the l component of the unipotent albanese map we prove a finiteness theorem for the local l p component of the qp unipotent albanese map for curves as an application we refine the non abelian selmer varieties arising in the study of global points and deduce thereby a new proof of siegel s theorem for affine curves over q of genus one with mordell weil rank at most one

massey products and ideal class groups we consider certain massey products in the cohomology of a galois extension of fields with coefficients in p power roots of unity we prove formulas for these products both in general and in the special case that the galois extension in question is the maximal extension of a number field unramified outside a set of primes s including those above p and any archimedean places we then consider those zp kummer extensions l of the maximal p cyclotomic extension k of a number field k that are unramified outside s we show that massey products describe the structure of a certain decomposition free quotient of a graded piece of the maximal unramified abelian pro p extension of l in which all primes above those in s split completely with the grading arising from the augmentation filtration on the group ring of the galois group of l k we explicitly describe examples of the maximal unramified abelian pro p extensions of unramified outside p kummer extensions of the cyclotomic field of all p power roots of unity for irregular primes p

idempotent completion of triangulated categories we show that the idempotent completion of a triangulated category has a natural structure of a triangulated category the idempotent completion of the bounded derived category of an exact category gives the derived category of the idempotent completion in particular the derived category of an idempotent complete exact category is idempotent complete

karoubianness of a triangulated category abstract we prove that triangulated categories with bounded t structures are karoubian consequently for an ext finite abelian category over a commutative noetherian complete local ring its bounded derived category is krull schmidt

hearts of twin cotorsion pairs on extriangulated categories abstract in this article we study the heart of a cotorsion pairs on an exact category and a triangulated category in a unified method by means of the notion of an extriangulated category we prove that the heart is abelian and construct a cohomological functor to the heart if the extriangulated category has enough projectives this functor gives an equivalence between the heart and the category of coherent functors over the coheart modulo projectives we also show how an n cluster tilting subcategory of an extriangulated category gives rise to a family of cotorsion pairs with equivalent hearts

a non abelian conjecture of tate shafarevich type for hyperbolic curves let x denote a hyperbolic curve over mathbb q and let p denote a prime of good reduction the third author s approach to integral points introduced in kim invent math 161 629 656 2005 publ res inst math sci 45 89 133 2009 endows x mathbb z p with a nested sequence of subsets x mathbb z p n which contain x mathbb z these sets have been computed in a range of special cases balakrishnan et al j am math soc 24 281 291 2011 dan cohen and wewers proc lond math soc 110 133 171 2015 dan cohen and wewers int math res not imrn 17 5291 5354 2016 kim j am math soc 23 725 747 2010 there is good reason to believe them to be practically computable in general in 2012 the third author announced the conjecture that for n sufficiently large x mathbb z x mathbb z p n this conjecture may be seen as a sort of compromise between the abelian confines of the bsd conjecture and the profinite world of the grothendieck section conjecture after stating the conjecture and explaining its relationship to these other conjectures we explore a range of special cases in which the new conjecture can be verified

appendix and erratum to massey products for elliptic curves of rank 1 the paper 6 contains a few errors in the basic assumptions as well as in the formula of corollary 0 2 first of all it should have been made clear at the outset that the regular model e for the elliptic curve e must be the minimal regular model and x the complement of the origin in the regular minimal model similarly the tangential base point b must be integral in that it is a z basis of the relative tangent space ete z it could also be an integral two torsion point for the arguments of the paper to hold verbatim the most significant error is in the contribution of the local terms at l 6 p that is lemma 1 2 the problem is that a point that is integral on x may not be integral on a smooth model over a field of good reduction as it stands the lemma will only apply to points that are integral in this stronger sense however to get immediate examples one can replace the lemma by

li p service an algorithm for computing p adic polylogarithms we describe an algorithm for computing coleman s p adic poly logarithms up to a given precision 2007 american mathematical society

