optimal mock jacobi theta functions abstract we classify the optimal mock jacobi forms of weight one with rational coefficients the space they span is thirty four dimensional and admits a distinguished basis parameterized by genus zero groups of isometries of the hyperbolic plane we show that their fourier coefficients can be expressed explicitly in terms of singular moduli and obtain positivity conditions which distinguish the optimal mock jacobi forms that appear in umbral moonshine we find that many of ramanujan s mock theta functions can be expressed simply in terms of the optimal mock jacobi forms with rational coefficients

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

on the non vanishing of cubic twists of automorphic l series let f be a normalised new form of weight 2 for ro n over q and f its base change lift to q v a sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the l function of f there is an algorithm to check the condition for any given form the new form of level 11 is used to illustrate our method 0 introduction there has been a tremendous amount of recent research on the nonvanishing of l functions on gl 2 the first results are due to shimura 15 who proved that a given l function can be twisted by a character of finite order so that the twisted l function does not vanish at a certain point shimura s nonvanishing results were generalized by rohrlich 14 the study of the nonvanishing of quadratic twists rather than arbitrary finite order twists was started by goldfeld hoffstein and patterson 7 for the cm case and in greater generality by waldspurger 16 this was later complemented by the work of bump friedberg and hoffstein 2 3 5 using metaplectic eisenstein series and the rankin selberg method alternately analytic number theoretic methods have been used by murty and murty 12 and by iwaniec 10 to obtain nonvanishing results the first results on the non vanishing of cubic twists were obtained by lieman 11 he applied the theory of automorphic forms on the cubic cover of gl 3 to the l series of the cm elliptic curve x3 y3 1 in this way he obtained a non vanishing result for cubic twists of the l series of the automorphic form corresponding to the curve however because the curve has complex multiplication the l series in this instance is a gl 1 hecke l series with grossencharacter it is this particular fact which made the gl 3 theory applicable in this paper we will give the first non vanishing results for cubic twists of autmorphic l series on gl 2 that are not lifts from gl 1 in particular we will show that if f is the new form of weight 2 and level 11 then infinitely many cubic twists of the l series of f are non zero at the center of the critical strip it then follows as an immediate corollary that there are infinitely many cubic extensions k of q vz w such that the analytic rank of e xo 11 over k is zero our approach gives a method of obtaining a similar result for any given gl 2 automorphic form however for reasons that will be described below there is one step in the computation which can be verified for any given form but which cannot yet be done in general received by the editors september 27 1996 and in revised form february 14 1997 1991 mathematics subject classification primary 11f66 secondary 11f70 11m41 11n75 d1999 american mathematical society 1075 this content downloaded from 157 55 39 145 on sat 02 jul 2016 05 14 40 utc all use subject to http about jstor org terms

extending the bruhat order and the length function from the weyl group to the weyl monoid abstract for a symmetrizable kac moody algebra the category of admissible representations of the category o is an analogue of the category of finite dimensional representations of a semisimple lie algebra the monoid which is associated to this category and the category of restricted duals by a tannaka krein reconstruction contains the kac moody group as open dense unit group it has similar properties as a reductive algebraic monoid in particular there are bruhat and birkhoff decompositions the weyl group replaced by the weyl monoid c mokler an analogue of a reductive algebraic monoid whose unit group is a kac moody group arxiv math ag 0204246 mem amer math soc in press we determine the closure relations of the bruhat and birkhoff cells which give extensions of the bruhat order from the weyl group to the weyl monoid we show that the bruhat and birkhoff cells are irreducible and principal open in their closures we give product decompositions of the bruhat and birkhoff cells we define extended length functions which are compatible with the extended bruhat orders we show a generalization of some of the tits axioms for twin bn pairs

