Showing papers in "Pure and Applied Mathematics Quarterly in 2006"
TL;DR: The local-global compatibility conjecture as mentioned in this paper is a conjecture on the multiplicities with which certain GQ-representations appear in the E-vector space of a topological manifold.
Abstract: 1.1. The local-global compatibility conjecture. Fix a prime p, as well as a finite extension E of Qp. If K is an open subgroup of GL2(Ẑ) (referred to as a “tame level”), then one can define a certain E-Banach space Ĥ(K)E , equipped with an action of GQ × GL2(Qp), by taking the inductive limit of the étale cohomology with coefficients in E of the modular curves of arbitrary p-power level and of tame level K, and then completing with respect to the norm induced by the OE-submodule of integral cohomology classes. Passing to the locally convex inductive limit over all tame levels K, we obtain a complete locally convex topological E-vector space ĤE equipped with a representation of GQ × GL2(Af ) that is (so to speak) “smooth in the prime-to-p-directions, but unitary Banach in the p-adic direction”. (See Subsection 7.2 for the precise definitions of these various topological vector spaces.) The object of this note is to explain a conjecture on the multiplicities with which certain GQ-representations appear in ĤE . This conjecture is in some sense the most optimistic possible, in light of what is already known, or believed, to be true. In order to state the conjecture, we must first admit the truth of a “local padic Langlands conjecture for GL2”. The idea that such a conjecture should (or even could) exist is due largely to Breuil, and has been extensively developed both by him and others. In what follows, we will take as given the most optimistic version of this conjecture, namely that to any continuous representation of GQp on a two dimensional E-vector space V there is associated in a natural manner an
137 citations
TL;DR: In this paper, the construction of joins and secant varieties is studied in the combinatorial context of monomial ideals, and the notion of delightful triangulations of convex polytopes is introduced.
Abstract: The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal of the secant ideal coincides with the secant ideal of the initial ideal. For toric varieties, this leads to the notion of delightful triangulations of convex polytopes.
89 citations
TL;DR: In this article, a self-contained treatment of the theory of crystals and the path model is presented, with a focus on the relationship between the general path model and the crystal operators of Lascoux and Schutzenberger.
Abstract: Together, Sections 2 and 5 of this paper form a self contained treatment of the theory of crystals and the path model. It is my hope that this will be useful to the many people who, over the years, have told me that they wished they understood crystals but have found the existing literature too daunting. One goal of the presentation here is to clarify the relationship between the general path model and the crystal operators of Lascoux and Schutzenberger used in the type A case [LS]. More specifically, Section 2 is a basic pictorial exposition of Weyl groups and affine Weyl groups and Section 5 is an exposition of the theory of (a) symmetric functions, (b) crystals and (c) the path model which is designed for readers whose only background is the material in Section 2. These two sections can be read independently of Sections 3 and 4.
75 citations
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TL;DR: The problem of local harmonic analysis is to classify the set Π(G) of equivalence classes of irreducible representations of G(F ) in a given packet Πφ.
Abstract: One of the fundamental problems of local harmonic analysis is to classify the set Π(G) of equivalence classes of irreducible representations of G(F ). The problem separates naturally into two parts. The first is to establish the local Langlands correspondence. This conjecture of Langlands asserts that Π(G) is a disjoint union of finite subsets Πφ, indexed by (equivalence classes of) Langlands parameters φ for G. The sets Πφ are called L-packets, since their constituents could then be equipped with a common set of L-functions and e-factors, by the constructions of [T, §3–4] and [B, §12]. The second part of the problem is to classify the representations in a given packet Πφ. One would like to characterize them directly in terms of data attached to φ.
68 citations
TL;DR: In this paper, the distribution of arithmetic Fuchsian groups for which the underlying surface of the orbifold H 2 = i is of genus zero is studied, i.e., i is a group acting on a hyperbolic 2-orbifold, with underlying space an orientable surface and a number of cone points.