torsion points on curves and p adic abelian integrals theorem a let f c j be an albanese morphism defined over a number field k of a s mooth curve of genus g into its jacobian suppose j has potential complex multiplication let s denote the set of primes p of k satisfying i p does not divide 2 or 3 ii k q is unramified at p iii c has good ordinary reduction over k then the set t of torsion points of a defined over an algebraic closure of k which lie on the image of c is defined over an extension of k unramified above s moreover to pg

quasi abelian hearts of twin cotorsion pairs on triangulated categories abstract we prove that under a mild assumption the heart h of a twin cotorsion pair s t u v on a triangulated category c is a quasi abelian category if c is also krull schmidt and t u we show that the heart of the cotorsion pair s t is equivalent to the gabriel zisman localisation of h at the class of its regular morphisms in particular suppose c is a cluster category with a rigid object r and x r the ideal of morphisms factoring through x r ker hom c r then applications of our results show that c x r is a quasi abelian category we also obtain a new proof of an equivalence between the localisation of this category at its class of regular morphisms and a certain subfactor category of c

grothendieck s problems concerning profinite completions and representations of groups in 1970 alexander grothendieck 6 posed the following problem let 1 and 2 be finitely presented residually finite groups and let u 1 2 be a homomorphism such that the induced map of profinite completions u 1 2 is an isomorphism does it follow that u is an isomorphism in this paper we settle this problem by exhibiting pairs of groups u p such that is a direct product of two residually finite hyperbolic groups p is a finitely presented subgroup of infinite index p is not

projective injective modules serre functors and symmetric algebras we describe serre functors for generalisations of the category o associated with a semisimple complex lie algebra in our approach projective injective modules that is modules which are both

hida families and rational points on elliptic curves let e be an elliptic curve over q of conductor n mp having a prime p n of multiplicative reduction because e is modular it corresponds to a normalised weight two eigenform on 0 n whose q expansion is denoted f n anq letx hom z p z p z p 1 z zp which contains z as a dense subset by associating to k z the character x xk 2 denote by a u the ring of cp valued p adic analytic functions on a compact open subset u of x hida s theory associates to f a neighborhood u of 2 x which can be assumed for simplicity to be contained in the residue class of 2 modulo p 1 and a formal q expansion

a note on morphisms determined by objects abstract we prove that a hom finite additive category having determined morphisms on both sides is a dualizing variety this complements a result by krause we prove that in a hom finite abelian category having serre duality a morphism is right determined by some object if and only if it is an epimorphism we give a characterization to abelian categories having serre duality via determined morphisms

p adic multiple zeta values ii tannakian interpretations we establish a tannakian formalism of p adic multiple polylogarithms and p adic multiple zeta values introduced in our previous paper via a comparison isomorphism between a de rham fundamental torsor and a rigid fundamental torsor of the projective line minus three points and also discuss its hodge and e tale analogues as an application we give a way to erase log poles of p adic multiple polylogarithms and introduce overconvergent p adic multiple polylogarithms which might be p adic multiple analogue of zagier s single valued complex polylogarithms

the bounded derived categories of an algebra with radical squared zero abstract let be an elementary locally bounded linear category over a field with radical squared zero we shall show that the bounded derived category d b mod b of finitely supported left modules admits a galois covering which is the bounded derived category of almost finitely co presented representations of a gradable quiver restricting to the bounded derived category d b mod b of finite dimensional left modules we shall be able to describe its indecomposable objects obtain a complete description of the shapes of its auslander reiten components and classify those such that d b mod b has only finitely many auslander reiten components

subgroup growth and congruence subgroups letk be a global field o its ring of integers g an almost simple simply connected connected algebraic subgroups ofgl m defined overk and g o which is assumed to be infinite let n resp n be the number of all resp congruence subgroups of index at mostn in we show a if char k 0 then i c 1 frac log 2 n log log n leqq log gamma n gamma leqq c 2 frac log 2 n log log n for suitable constantsc1 andc2 under some mild assumptions we also have ii has the congruence subgroup property if and only if log n o log2n iii if is boundedly generated the has the congruence subgroup property this confirms a conjecture of rapinchuk r1 which was also proved by platonov and rapinchuk pr3 b if char k 0 and under somewhat stronger conditions ong then for suitable constantsc3 andc4 c3 log2n log n c4log3n