p converse to a theorem of gross zagier kolyvagin and rubin let e be a cm elliptic curve over the rationals and p 3 a good ordinary prime for e we show that begin aligned mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 implies mathrm ord s 1 l s e mathbb q 1 end aligned for the p infty selmer group mathrm sel p infty e mathbb q and the complex l function l s e mathbb q in particular the tate shafarevich group hbox x e mathbb q is finite whenever mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 we also prove an analogous p converse for cm abelian varieties arising from weight two elliptic cm modular forms with trivial central character for non cm elliptic curves over the rationals first general results towards such a p converse theorem are independently due to skinner a converse to a theorem of gross zagier and kolyvagin arxiv 1405 7294 2014 and zhang camb j math 2 2 191 253 2014

mean square values of hecke l series formed with r th order characters let l be a number field containing the r th roots of unity starting with the rankin selberg convolution of a metaplectic eisenstein series on the r fold cover of gl 2 with itself we construct a dirichlet series defined over l whose coefficients involve the r th order twists of a fixed hecke l function we then observe that a group of functional equations can be naturally associated with this construction combining this with the convexity theorem for holomorphic functions of several complex variables we show that this object as a function of two complex variables admits meromorphic continuation to 2 as an application we obtain asymptotic formulae for mean square values of the r th order twists of an arbitrary hecke l function defined over l

pullbacks of saito kurokawa lifts pullbacks of siegel eisenstein series have been studied by garrett 17 18 bocherer 6 7 heim 26 and play a key role in the proof of the algebraicity of critical values of certain automorphic l functions more generally one might consider pullbacks of siegel cusp forms for example ikeda 30 gave a conjectural formula for pullbacks of ikeda lifts 29 in terms of critical values of l functions for spn gl2 also the gross prasad conjecture 22 23 8 27 8 would relate pullbacks of siegel cusp forms of degree 2 to central critical values of l functions for gsp2 gl2 gl2 indeed bocherer furusawa and schulze pillot 8 gave an explicit formula for pullbacks of yoshida lifts 57 in this paper we give an explicit formula for pullbacks of saito kurokawa lifts and prove the algebraicity of central critical values of certain l functions for sp1 gl2 to be precise let be an odd positive integer let f s2 sl2 z be a normalized hecke eigenform and h s 1 2 0 4 a hecke eigenform associated to f by the shimura correspondence let f s 1 sp2 z be the saito kurokawa lift of h for each normalized hecke eigenform g s 1 sl2 z we consider the period integral f h h g g given by f h h g g

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

non vanishing of high derivatives of automorphic l functions at the center of the critical strip keywords modular l functions mollification nonvanishing results central value of high derivatives analytic rank jacobian variety reference tan article 2000 004doi 10 1515 crll 2000 074 record created on 2008 11 14 modified on 2017 05 12

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

conjugacy decomposition of reductive monoids we study the decomposition of a reductive monoid related to conjugacy classes and obtain a description of the associated finite poset open image in new window this poset is constructed via the gauss jordan elements of the renner monoid r for mn k the poset can be identified with the lattice of all partitions of m 0 m n where the order is an extension of the dominance order on partitions of n

selmer groups of quadratic twists of elliptic curves 1 1 e y a1xy a3y x 3 a2x 2 a4x a6 where a1 a2 a3 a4 a6 z let n e denote the conductor of e j e the j invariant of e and l e s n 1 a n n s the hasse weil l function of e if e is modular then let fe z n 1 ae n q n s2 n e 1 be the associated weight 2 cusp form here 1 denotes the trivial dirichlet character throughout d will denote a square free integer and d shall denote the kronecker character for the field q d let h d denote the order of cl d the ideal class group of q d if l is prime then define h d l by

sandwich matrices solomon algebras and kazhdan lusztig polynomials sandwich matrices have proved to be of importance in semigroup theory for the last 50 years the work of the author on algebraic monoids leads to sandwich matrices in group theory in this paper we find some connections between sandwich matrices and the hecke algebras for monoids introduced recently by louis solomon at the local level we then obtain an explicit isomorphism between solomon s hecke algebra and the complex monoid algebra of the renner monoid in the simplest case of monoids associated with a borel subgroup we find that the entries of the inverse of the sandwich matrix as well as those of the related structure matrix of solomon s hecke algebra are almost the polynomials r x y associated with the kazhdan lusztig polynomials