Abstract: If i is a flnite co-area Fuchsian group acting on H 2 , then the quotient H 2 =i is a hyperbolic 2-orbifold, with underlying space an orientable surface (possibly with punctures) and a flnite number of cone points. Through their close connections with number theory and the theory of automorphic forms, arithmetic Fuchsian groups form a widely studied and interesting subclass of flnite co-area Fuchsian groups. This paper is concerned with the distribution of arithmetic Fuchsian groups i for which the underlying surface of the orbifold H 2 =i is of genus zero; for short we say i is of genus zero. The motivation for the study of these groups comes from many difierent view
66 citations
TL;DR: This paper presented a self-contained overview of basic properties of nested complexes and their dual polyhedral realizations: as complete simplicial fans, and as simple polytopes, and showed a striking similarity between nested complex and associated fans and poly-topes on one side, and cluster complexes and generalized associahedra introduced and studied in hep-th/0111053, math.co/0202004 on the other side.
Abstract: This note which can be viewed as a complement to Alex Postnikov's paper math.CO/0507163, presents a self-contained overview of basic properties of nested complexes and their two dual polyhedral realizations: as complete simplicial fans, and as simple polytopes. Most of the results are not new; our aim is to bring into focus a striking similarity between nested complexes and associated fans and polytopes on one side, and cluster complexes and generalized associahedra introduced and studied in hep-th/0111053, math.CO/0202004, on the other side.
65 citations
TL;DR: In this paper, the authors give the state of the art on the question of the arithmetic nature of the real numbers and complex numbers and give a countable set of real and complex real numbers.
Abstract: The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are ``interesting'', which appear ``naturally'', which deserve our attention, form a countable set. Starting from this point of view we are interested in the periods as defined by M.~Kontsevich and D.~Zagier. We give the state of the art on the question of the arithmetic nature of these numbers: to decide whether a period is a rational number, an irrational algebraic number or else a transcendental number is the object of a few theorems and of many conjectures. We also consider the approximation of such numbers by rational or algebraic numbers.
59 citations
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TL;DR: The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and Swinnerton-Dyer as mentioned in this paper, and it is known to be an odd integer.
Abstract: The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and Swinnerton-Dyer. One conjectures that this constant is 1; it is known to be an integer. After surveying what is known about the Manin constant, we establish a new sufficient condition that ensures that the Manin constant is an odd integer. Next, we generalize the notion of the Manin constant to certain abelian variety quotients of the Jacobians of modular curves; these quotients are attached to ideals of Hecke algebras. We also generalize many of the results for elliptic curves to quotients of the new part of J0(N), and conjecture that the generalized Manin constant is 1 for newform quotients. Finally an appendix by John Cremona discusses computation of the Manin constant for all elliptic curves of conductor up to 130000.
50 citations
TL;DR: In this article, an enriched notion of Chow groups for algebraic varieties is defined, which allows us to glue intersection-theoretic information across elements of a stratification of a variety; we illustrate this operation by giving a direct construction of Chern-Schwartz-MacPherson classes of singular varieties.
Abstract: We define an `enriched' notion of Chow groups for algebraic varieties, agreeing with the conventional notion for complete varieties, but enjoying a functorial push-forward for arbitrary maps. This tool allows us to glue intersection-theoretic information across elements of a stratification of a variety; we illustrate this operation by giving a direct construction of Chern-Schwartz-MacPherson classes of singular varieties, providing a new proof of an old (and long since settled) conjecture of Deligne and Grothendieck.
44 citations
TL;DR: The authors generalizes the classical theory of Newton polygons from the case of general linear groups to split reductive groups and gives a root-theoretic formula for dimensions of Newton strata in the adjoint quotients of reductive group.
Abstract: This paper generalizes the classical theory of Newton polygons from the case of general linear groups to the case of split reductive groups. It also gives a root-theoretic formula for dimensions of Newton strata in the adjoint quotients of reductive groups.
40 citations
TL;DR: In this article, a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subva- riety in the torus integral cohomology ring of the Grassmannian was given.
