invariants is a vector modular form of weight 3 2 as predicted in s duality

an introduction to motivic hall algebras abstract we give an introduction to joyce s construction of the motivic hall algebra of coherent sheaves on a variety m when m is a calabi yau threefold we define a semi classical integration map from a poisson subalgebra of this hall algebra to the ring of functions on a symplectic torus this material will be used in bridgeland 2011 3 to prove some basic properties of donaldson thomas curve counting invariants on calabi yau threefolds

levels of positive definite ternary quadratic forms the level n and squarefree character q of a positive definite ternary quadratic form are defined so that its associated modular form has level n and character xq we define a collection of correspondences between classes of quadratic forms having the same level and different discriminants this makes practical a method for finding representatives of all classes of ternary forms having a given level we also give a formula for the number of genera of ternary forms with a given level and character

canonical rational equivalence of intersections of divisors one way to define an operation in intersection theory is to define a map on the group of algebraic cycles together with a map on the group of rational equivalences which commutes with the boundary operation assuming the maps commute with smooth pullback the extension of the operation to the setting of algebraic stacks is automatic the goal of section 2 of this paper is to present the operation of intersecting with a principal cartier divisor in this light we then show how this operation lets us obtain a rational equivalence which is fundamental to intersection theory a one dimensional family of cycles on an algebraic variety always admits a unique limiting cycle but a family of cycles over the punctured affine plane may yield different limiting cycles if one approaches the origin from different directions an important step in the historical development of intersection theory was realizing how to prove that any two such limiting cycles are rationally equivalent the results of section 2 yield as a corollary a new explicit formula for this rational equivalence another important rational equivalence in intersection theory is the one that is used to demonstrate commutativity of gysin maps associated to regularly embedded subschemes in section 3 we exhibit a two dimensional family of cycles such that the cycles we obtain from specializing in two different ways are precisely the ones we need to show to be rationally equivalent to obtain the commutativity result our explicit rational equivalence respects smooth pullback and hence the generalization to stacks is automatic this simplifies intersection theory on deligne mumford stacks as in 8 where construction of such a rational equivalence fills the most difficult section of that important paper since our rational equivalence arises by considering families of cycles on a larger total space we are able to deduce section 4 that the rational equivalence is invariant under a certain naturally arising group action the key observation is that we can manipulate the situation so that the group action extends to the total space this equivariance result is used but appears with mistaken proof in 2 where an important new tool of modern intersection theory the theory of virtual fundamental classes is developed acknowledgement the author would like to thank s bloch w fulton t graber and r pandharipande for helpful advice and the organizers and staff of the mittagleffler institute for hospitality during the 1996 97 program in algebraic geometry

the integral orbifold chow ring of toric deligne mumford stacks in this paper we study the integral chow ring of toric deligne mumford stacks we prove that the integral chow ring of a semi projective toric deligne mumford stack is isomorphic to the stanley reisner ring of the associated stacky fan the integral orbifold chow ring is also computed our results are illustrated with several examples

cycle groups for artin stacks we construct an algebraic homology functor for artin stacks of finite type over a field and we develop intersection theoretic properties

gromov witten theory of deligne mumford stacks given a smooth complex deligne mumford stack cal x with a projective coarse moduli space we introduce gromov witten invariants of cal x and prove some of their basic properties including the wdvv equation

witten s top chern class via cosection localization for a landau ginzburg space mathbb c n g w we construct witten s top chern class as an algebraic cycle using cosection localized virtual cycles in the case where all sectors are narrow verify all axioms of this class and derive an explicit formula for it in the free case we prove that this construction is equivalent to the constructions of polishchuk vaintrob chiodo and fan jarvis ruan

the size of selmer groups for the congruent number problem where np is the number of solutions of the congruence y x d2x mod p then ld s has an analytic continuation as an entire function on the complex plane the conjecture of birch and swinnerton dyer then states in particular that the rank r d of ed is equal to the order r d of ld s at s 1 this being the so called analytic rank while we cannot at present find r d in all cases we can at least determine whether or not ld 1 0 and hence conjecturally whether or not r d 0 moreover one has a functional equation for ld s which relates its values at s and 2 s via a sign change ed 1 one may deduce that 1 r d ed it would then follow from the conjecture of birch and swinnerton dyer that the rank is positive whenever ed 1 according to the calculations of birch and stephens 2 one has

nonvanishing of l series associated to cubic twists of elliptic curves has a rational solution an extensive compilation of the older history of the problem is given in dickson 6 within the past century researchers have tried to exploit either explicitly or unknowingly the fact that the curve 0 1 is in fact an elliptic curve during the nineteenth century lucas and later sylvester used a descent argument to prove that 0 1 had no solution for infinitely many d in certain congruence classes mod 9 and 18 see 6 ch xxi zagier and kramarz 19 have produced a great deal of numerical evidence about the l series of the curves based on these computations they have argued heuristically that for