on the ramanujan conjecture and finiteness of poles for certain l functions in this paper we prove a number of general results about automorphic l functions which appear in the constant terms of eisenstein series for an arbitrary quasi split connected reductive algebraic group over a number field these l functions were first introduced by langlands for chevalley groups 20 and through them he was led to make some of his deep conjectures 21 one significance of these l functions is that all the automorphic l functions studied so far are among them there are three general results first we obtain a uniform line of absolute convergence for all of these l functions theorem 5 1 a uniform estimate for the hecke eigenvalues of generic cusp forms on many absolutely simple quasi split connected reductive algebraic groups over number fields corollary 5 4 and an improvement on the best available estimate for the fourier coefficients of maass wave forms corollary 5 5 also follow next we establish a meromorphic continuation and functional equation for each of these l functions theorem 6 1 finally in theorem 6 2 we prove finiteness of poles on the whole complex plane for an important class of these l functions corollaries 6 6 through 6 10 more precisely let g be a quasi split connected reductive algebraic group over a number field f set g g a f where a f is the ring of adeles of f fix a borel subgroup b of g over f and let u be its unipotent radical let p be a maximal f parabolic subgroup of g with p d b in the context of the problems studied here nothing new will be obtained if one drops the maximality condition on p write p mn a levi decomposition n c u and let b u p m and n be the corresponding groups of adelic points for every place v of f let gv g fj similarly we have bv us pv me and nv let x xv be a generic character of u f u cf section 3 then each xv is generic let r ov 7tv be a cusp form on m we shall say 7t is

central critical values of modular l functions and coefficients of half integral weight modular forms modulo l if f z is a newform of weight 2 and d is a fundamental discriminant then let l f d s be the usual twisted l series we study the algebraic parts of the central critical values of these twisted l series modulo primes we show that if there are two d subject to some local conditions for which the algebraic part of l f d is not 0 mod l then there are infinitely many such d these results depend on precise nonvanishing results for the fourier coefficients of half integral weight modular forms modulo l which are of independent interest

hirzebruch zagier cycles and twisted triple product selmer groups let e be an elliptic curve over mathbb q and a another elliptic curve over a real quadratic number field we construct a mathbb q motive of rank 8 together with a distinguished class in the associated bloch kato selmer group using hirzebruch zagier cycles that is graphs of hirzebruch zagier morphisms we show that under certain assumptions on e and a the non vanishing of the central critical value of the twisted triple product l function attached to e a implies that the dimension of the associated bloch kato selmer group of the motive is 0 and the non vanishing of the distinguished class implies that the dimension of the associated bloch kato selmer group of the motive is 1 this can be viewed as the triple product version of kolyvagin s work on bounding selmer groups of a single elliptic curve using heegner points

the dirichlet problem for weingarten hypersurfaces in lorentz manifolds we solve the dirichlet problem for strictly convex spacelike hypersurfaces of prescribed weingarten curvature under the main assumption that there exists an upper barrier we consider curvature functions that correspond in general to fully nonlinear elliptic partial differential equations which are not uniformly elliptic

cube sum problem and an explicit gross zagier formula a nonzero rational number is called a it cube sum if it is of the form a 3 b 3 with a b in bbb q times in this paper we prove that for any odd integer k geq 1 there exist infinitely many cube free odd integers n with exactly k distinct prime factors such that 2n is a cube sum resp not a cube sum we present also a general construction of heegner points and obtain an explicit gross zagier formula which is used to prove the birch and swinnerton dyer conjecture for certain elliptic curves related to the cube sum problem

un analogue du calcul de heegner on demontre les deux resultats suivants a si p est un nombre premier impair congru a 2 modulo 9 la courbe elliptique x 3 y 3 2pz 3 possede une infinite de points rationnels b si p est un nombre premier congru a 5 modulo 9 la courbe elliptique x 3 y 3 2p 2 z 3 possede une infinite de points rationnels