quadratic chabauty for modular curves and modular forms of rank one in this paper we provide refined sufficient conditions for the quadratic chabauty method on a curve x to produce an effective finite set of points containing the rational points x mathbb q with the condition on the rank of the jacobian of x replaced by condition on the rank of a quotient of the jacobian plus an associated space of chow heegner points we then apply this condition to prove the effective finiteness of x mathbb q for any modular curve x x 0 n or x mathrm ns n of genus at least 2 with n prime the proof relies on the existence of a quotient of their jacobians whose mordell weil rank is equal to its dimension and at least 2 which is proven via analytic estimates for orders of vanishing of l functions of modular forms thanks to a kolyvagin logachev type result

groups of oscillating intermediate growth we construct an uncountable family of nitely generated groups of intermediate growth with growth functions of new type these functions can have large oscillations between lower and upper bounds both of which come from a wide class of functions in particular we can have growth oscillating between e n and any prescribed function growing as rapidly as desired our construction is built on top of any of the grigorchuk groups of intermediate growth and is a variation on the limit of permutational wreath product

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

smashing subcategories and the telescope conjecture an algebraic approach we prove a modified version of ravenel s telescope conjecture it is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra this result is based on a new characterization of smashing subcategories which leads in addition to a classification of these subcategories in terms of the category of finite spectra the approach presented here is purely algebraic it is based on an analysis of pure injective objects in a compactly generated triangulated category and covers therefore also situations arising in algebraic geometry and representation theory

the cuspidalisation of sections of arithmetic fundamental groups abstract in this paper which is a sequel to 7 we investigate the theory of cuspidalisation of sections of arithmetic fundamental groups of hyperbolic curves to cuspidally i th and 2 p th step prosolvable arithmetic fundamental groups as a consequence we exhibit two necessary and sufficient conditions for sections of arithmetic fundamental groups of hyperbolic curves over p adic local fields to arise from rational points we also exhibit a class of sections of arithmetic fundamental groups of p adic curves which are orthogonal to pic and which satisfy unconditionally one of the above conditions

generalized serre duality abstract we introduce a notion of generalized serre duality on a hom finite krull schmidt triangulated category t this duality induces the generalized serre functor on t which is a linear triangle equivalence between two thick triangulated subcategories of t moreover the domain of the generalized serre functor is the smallest additive subcategory of t containing all the indecomposable objects which appear as the third term of an auslander reiten triangle in t dually the range of the generalized serre functor is the smallest additive subcategory of t containing all the indecomposable objects which appear as the first term of an auslander reiten triangle in t we compute explicitly the generalized serre duality on the bounded derived categories of artin algebras and of certain noncommutative projective schemes in the sense of artin and zhang we obtain a characterization of gorenstein algebras an artin algebra a is gorenstein if and only if the bounded homotopy category of finitely generated projective a modules has serre duality in the sense of bondal and kapranov

arbitrarily large residual finiteness growth the residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients in this paper we construct groups with arbitrarily large residual finiteness growth we also demonstrate a new relationship between residual finiteness growth and some decision problems in groups which we apply to our new groups let be a residually finite group generated by a finite set x an element is detected by a finite quotient q of if there exists a homomorphism q such that ker set f x n to be the minimal natural number n such that any element of word length n with respect to x can be detected in a quotient q of cardinality n the study of f x the residual finiteness growth is an area of active research see for instance b10 b11 bm1 bk10 and km11 our main result is the following theorem 1 for any function f n n there exists a residually finite group bf and a two element generating set x for bf such that fbf x n f n for all n 8 our proof is constructive elementary and demonstrates the versatility of b h neumann s construction of a continuous family of non isomorphic two generated groups n37 our work is of the same flavor as lps96 and p04 where neumann s construction is modified to find groups with interesting subgroup growth properties our remaining results exhibit how residual finiteness growth may be applied to illicit decision problem data on groups the first of these shows that a variant of theorem 1 does not exist for groups that are finitely presented in particular the main result on residual finiteness growth stated in kms13 is the strongest of its kind for finitely presented groups proposition 1 if is a residually finite finitely presented group then the residual finiteness growth function for is computable o kharlampovich showed that there exists a finitely presented solvable group with undecidable word problem k81 of which many variations have been found see b83 in ls77 iv 4 3 theorem 19 it is shown that f2 f2 does not have decidable membership problem with respect to finitely generated subgroups we address a weaker membership problem for the groups bf for a finitely generated group g we say the finite index normal subgroup membership problem in g is solvable if there exists an algorithm which for every element g g and every finiteindex normal subgroup h g will determine in in a finite amount of time whether g h 1991 mathematics subject classification 20e26

