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

a vanishing theorem of kollar ohsawa type for proper surjective holomorphic maps from kahler manifolds to analytic spaces we give a decomposition theorem for the cohomology groups of the canonical bundle twisted by nakano semi positive vector bundles by means of the higher direct image sheaves by using the theory of harmonic integrals developed by takegoshi as an application we prove a vanishing theorem of kollar ohsawa type by combining the l 2 method for the overline partial equation

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

an extension of knopp s core theorem abstract in s l devi j london math soc 2 12 1976 397 401 an inequality sharper than that of knopp s core theorem has been established the purpose of this note is to generalize knopp s core theorem as well as the results obtained by devi

σ core and i core of bounded sequences in this paper we characterize matrices that map every bounded sequence into one whose core is a subset of the i core of the original sequence

iterated elliptic and hypergeometric integrals for feynman diagrams we calculate 3 loop master integrals for heavy quark correlators and the 3 loop quantum chromodynamics corrections to the parameter they obey non factorizing differential equations of second order with more than three singularities which cannot be factorized in mellin n space either the solution of the homogeneous equations is possible in terms of 2f1 gaus hypergeometric functions at rational argument in some cases integrals of this type can be mapped to complete elliptic integrals at rational argument this class of functions appears to be the next one arising in the calculation of more complicated feynman integrals following the harmonic polylogarithms generalized polylogarithms cyclotomic harmonic polylogarithms square root valued iterated integrals and combinations thereof which appear in simpler cases the inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals where the new innermost letter itself is not an iterative integral a new class of iterative integrals is introduced containing letters in which multiple definite integrals appear as factors for the elliptic case we also derive the solution in terms of integrals over modular functions and also modular forms using q product and series representations implied by jacobi s i functions and dedekind s function the corresponding representations can be traced back to polynomials out of lambert eisenstein series having representations also as elliptic polylogarithms a q factorial 1 k logarithms and polylogarithms of q and their q integrals due to the specific form of the physical variable x q for different processes different representations do usually appear numerical results are also presented

weak leopoldt s conjecture for hecke characters of imaginary quadratic fields we give a proof of weak leopoldt s conjecture a la perrin riou under some technical condition for the p adic realizations of the motive associated to hecke characters over an imaginary quadratic fleld k of class number 1 where p is a prime 3 and where the cm elliptic curve associated to the hecke character has good reduction at the primes above p in k this proof makes use of the 2 variable iwasawa main conjecture proved by rubin thus we prove the jannsen conjecture for the above p adic realizations for almost all tate twists 1 the formulation of weak leopoldt s conjecture and weak jannsen s conjecture for hecke characters let us flrst introduce the p adic integer realizations associated to hecke characters the aim of this paper is verify for these realizations the weak leopoldt conjecture a la perrin riou and jannsen s conjecture for almost all tate twists let e be a flxed elliptic curve with cm which is deflned over an imaginary quadratic fleld k with cm by ok the ring of integers of k note that this implies cl k 1 associated to this elliptic curve there is a cm character of the imaginary quadratic fleld k with conductor f an ideal of ok that coincides with the conductor of the elliptic curve e let us consider once for all p be an odd prime where e has good reduction at the primes above p in k ordinary or supersingular we impose p 6 3 in order that any primitive p root of unity is in k thus in particular p dk the discriminant of k let us introduce the motives and the p adic realizations on which we study the conjectures consider the category of chow motives mq k over k with morphisms induced by graded correspondences in chow theory tensored with q then the motive of the elliptic curve e has a canonical decomposition h e q h 0 e q h 1 e q h 2 e q the motive h 1 e q has a multiplication by k 5 let us consider the motive wh 1 e q for w a strictly positive integer which has multiplication by tw wk observe that tw has a decomposition q

zagier s conjecture on l e 2 in this paper we introduce an elliptic analog of the bloch suslin complex and prove that it essentially computes the weight two parts of the groups k 2 e and k 1 e for an elliptic curve e over an arbitrary field k combining this with the results of bloch and beilinson we proved zagier s conjecture on l e 2 for modular elliptic curves over