classification of semisimple algebraic monoids let x be a semisimple algebraic monoid with unit group g associated with e is its polyhedral root system x c where x x t is the character group of the maximal torus t c g c x t is the set of roots and c x t is the character monoid of t c e zariski closure the correspondence e x c is a complete and discriminating invariant of the semisimple monoid e extending the well known classification of semisimple groups in establishing this result monoids with preassigned root data are first constructed from linear representations of g that done we then show that any other semisimple monoid must be isomorphic to one of those constructed to do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory thls ultimately yields the desired conclusions consider the classification problem for semisimple algebraic monoids over the algebraically closed field k what sort of problem is this first the definition a semisimple algebraic monoid is an irreducible affine algebraic variety e defined over k together with an associative morphlsm m e x e e and a two sided unit 1 e e for m we assume further that e has a 0 the unit group g which is always linear algebraic and dense in e is reductive e g gl k dim zg 1and e is a normal variety the problem then presents us with two familiar objects let t c g be a maximal torus then z t c e zariski closure is an affine torus embedding and g c e is a reductive algebraic group torus embeddings have been introduced by demazure in 6 in his study of cremona groups and are classified numerically using rational polyhedral cones 14 on the other hand reductive groups have been studied at least in principle since the nineteenth century their classification in the modern sense being achieved largely by chevalley 4 that numerical classification uses the now familiar root systems l l chapter 31 of killing that were introduced by him 15 in his penetrating formulation of the classification e cartan s of semisimple lie algebras thus in the classification of semisimple monoids we are compelled to consider the root system x x t t of g and the polyhedral cone c x z c x t of t c z the two objects are canonically related by the weyl group action on x which leaves c stable received by the editors november 16 1984 1980 mathematics subject classification primary 14m99 secondary 20m99

weyl group multiple dirichlet series constructed from quadratic characters we construct multiple dirichlet series in several complex variables whose coefficients involve quadratic residue symbols the series are shown to have an analytic continuation and satisfy a certain group of functional equations these are the first examples of an infinite collection of unstable weyl group multiple dirichlet series in greater than two variables having the properties predicted in 2

rankin selberg l functions in the level aspect keywords moments rankin selberg convolution level aspect convexity breaking reference tan article 2002 003doi 10 1215 s0012 7094 02 11416 1 record created on 2008 11 14 modified on 2017 05 12

on the birch swinnerton dyer quotients modulo squares let a be an abelian variety over a number field k an identity between the l functions l a k i s for extensions k i of k induces a conjectural relation between the birch swinnerton dyer quotients we prove these relations modulo finiteness of lii and give an analogous statement for selmer groups based on this we develop a method for determining the parity of various combinations of ranks of a over extensions of k as one of the applications we establish the parity conjecture for elliptic curves assuming finiteness of iii e k e 2 6 and some restrictions on the reduction at primes above 2 and 3 the parity of the mordell weil rank of e k agrees with the parity of the analytic rank as determined by the root number we also prove the p parity conjecture for all elliptic curves over q and all primes p the parities of the p selmer rank and the analytic rank agree

p adic interpolation of square roots of central l values of modular forms we construct a meromorphic function on part of the eigencurve that interpolates a square root of a ratio of quadratic twists of the central modular l value

non vanishing of quadratic twists of modular l functions k have been the subject of much study both because of their intrinsicinterest and because of the prominent role they have played inkolyvagin s work on the birch and swinnerton dyer conjecture see b f h i j ko ma m m m1 o s p p waldspurger proved a fundamental theorem the ore me 1 w1 relating these central critical values to the fourier coe cients of half integral weight cusp forms for notational convenience if d is afundamental discriminant of a quadratic number eld then de ne d

a refined beilinson bloch conjecture for motives of modular forms we propose a refined version of the beilinson bloch conjecture for the motive associated with a modular form of even weight this conjecture relates the dimension of the image of the relevant p adic abel jacobi map to certain combinations of heegner cycles on kuga sato varieties we prove theorems in the direction of the conjecture and in doing so obtain higher weight analogues of results for elliptic curves due to darmon

closed weingarten hypersurfaces in semi riemannian manifolds the existence of closed hypersurfaces of prescribed curvature in semi riemannian manifolds is proved provided there are barriers