dualizing complexes over noncommutative graded algebras the motivation for the work presented in this paper was two questions posed by m artin let a be a d dimensional regular graded algebra over the field k with augmentation ideal m see def 4 13 denote by hi a resp hi a the dth left resp right local cohomology of a at the ideal m i e hi a lim extd a m a and similarly for hi a as left modules and as right modules hi a and h a are both isomorphic to a hom a k up to a twist in degrees all operations are taking place in the graded category

lower bounds on the arithmetic self intersection number of the relative dualizing sheaf on arithmetic surfaces we give an explicitly computable lower bound for the arithmetic self intersection number of the dualizing sheaf on a large class of arithmetic surfaces if some technical conditions are satisfied then this lower bound is positive in particular these technical conditions are always satisfied for minimal arithmetic surfaces with simple multiplicities and at least one reducible fiber but we have also used our techniques to obtain lower bounds for some arithmetic surfaces with non reduced fibers

derived equivalences for hereditary artin algebras we study the role of the serre functor in the theory of derived equivalences let a a be an abelian category and let u v u v be a t structure on the bounded derived category d b a d b a with heart h h we investigate when the natural embedding h d b a h d b a can be extended to a triangle equivalence d b h d b a d b h d b a our focus of study is the case where a a is the category of finite dimensional modules over a finite dimensional hereditary algebra in this case we prove that such an extension exists if and only if the t structure is bounded and the aisle u u of the t structure is closed under the serre functor

symmetric groups and expander graphs we construct explicit generating sets sn and tilde s n of the alternating and the symmetric groups which turn the cayley graphs mathcal c text alt n s n and mathcal c text sym n tilde s n into a family of bounded degree expanders for all n

p adic multiple zeta values i our main aim in this paper is to give a foundation of the theory of p adic multiple zeta values we introduce one variable p adic multiple polylogarithms by coleman s p adic iterated integration theory we define p adic multiple zeta values to be special values of p adic multiple polylogarithms we consider the formal p adic kz equation and introduce the p adic drinfel d associator by using certain two fundamental solutions of the p adic kz equation we show that our p adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the p adic kz equation and our p adic multiple zeta values appear as coefficients of the p adic drinfel d associator we show various properties of p adic multiple zeta values which are sometimes analogous to the complex case and are sometimes peculiar to the p adic case via the p adic kz equation