statistical core theorems abstract based on the recently introduced concept of a statistical cluster point a definition is given for the statistical core of a complex number sequence which is analogous to the knopp core the theorems proved here are statistical analogues of theorems about the knopp core concerning its containment or invariance under a matrix transformation

on relation between the ideal core and ideal cluster points abstract this article is motivated by the recent paper characterization of the ideal core by paolo leonetti jmaa 2019 we show that in the main result of that paper the first countability condition can be omitted and the compactness condition may be substituted by a strictly weaker one

zeta polynomials for modular form periods abstract answering problems of manin we use the critical l values of even weight k 4 newforms f s k 0 n to define zeta polynomials z f s which satisfy the functional equation z f s z f 1 s and which obey the riemann hypothesis if z f 0 then re 1 2 the zeros of the z f s on the critical line in t aspect are distributed in a manner which is somewhat analogous to those of classical zeta functions these polynomials are assembled using signed stirling numbers and weighted moments of critical l values in analogy with ehrhart polynomials which keep track of integer points in polytopes the z f s encode arithmetic information assuming the bloch kato tamagawa number conjecture they encode the arithmetic of a combinatorial arithmetic geometric object which we call the bloch kato complex for f loosely speaking these are graded sums of weighted moments of orders of safarevic tate groups associated to the tate twists of the modular motives

p adic elliptic polylogarithm p adic eisenstein series and katz measure the specializations of the motivic elliptic polylogarithm on the universal elliptic curve to the modular curve are referred to as eisenstein classes in this paper we prove that the syntomic realization of the eisenstein classes restricted to the ordinary locus of the modular curve may be expressed using p adic eisenstein series of negative weight which are p adic modular forms defined using the two variable p adic measure with values in p adic modular forms constructed by katz the motivation of our research is the p adic beilinson conjecture formulated by perrin riou

rankin eisenstein classes for modular forms in this paper we make a systematic study of certain motivic cohomology classes rankin eisenstein classes attached to the rankin selberg convolution of two modular forms of weight 2 the main result is the computation of the p adic syntomic regulators of these classes as a consequence we prove many cases of the perrin riou conjecture for rankin selberg convolutions of cusp forms

core equality results for sequences abstract given densities and v we characterize nonnegative matrices t such that the statistical core of x equals the v statistical core of tx for every real valued bounded sequence

some new paranormed difference sequence spaces and weighted core in this study we define new paranormed sequence spaces by combining a generalized weighted mean and a difference operator furthermore we compute the a b email protected duals and obtain bases for these sequence spaces besides this we characterize the matrix transformations from the new paranormed sequence spaces to the spaces c 0 q c q q and q finally weighted core of a complex valued sequence has been introduced and we prove some inclusion theorems related to this new type of core

a kollar type vanishing theorem let f x rightarrow y be a smooth fibration between two complex manifolds x and y and let l be a pseudo effective line bundle on x we obtain a sufficient condition for r q f k x y otimes l to be reflexive and hence derive a kollar type vanishing theorem

a note on polylogarithms on curves and abelian schemes in this note we study the polylogarithm extension on curves and abelian schemes in the etale realization the main result shows that the polylog on the jacobian of a curve is given as the cup product of the polylog on the curve with the fundamental class of the curve

the hodge realization of the polylogarithm on the product of multiplicative groups the purpose of this article is to describe explicitly the polylogarithm class in absolute hodge cohomology of a product of multiplicative groups in terms of the bloch wigner ramakrishnan polylogarithm functions we will use the logarithmic dolbeault complex defined by burgos to calculate the corresponding absolute hodge cohomology groups

deligne periods of mixed motives theory and the entropy of certain ℤⁿ actions for a coherent sheaf m on the split n torus gn a spec a 7n over a commutative ring a there is a natural en action on its group of global sections f m f g aa m if we give f m the discrete topology we obtain a en action on the compact abelian pontrjagin dual f m the dynamical properties of this action for a z have been investigated for some time especially by k schmidt his book sch gives a comprehensive account of the theory developed so far in particular one may ask for the entropy 0 h m oc of the en action on f m i which measures to what extent repeated application of the action scatters around points note that for any action of zn on a compact abelian group topological entropy and metric entropy with respect to haar measure coincide the results of lind schmidt and ward lsw see also sch chap v can be formulated as follows a z the entropy of m is finite if and only if m belongs to the category t of coherent torsion sheaves on gn by yuzvinskii s addition formula sch th 14 1 entropy defines a homomorphism from the grothendieck group of t to the reals h ko t ri so that it suffices to calculate h oz for irreducible closed subschemes z c n with z g n if z is not defined by a laurent polynomial then h oz 0 on the other hand for any