Abstract: The main result of the paper is a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subva- riety in the torus equivariant integral cohomology ring of the Grassmannian. As a corollary, we obtain an equivariant version of the Giambelli formula. The (torus) equivariant cohomology rings of flag varieties in general and of the Grassmannian in particular have recently attracted much interest. Here we con- sider the equivariant integral cohomology ring of the Grassmannian. Just as the ordinary Schubert classes form a module basis over the ordinary cohomology ring of a point (namely the ring of integers) for the ordinary integral cohomology ring of the Grassmannian, so do the equivariant Schubert classes form a basis over the equivariant cohomology of a point (namely the ordinary cohomology ring of the classifying space of the torus) for the equivariant cohomology ring (this is true for any generalized flag variety of any type, not just the Grassmannian). Again as in the ordinary case, computing the structure constants of the multiplication with respect to this basis is an interesting problem that goes by the name of Schubert calculus. There is a forgetful functor from equivariant cohomology to ordinary cohomology so that results about the former specialize to those about the latter. Knutson-Tao-Woodward (5) and Knutson-Tao (6) show that the structure con- stants, both ordinary and equivariant, count solutions to certain jigsaw puzzles, thereby showing that they are "manifestly" positive. In the present paper we take a very different route to computing the equivariant structure constants. Namely, we try to extend to the equivariant case the classical approach by means of the Pieri and Giambelli formulas. Recall, from (3, Eq.(10), p.146) for example, that the Gi- ambelli formula expresses an arbitrary Schubert class as a polynomial with integral coefficients in certain "special" Schubert classes—the Chern classes of the tautolog- ical quotient bundle—and that the Pieri formula expresses as a linear combination of the Schubert classes the product of a special Schubert class with an arbitrary Schubert class. Together they can be used to compute the structure constants. We only partially succeed in our attempt: the first of the three theorems of this paper—see §2 below—is an equivariant Giambelli formula that specializes to the ordinary Giambelli formula as in (3, Eq.(10), p.146), but we still do not have a satisfactory equivariant Pieri formula—see, however, §7 below. The derivation in Fulton (2, §14.3) of the Giambelli formula can perhaps be extended to the equi- variant case, but this is not what we do. Instead, we deduce the Giambelli formula from our second theorem which gives a certain closed-form determinantal formula for the restriction to a torus fixed point of an equivariant Schubert class.
TL;DR: The theory of combinatorial intersection cohomology of fans developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu is reviewed in this article.
Abstract: We review the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for non-rational fans, which do not define a toric variety. As a result, a number of interesting results on the toric g and h polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two complex gives the rigidity complex previously used by Kalai to study g2. We present several new results which follow from these methods, as well as previously unpublished proofs of Kalai that gk(P ) = 0 implies gk(P ∗) = 0 and gk+1(P ) = 0. For a d-dimensional convex polytope P , Stanley [St2] defined a polynomial invariant h(P, t) = ∑d k=0 hk(P )t k of its face lattice, which is usually called the “generalized” or “toric” h-polynomial of P . It is “generalized” in that it extends a previous definition from simplicial polytopes to general polytopes, while the adjective “toric” refers to the fact that if P is a rational polytope (meaning that all its vertices have all coordinates in Q), then the coefficients of h(P, t) are intersection cohomology Betti numbers of an associated projective toric variety XP : (1) hk(P ) = dimR IH(XP ;R). In fact, these Betti numbers had been computed independently by several people, including Robert MacPherson, and these calculations inspired Stanley’s definition Received November 20, 2005. The author was supported in part by NSF grant DMS-0201823.
TL;DR: In this paper, the authors give a purely rigid-analytic development of the theory of canonical subgroups with arbitrary torsion-level in generalized elliptic curves over rigid spaces over k (using Lubin's p-torsion theory only over valuation rings, which is to say on fibers).