computing heights on elliptic curves we describe how to compute the canonical height of points orn elliptic curves tate has given a rapidly converging series for archimedean local heights over r we describe a modified version of tate s series which also converges over c and give an efficient procedure for calculating local heights at non archimedean places in this way we can calculate heights over number fields having complex embeddings we also give explicit estimates for the tail of our series and present several examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 alxy a3y x3 a2x2 a4x a6 the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 11 viii section 9 or 6 chapter vi the canonical height is an extremely important theoretical tool in the arithmetic theory of elliptic curves being used for such diverse purposes as studying values of l functions 5 numbers of integral points 12 and transcendence theory 9 it is also important as a computational tool such as its use in zagier s algorithm for finding integral points up to large bounds 18 it is thus of interest to have an efficient method for calculating the canonical height of a point the usual definition of h as a limit h p limn 0 4 nh x 2np is not practical for computation instead one uses the fact that the canonical height can be written as a sum of local heights one term for each distinct absolute value on k 2 h p e na p vemk for example if k q then mk can be identified with the set of rational primes together with the usual absolute value on q the multiplicities nv are chosen so that the product formula holds and so that h is independent of the choice of the field k the local height corresponding to a non archimedean absolute value is given by intersection theory in a well known manner see e g 2 4 or 7 chapter 11 section 5 we will describe a quick way to compute non archimedean local heights in section 5 the local height for an archimedean absolute value is given by a transcendental function and so efficient computation is somewhat more difficult j tate 15 1988 american mathematical society 0025 5718 88 1 00 25 per page 339 received august 20 1987 revised october 21 1987 1980 mathematics subject classification 1985 revision primary 11g05 14k07 11d25 this work was partially supported by nsf grant dms 8612393 current address mathematics department brown university providence ri 02912 this content downloaded from 207 46 13 103 on thu 20 oct 2016 04 37 48 utc all use subject to http about jstor org terms 340 joseph h silverman has given an easily computed power series which works for real absolute values precisely for a given curve e and point p x y he gives a sequence of easily computed numbers co cl so that

perpendicular categories with applications to representations and sheaves this paper is concerned with the omnipresence of the formation of the subcategories right left perpendicular to a subcategory of objects in an abelian category we encounter these subcategories in various contexts the formation of quotient categories with respect to localizing subcategories cf section 2 the deletion of vertices and shrinking of arrows see 37 in the representation theory of finite dimensional algebras cf section 5 the comparison of the representation theories of different extended dynkin quivers cf section 10 the theory of tilting cf sections 4 and 6 the study of homological epimorphisms of rings cf section 4 the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme cf section 7 and 21 the study of maximal cohen macaulay modules over surface singularities cf sections the comparison of weighted projective lines for different weight sequences cf section 9 the formation of affine and local algebras attached to path algebras of extended dynkin quivers canonical algebras and weighted projective lines cf section 11 and 21 and the concept of universal localization in 40 formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization this applies in particular to applications in the domain of finite dimensional algebras and their representations several applications of the methods presented in this paper are already in existence partly published or appearing in print in the near future see for instance 40 39 4 45 26 49 46 and have show the versatility of the notion of a perpendicular category it seems that right perpendicular categories first appeared as the subcategories of so called closed objects in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing serre subcategory see 18 47 34 another natural occurrence is encountered in commutative algebra forming the possibly infinitely generated modules of depth 2 cf section 7 the concept and some of the central applications were first presented in a talk given by the first author at the honnef meeting in january 1985 we also note that the perfectly matching nomination perpendicular category was coined by a schofield who discovered independently the usefulness of this concept in dealing with hereditary algebras see 39 cf also section 7 the authors further acknowledge the support of the deutsche forschungsgemeinschaft spp darstellungstheorie von endlichen gruppen und endlichdimensionalen algebren throughout this paper rings are associative with unit and modules are unitary right modules mod r respectively mod r denotes the category of all respectively all finitely presented right r modules

coxeter transformation and inverses of cartan matrices for coalgebras abstract let c be a coalgebra and let z i c z i c z i c be the grothendieck groups of the category c o p inj and c inj of the socle finite injective right and left c comodules respectively one of the main aims of the paper is to study the coxeter transformation c z i c z i c and its dual c z i c z i c of a pointed sharp euler coalgebra c and to relate the action of c and c on a class of indecomposable finitely cogenerated c comodules n with the ends of almost split sequences starting with n or ending at n by applying chin kleiner and quinn 2002 5 we also show that if c is a pointed k coalgebra such that the every vertex of the left gabriel quiver q c of c has only finitely many neighbours then for any indecomposable non projective left c comodule n of finite k dimension there exists a unique almost split sequence 0 c n n n 0 in the category c comod f c of finitely cogenerated left c comodules with an indecomposable comodule c n we show that dim c n c dim n if c is hereditary or more generally if inj dim d n 1 and hom c c d n 0