motives and siegel modular forms for a motive m over q we define the fundamental periods of m using invariant theory our definition generalizes deligne s periods we show that if a motive m is constructed from motives m 1 m 2 m n by a standard algebraic operation then the fundamental periods of m can be expressed as monomials of the fundamental periods of m 1 m 2 m n applying this theory we discuss two hypothetical motives attached to a siegel modular form we show that a siegel modular form of degree m has at most m 1 period invariants

algebraic cycles modular forms and euler systems fix a squarefree integer n and let f be a newform of weight 2 for 0 n we assume that f does not have complex multiplication it was shown in 14 and 15 that for a set of primes l of density 1 the naive deformation theory of the mod l galois representation associated to f is unobstructed in the sense that the universal deformation ring is a power series ring over the witt vectors in 31 these methods were modified to obtain results on the deformation problems studied by taylor wiles in this paper we extend the results of flach and mazur to the case of newforms f of weight 2 for 1 n we now state our results more precisely fix l max 5 1 let f be as above and let h be the associated l adic representation h is a free module of rank 2 over a certain hecke algebra a which itself is a finite flat local gorenstein zl algebra let t be the tate twist endah 1 of the module of trace zero endomorphisms of h using techniques of flach we construct a collection of cohomology classes c in h q t with tightly controlled ramification with some mild additional hypotheses applying the methods of kolyvagin to these classes yields a certain annihilator a of the selmer group h f q t of the cartier dual of t this selmer group is dual to the differentials r ra where r is the universal minimally ramified deformation ring of the residual representation of h in the case that is a unit this then implies that both r and a are isomorphic to the ring of witt vectors over the residue field of a in the general case following mazur we show that our construction yields a derivation from a to the selmer group h f q t t it follows by a formal argument that the natural surjection r a induces an isomorphism r ra a although not the strongest possible result this does provide a great deal of information on the structure of the ring r it is possible that any such map r a must be an isomorphism although as far as i know this question remains open we also show that the isomorphism r ra a is characterized by the fact that a r ra homzl h f q t ql zl

on special elements in higher algebraic k theory and the lichtenbaum gross conjecture abstract we conjecture the existence of special elements in odd degree higher algebraic k groups of number fields that are related in a precise way to the values at strictly negative integers of the derivatives of artin l functions of finite dimensional irreducible complex representations we prove this conjecture for an important family of examples and also provide other evidence both theoretical and numerical in its support

an algebraic integration for mahler measure there are many examples of several variable polynomials whose mahler measure is expressed in terms of special values of polylogarithms these examples are expected to be related to computations of regulators as observed by deninger d and later by rodriguez villegas r and maillot m while rodriguez villegas made this relationship explicit for the two variable case it is our goal to understand the three variable case and shed some light on the examples with more variables

weyl groups and elliptic solutions of the wdvv equations abstract a functional ansatz is developed which gives certain elliptic solutions of the witten dijkgraaf verlinde verlinde or wdvv equations this ansatz is based on the elliptic trilogarithm function introduced by beilinson and levin for this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz this providing an elliptic version of the idea introduced by veselov of a system rational and trigonometric limits are studied together with examples of elliptic systems based on various weyl groups jacobi group orbit spaces are studied these carry the structure of a frobenius manifold the corresponding almost dual structure is shown in the a n and b n cases and conjecturally for an arbitrary weyl group to correspond to the elliptic solutions of the wdvv equations transformation properties under the jacobi group of the elliptic trilogarithm are derived together with various functional identities which generalize the classical frobenius stickelberger relations