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

l series with nonzero central critical value suppose that f zn 1 an f qtn is a cusp form of weight 2k k e n we denote by l f s the l function of f for re s sufficiently large the value of l f s is given by l f s en 1 an and one can show that l f s has analytic continuation to the entire complex plane the value of l f s at s k will be of particular interest to us and we will refer to this value as the central critical value of l f s let xd denote the dirichlet character associated to the extension q d q that is xd n va where ad denotes the discriminant of q d q define the dth quadratic twist of f to be fxd en 1 an f xd n qn for any integer d the l function of fx is the dth quadratic twist of l f s that is l fxd s zn aan f xd n we will be interested in determining how often l fxd s has nonzero central critical value as d varies over all integers since xd 2 xd we will restrict our attention to the square free integers d we expect that as we let d vary over all of the square free integers a positive proportion of the l functions l fxd s will have nonzero central critical value indeed goldfeld 7 conjectures that for newforms f of weight 2 l fxd 1 78 0 for 1 of the square free integers given an elliptic curve e y2 x3 ax2 bx c a b c e z with conductor ne and an integer d we define the dth quadratic twist of e to be the curve ed y2 x3 adx2 bd2x cd3 let l ed s denote the l function associated to ed for square free d coprime to 2ne l ed s is simply the dth quadratic twist of l ei s if f e s2 n is a newform with integer coefecients we know via the theory of eichler and shimura that there is an elliptic curve e over q having conductor n so that l e s l f s thus if d is coprime to 2n then l ed s l fxd s also one knows from the work of kolyvagin 13 as supplemented by the work of murty and murty 17 or that of bump friedberg and hoffstein 3 see also 10 for a shorter proof that if e is a modular elliptic curve and if l e 1 7 0 then the rank of e is 0 thus if f has the property that a positive proportion of the twists of l f s have nonzero central critical value then this implies that a positive density of the quadratic twists ed have rank 0 there have been many papers which have proved results in this direction for example in 2 3 6 10 16 17 19 28 one can find general theorems on

canonical periods and congruence formulae the purpose of this article is to show how congruences between the fourier coefficients of hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated l functions this study was initiated by b mazur in his fundamental work on the eisenstein ideal see maz77 and maz79 where it was made clear that congruences for analytic l values were closely related to the integral structure of certain hecke rings and cohomology groups the results of maz79 also showed that congruences were useful in the study of nonvanishing of l functions this idea was then further developed by stevens ste82 and rubin wiles rw82 the work of rubin and wiles in particular used congruences to study the behavior of elliptic curves in towers of cyclotomic fields a key ingredient here was a theorem of washington which states roughly that almost l values in certain families are nonzero modulo p this theme has recently been taken up again in the work of ono skinner osa osb james jam and kohnen koh97 while the earlier history was primarily concerned with cyclotomic twists the current emphasis is on families of twists by quadratic characters here one wants quantitative estimates for the number of quadratic twists of a given modular form which have nonvanishing l function at s 1 we continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three this generalizes a beautiful example due to kevin james and provides new evidence for a conjecture of goldfeld gol79 it should however be pointed out that even the study of quadratic twists may be traced back to mazur the reader is urged to look at pages 212 213 of maz79 and especially at the footnote at the bottom of page 213 the theorems of davenport heilbronn dh71 and washington was78 which are crucial in this paper are both mentioned in mazur s article we want to begin by discussing the congruences that lie at the heart of this article thus let f anq n be an elliptic modular cuspform of level m and weight k 2 assume that f is a simultaneous eigenform for all the hecke operators and that a1 f 1 the l function associated to f is defined by the dirichlet series l s f ann s which converges for the real part of s sufficiently large and has analytic continuation to s c a fundamental theorem of shimura shi76 states that l s f enjoys the following algebraicity property

collectivity and geometry iv sp 6 sp 2 o 3 basis states for open shells using generating function methods branching rules for sp 6 sp 2 o 3 are derived the branching rules suggest an integrity basis or set of elementary permissible diagrams in terms of which the subgroup basis states are defined they correspond to vibrational or bohr mottelson type states in the nuclear symplectic model