commensurability growths of algebraic groups fixing a subgroup vargamma in a group g the full commensurability growth function assigns to each n the cardinality of the set of subgroups vardelta of g with vargamma vargamma cap vardelta vardelta vargamma cap vardelta le n for pairs vargamma le g where g is a higher rank chevalley group scheme defined over mathbb z and vargamma is an arithmetic lattice in g we give precise estimates for the full commensurability growth relating it to subgroup growth and a computable invariant that depends only on g

general heart construction for twin torsion pairs on triangulated categories abstract in our previous article we constructed an abelian category from any torsion pair on a triangulated category this generalizes the heart of a t structure and the ideal quotient by a cluster tilting subcategory recently generalizing the quotient by a cluster tilting subcategory buan and marsh showed that an integral preabelian category can be constructed as a quotient from a rigid object in a triangulated category with some conditions in this article by considering a pair of torsion pairs we make a simultaneous generalization of these two constructions

hearts of twin cotorsion pairs on exact categories abstract in the papers of nakaoka he introduced the notion of hearts of twin cotorsion pairs on triangulated categories and showed that they have structures of semi abelian categories we study in this article a twin cotorsion pair s t u v on an exact category b with enough projectives and injectives and introduce a notion of the heart first we show that its heart is preabelian moreover we show the heart of a single cotorsion pair is abelian these results are analog of nakaoka s results in triangulated categories we also consider special cases where the heart has nicer structure by our results the heart of a special twin cotorsion pair s t t v is integral and almost abelian finally we show that the gabriel zisman localization of the heart at the class of regular morphisms is abelian and moreover it is equivalent to the category of finitely presented modules over a stable subcategory of b

frobenius n exangulated categories abstract herschend liu nakaoka introduced the notion of n exangulated categories as higher dimensional analogues of extriangulated categories defined by nakaoka palu the class of n exangulated categories contains n exact categories and n 2 angulated categories as examples in this article we introduce a notion of frobenius n exangulated categories which are a generalization of frobenius n exact categories we show that the stable category of a frobenius n exangulated category is an n 2 angulated category as an application this result generalizes the work by jasso we provide a class of n exangulated categories which are neither n exact categories nor n 2 angulated categories finally we discuss an application of the main results and give some examples illustrating it

nonarithmetic superrigid groups counterexamples to platonov s conjecture margulis showed that most arithmetic groups are superrigid platonov conjectured conversely that finitely generated linear groups which are superrigid must be of arithmetic type we construct counterexamples to platonov s conjecture

decision problems and profinite completions of groups abstract we consider pairs of finitely presented residually finite groups p for which the induced map of profinite completions p is an isomorphism we prove that there is no algorithm that given an arbitrary such pair can determine whether or not p is isomorphic to we construct pairs for which the conjugacy problem in can be solved in quadratic time but the conjugacy problem in p is unsolvable let j be the class of super perfect groups that have a compact classifying space and no proper subgroups of finite index we prove that there does not exist an algorithm that given a finite presentation of a group and a guarantee that j can determine whether or not 1 we construct a finitely presented acyclic group h and an integer k such that there is no algorithm that can determine which k generator subgroups of h are perfect

almost split triangles and morphisms determined by objects in extriangulated categories abstract let c e s be an ext finite krull schmidt and k linear extriangulated category with k a commutative artinian ring we define an additive subcategory c r respectively c l of c in terms of the representable functors from the stable category of c modulo s injectives respectively s projectives to k modules which consists of all s projective respectively s injective objects and objects isomorphic to direct summands of finite direct sums of all third respectively first terms of almost split s triangles we investigate the subcategories c r and c l in terms of morphisms determined by objects and then give equivalent characterizations on the existence of almost split s triangles

massey products for elliptic curves of rank 1 for an elliptic curve over q of analytic rank 1 we use the level two selmer variety and secondary cohomology products to find explicit analytic defining equations for global integral points inside the set of p adic points