algebraic cycles and the hodge structure of a kuga fiber variety let a denote a smooth compactification of the k fold fiber product of the universal family a 1 m of elliptic curves with level n structure the purpose of this paper is to completely describe the algebraic cycles in and the hodge structure of the betti cohomology h a q of a for by doing so we are able a to verify both the usual and generalized hodge conjectures for a b to describe both the kernel and the image of the abel jacobi map from algebraic cycles algebraically equivalent to zero modulo rational equivalence into the griffiths intermediate jacobian and c to verify tate s conjecture concerning the algebraic cycles in the etale cohomology h et a q q l

on tensor product l functions for quaternion unitary groups and gl 2 over totally real number fields mixed weight cases abstract we prove an algebraicity of special values of tensor product l functions for quaternion unitary groups and gl 2 in the mixed weight cases over any totally real number field as a consequence we can prove a period relation for automorphic forms of quaternion unitary groups of degree 2 and gsp 4 when these automorphic forms correspond under the conjectural jacquet langlands correspondence and a period relation for endoscopic liftings of gsp 4 in very general situations

p adic l functions and the coniveau filtration on chow groups we give evidence for the refined version of the bloch beilinson conjecture involving coniveau filtrations by studying several infinite families of cm motives indexed by the integers r 1 that are irreducible of hodge type 2r 1 0 0 2r 1 and whose l functions vanish at the center in each case we construct a corresponding algebraic cycle that is homologically trivial but nontrivial in the top graded piece for the coniveau filtration consequently we obtain new explicit examples of cycles that are nontorsion in the griffiths group the proof of the main result relies crucially on p adic hodge theoretic methods and in particular on the relation between images of homologically trivial cycles under the p adic abel jacobi map and the values of p adic l functions

characterizations of the ideal core abstract given an ideal i on and a sequence x in a topological vector space we let the i core of x be the least closed convex set containing x n n i for all i i we show two characterizations of the i core this implies that the i core of a bounded sequence in r k is simply the convex hull of its i cluster points as applications we simplify and extend several results in the context of pringsheim convergence and e convergence of double sequences

on the bloch kato conjecture for elliptic modular forms of square free level let kappa ge 6 be an even integer m an odd square free integer and f in s 2 kappa 2 gamma 0 m a newform we prove that under some reasonable assumptions that half of the lambda part of the bloch kato conjecture for the near central critical value l kappa f is true we do this by bounding the ell valuation of the order of the appropriate bloch kato selmer group below by the ell valuation of algebraic part of l kappa f we prove this by constructing a congruence between the saito kurokawa lift of f and a cuspidal siegel modular form

intermediate extension of chow motives of abelian type abstract in this article we give an unconditional construction of a motivic analogue of the intermediate extension in the context of chow motives of abelian type our main application concerns intermediate extensions of chow motives associated to kuga families to the baily borel compactification of a shimura variety

matrix transformation and statistical convergence abstract in this paper we will say that a sequence xk is a statistically convergent if for every e 0 lim n 1 n k i n ax k l e 0 with in n n 1 n where a is an infinite matrix and a strictly increasing sequence of positive numbers tending to infinity such that 1 1 and n 1 n 1 for all n using the banach algebra w0 w0 we get sufficient conditions to have a sequence a 1 statistically convergent then we deduce conditions for a sequence to be n q statistically convergent finally we get results in the cases when a is the operator c and the cesaro operator

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

dirac operators in tensor categories and the motive of quaternionic modular forms we define a motive whose realizations afford modular forms of arbitrary weight on an indefinite division quaternion algebra this generalizes work of iovita spiess to odd weights in the spirit of jordan livne it also generalizes a construction of scholl to indefinite division quaternion algebras and provides the first motivic construction of new subspaces of modular forms

a hierarchic array of integrable models motivated by harish chandra theory starting from a simple cdd pole s matrix a hierarchy of new s matrices involving ever higher in the sense of barnes gamma functions are constructed these new s matrices correspond to scattering of excitations in ever more complex integrable models from each of these models new ones are obtained either by q deformation or by considering the selberg type euler products of which they represent the infinite place a hierarchic array of integrable models is thus obtained a remarkable diagonal link in this array is established though many entries in this array correspond to familiar integrable models the array also leads to new models setting up this array led to new results on the q gamma function and on the q deformed bloch wigner function