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

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

quadratic twists of elliptic curves in this paper we give the method of constructing non torsion points on elliptic curves which generalizes the classical birch lemma as an application we get more quadratic twist families of the elliptic curve x 0 49 which have rank one this report is a combination of the two joint works coates li tian zhai proc lond math soc 110 2 357 394 2015 4 cai li wang sci china math 59 7 1307 1326 2016 2

on ranks of twists of elliptic curves and power free values of binary forms let e be an elliptic curve over q by the rank of e we shall mean the rank of the group of rational points of e mestre 31 improving on the work of neron 34 cf 13 39 and 46 has shown that there is an infinite family of elliptic curves over q with rank at least 12 however computational work see 3 4 7 and 47 suggests that a typical elliptic curve will have much smaller rank with curves of rank 0 or 1 being predominant indeed brumer 6 has proved subject to the birch swinnerton dyer conjecture the shimura taniyama weil conjecture and the generalized riemann hypothesis that the average rank of an elliptic curve ordered according to its faltings height is at most 2 3 in this article we shall study the behaviour of the rank as we run over twists of a given elliptic curve over q that is we shall restrict our attention to families of elliptic curves defined over q which are isomorphic over c there are families of quadratic cubic quartic and sextic twists see for example proposition 5 4 of chapter x of 42 let e be an elliptic curve over q with weierstrass equation y2 x ax b and for any non zero integer d let ed 2 3 denote a quadratic twist of e given by the equation dy2 x ax b let r d denote the rank of ed note that if d1 and d2 are non zero integers then ed is isomorphic to ed over q if and only if d1 d2 is the square of a rational number subject to the conjectures of birch and swinnerton dyer and of shimura taniyama and weil goldfeld 14 conjectured in 1979 that

generic hecke algebra for renner monoids abstract to each renner monoid r we associate a generic hecke algebra h r over z q which is a deformation of the monoid z algebra of r if m is a finite reductive monoid with borel subgroup b and associated renner monoid r then we obtain the associated iwahori hecke algebra h m b by specialising q in h r and tensoring by c over z as in the classical case of finite reductive groups

analog of the bruhat decomposition for algebraic monoids ii the length function and the trichotomy 0 introduction the purpose of this paper is to continue the study of b b orbits on a reductive monoidm this was first taken up systematically by the author in 9 where several basic properties analogous to the case of groups were established the b b orbits were identified with a finite monoid r defined in terms of the normalizer of a maximal torus and a direct analysis of these orbits led to an extension of tits axiom br brb b rb to the case of reductive monoids however there is more to this axiom when dealing with monoids obviously

faltings heights of cm cycles and derivatives of l functions we study the faltings height pairing of arithmetic special divisors and cm cycles on shimura varieties associated to orthogonal groups we compute the archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a rankin l function we prove the conjecture in certain cases where the shimura variety has dimension 0 1 or 2 in particular we obtain a new proof of the gross zagier formula

central derivatives of l functions in hida families we prove a result of the following type given a hida family of modular forms if there exists a weight two form in the family whose l function vanishes to exact order one at s 1 then all but finitely many weight two forms in the family enjoy this same property the analogous result for order of vanishing zero is also true and is an easy consequence of the existence of the mazur kitagawa two variable p adic l function

the square free sieve and the rank of elliptic curves let e be an elliptic curve over q a celebrated theorem of mordell asserts that e q the abelian group of rational points of e is finitely generated by the rank of e we mean the rank of e q thus the rank of e is positive if and only if e possesses an infinity of rational points relatively few general qualitative assertions can be made about the rank as e varies how large can the rank get although we expect that there are elliptic curves over q with arbitrarily high ranks this is presently unknown what is the average size of the rank recently brumer and mcguinness have reported on their study of 310 716 elliptic curves of prime conductor less than 108 where they have found that 20 06 of those curves have even rank 2 cf bm also see forthcoming work of brumer where subject to a number of standard conjectures he shows that 2 3 is an upper bound for the average rank for all elliptic curves over q ordered in terms of their faltings height what is the behavior of the rank over the family of twists of a given elliptic curve here one has three natural kinds of families of twists 1 quadratic twists one can take any elliptic curve and systematically twist it by all quadratic characters this is the type of family of elliptic curves we are concerned with in this paper specifically if e is an elliptic curve over q given by the weierstrass equation y2 x a x b and d is any squarefree integer the quadratic twist of e by d ed is given by the equation