proper classes and gorensteinness in extriangulated categories abstract extriangulated categories were introduced by nakaoka and palu as a simultaneous generalization of exact categories and triangulated categories a notion of proper class in an extriangulated category is defined in this paper let c be an extriangulated category and a proper class in c we prove that c admits a new extriangulated structure this construction gives extriangulated categories which are neither exact categories nor triangulated categories moreover we introduce and study gorenstein projective objects in c and demonstrate that gorenstein projective objects share some basic properties with gorenstein projective objects in module categories or in triangulated categories in particular we refine a result of asadollahi and salarian 2004 1 as an application the admissible model structure on extriangulated categories is obtained

auslander type conditions and cotorsion pairs we study the properties of rings satisfying auslander type conditions if an artin algebra satisfies the auslander condition that is is an gorenstein artin algebra then we construct two kinds of subcategories which form functorially finite cotorsion pairs

tangential localization for selmer varieties this paper proposes a tangential version of the theory of selmer varieties together with a formulation of cohomological duality in families of lie algebras indexed by nonabelian cohomology this theory allows one to consider deformations of cohomology classes as one moves over the selmer variety and suggests an approach for generalizing to number fields the homotopical techniques for proving diophantine finiteness that were developed over q the utility of this perspective is demonstrated by way of a new proof of siegel s theorem on finiteness of s integral points for the projective line minus three points over a totally real field

hereditary categories with serre duality which are generated by preprojectives abstract we show that every k linear abelian ext finite hereditary category with serre duality which is generated by preprojective objects is derived equivalent to the category of representations of a strongly locally finite thread quiver

p adic l functions and selmer varieties associated to elliptic curves with complex multiplication we show how the finiteness of integral points on an elliptic curve over q with complex multiplication can be accounted for by the nonvanishing of l functions that leads to bounds for dimensions of selmer varieties

abelian quotients of categories of short exact sequences abstract we mainly investigate abelian quotients of categories of short exact sequences the natural framework to consider the question is via identifying quotients of morphism categories as module categories these ideas not only can be used to recover the abelian quotients produced by cluster tilting subcategories of both exact categories and triangulated categories but also can be used to reach our goal let c e be an exact category we denote by e c the category of bounded complexes whose objects are given by short exact sequences in e and by s e c the full subcategory formed by split short exact sequences in general e c is just an exact category but the quotient e c s e c turns out to be abelian in particular if c e is frobenius we present three equivalent abelian quotients of e c and point out that the equivalences are actually given by left and right rotations the abelian quotient e c s e c admits some nice properties we explicitly describe the abelian structure projective objects injective objects and simple objects which provide a new viewpoint to understanding hilton rees theorem and auslander reiten theory furthermore we present some analogous results for triangulated versions

cotorsion pairs model category structures and representation theory we make a general study of quillen model structures on abelian categories we show that they are closely related to cotorsion pairs which were introduced by salce sal79 and have been much studied recently by enochs and coauthors ej00 this gives a method of constructing model structures on abelian categories which we illustrate by building two model structures on the category of modules over a possibly noncommutative gorenstein ring the homotopy category of these model structures is a generalization of the stable module category much used in modular representation theory this stable module category has also been studied by benson ben97

the motivic fundamental group of p1 0 1 and the theorem of siegel in this paper we establish a link between the structure theory of the pro unipotent motivic fundamental group of the projective line minus three points and diophantine geometry in particular we give a p adic proof of siegel s theorem

noetherian hereditary abelian categories satisfying serre duality norwegian univ sci technol dept math sci n 7491 trondheim norway limburgs univ ctr dept wni b 3590 diepenbeek belgium reiten i norwegian univ sci technol dept math sci n 7491 trondheim norway

ideal classes of three dimensional sklyanin algebras abstract in this paper we classify graded reflexive ideals up to isomorphism and shift in certain three dimensional artin schelter regular algebras this classification is similar to the classification of right ideals in the first weyl algebra a problem that was completely settled recently the situation we consider is substantially more complicated however