on the p adic height of heegner cycles 0 1 in gr za gross and zagier proved a remarkable formula which relates the first derivative of the l function of a modular form f of weight 2 on 0 n over a suitable imaginary quadratic field and the neron tate height of a heegner point over the same quadratic field on the f part of the jacobian of the modular curve x0 n later perrin riou pr 2 proved a p adic version of this formula kolyvagin s method of euler systems ko combined with the formula of gross zagier proves the conjecture of birch and swinnerton dyer up to a controlled rational factor for all modular elliptic curves over q with analytic rank 1

the saito kurokawa lifting and darmon points let e mathbb q be an elliptic curve of conductor np with p not mid n and let f be its associated newform of weight 2 denote by f infty the p adic hida family passing though f and by f infty its varlambda adic saito kurokawa lift the p adic family f infty of siegel modular forms admits a formal fourier expansion from which we can define a family of normalized fourier coefficients widetilde a t k t indexed by positive definite symmetric half integral matrices t of size 2 times 2 we relate explicitly certain global points on e coming from the theory of darmon points with the values of these fourier coefficients and of their p adic derivatives evaluated at weight k 2

weingarten hypersurfaces with prescribed gradient image we prove the existence of globally smooth convex solutions u of a class of curvature equations subject to the boundary condition du omega omega where omega and omega are smooth uniformly convex domains in mathbb r n the results generalize some of our previous work on the two dimensional case and on hessian equations in all dimensions

shellability in reductive monoids the purpose of this paper is to extend to monoids the work of bjorner wachs and proctor on the shellability of the bruhat chevalley order on weyl groups let m be a reductive monoid with unit group g borel subgroup b and weyl group w we study the partially ordered set of b x b orbits with respect to zariski closure inclusion within a g x g orbit of m this is the same as studying a w x w orbit in the renner monoid r such an orbit is the retract of a universal orbit which is shown to be lexicograhically shellable in the sense of bjorner and wachs

computational verification of the birch and swinnerton dyer conjecture for individual elliptic curves we describe theorems and computational methods for verifying the birch and swinnerton dyer conjectural formula for specific elliptic curves over q of analytic ranks 0 and 1 we apply our techniques to show that if e is a non cm elliptic curve over q of conductor 1000 and rank 0 or 1 then the birch and swinnerton dyer conjectural formula for the leading coefficient of the l series is true for e up to odd primes that divide either tamagawa numbers of e or the degree of some rational cyclic isogeny with domain e since the rank part of the birch and swinnerton dyer conjecture is a theorem for curves of analytic rank 0 or 1 this completely verifies the full conjecture for these curves up to the primes excluded above

solvable points on genus one curves a genus one curve defined over q which has points over qp for all primes p may not have a rational point it is natural to study the classes of q extensions over which all such curves obtain a global point in this article we show that every such genus one curve with semistable jacobian has a point defined over a solvable extension of q

analogue of the bruhat chevalley order for reductive monoids the purpose of this paper is to describe the adherence order of w x b b orbits on a reductive algebraic monoid by the results of 10 there is already a perfect analogue of the bruhat decomposition for reductive w x monoids indeed by 10 corollary 5 8 if m is reductive with unit group g and maximal torus t g with borel subgroup b t then the set of two sided b orbits b r mrb is canonically identified with a certain finite inverse monoid r in fact r s n t rt the orbit monoid of the zariski g closure of n t in m by definition for s t g r we define the adherg ence order by s f t if bs b bt b where bt b denotes the zariski closure in m of bt b