Abstract: 1.1. Motivation. In the original work of Katz on p-adic modular forms [Kz], a key insight is the use of Lubin’s work on canonical subgroups in 1-parameter formal groups to define a relative theory of a “canonical subgroup” in p-adic families of elliptic curves whose reduction types are good but not too supersingular. The theory initiated by Katz has been refined in various directions (as in [AG], [AM], [Bu], [GK], [G], [KL]). The philosophy emphasized in this paper and its sequel [C4] in the higher-dimensional case is that by making fuller use of techniques in rigid geometry, the definitions and results in the theory can be made applicable to families over rather general rigid-analytic spaces over arbitrary analytic extensions k/Qp (including base fields such as Cp, for which Galois-theoretic techniques as in [AM] are not applicable). The aim of this paper is to give a purely rigid-analytic development of the theory of canonical subgroups with arbitrary torsion-level in generalized elliptic curves over rigid spaces over k (using Lubin’s p-torsion theory only over valuation rings, which is to say on fibers), with an eye toward the development of a general theory of canonical subgroups in p-adic analytic families of abelian varieties that we shall discuss in [C4]. An essential feature is to work with rigid spaces and not with formal models in the fundamental definitions and theorems, and to avoid unnecessary reliance on the fine structure of integral models of modular curves. The 1-dimensional case exhibits special features (such as moduli-theoretic compactification and formal groups in one rather than several parameters) and it is technically simpler, so the results in this case are more precise than seems possible in the higher-dimensional case. It is therefore worthwhile to give a separate treatement in the case of relative dimension 1 as we do in the present paper.
TL;DR: In this article, the notion of hypersymmetric abelian varieties over a field of positive characteristic p was introduced, and it was shown that every symmetric Newton polygon admits a hypersymptotic Abelian variety having that Newton polygomorphism.
Abstract: We introduce the notion of a hypersymmetric abelian variety over a field of positive characteristic p. We show that every symmetric Newton polygon admits a hypersymmetric abelian variety having that Newton polygon; see 2.5 and 4.8. Isogeny classes of absolutely simple hypersymmetric abelian varieties are classified in terms of their endomorphism algebras and Newton polygons. We also discuss connections with abelian varieties of PEL-type, i.e. abelian varieties with extra symmetries, especially abelian varieties with real multiplications.
TL;DR: Certain double character sums over points of an elliptic curve and in the multiplicative subgroup of a finite field are estimated and bounds are improved that improve and extend the scope of a series of previous results.
Abstract: We estimate certain double character sums over points of an elliptic curve and in the multiplicative subgroup of a finite field. These bounds both improve and extend the scope of a series of previous results. We apply these results to estimate the related sums over primes, and to derive new uniformity of distribution results for the elliptic curve pseudorandom number power generator. We also mention some further applications, both to cryptography and to smooth number distribution. Subject Classification (2000) Primary 11L07, 11T23 Secondary 11G20
TL;DR: In this article, the authors describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the prime sets contain arbitrarily long arithmetic progressions.
Abstract: In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving ``randomly'', and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.
TL;DR: In this paper, the connection between Hodge purity of algebraic varieties over fields of different characteristics was studied. And the authors gave sufficient conditions for varieties over number fields to satisfy this property in terms of their rational points over finite fields.
Abstract: We study the connection between Hodge purity of the cohomology of algebraic varieties over fields of different characteristics. Specifically, we study varieties over number fields, whose cohomology in some fixed degree 2i consists entirely of Hodge classes, that is, whose Hodge cohomology in degree 2i is entirely of type (i, i). Among other things, we give sufficient conditions for varieties over number fields to satisfy this property in terms of their rational points over finite fields.
TL;DR: In this article, the moment map image of the closure of an orbit of a complex torus action is shown to be convex and Brion generalized this result to actions of a maximal solvable subgroup.
Abstract: Atiyah proved that the moment map image of the closure of an orbit of a complex torus action is convex. Brion generalized this result to actions of a complex reductive group. We extend their results to actions of a maximal solvable subgroup.
TL;DR: In this article, the authors studied the action of the convolution functors with the central sheaves on the affine flag scheme G((t))/I. They showed that each object of their category is an eigen-module with respect to these functors.
Abstract: Let g be a semi-simple Lie algebra, and let g^ be the corresponding affine Kac-Moody algebra. Consider the category of g^-modules at the critical level, on which the action of the Iwahori subalgebra integrates to algebraic action of the Iwahori subgroup I. We study the action on this category of the convolution functors with the "central" sheaves on the affine flag scheme G((t))/I. We show that each object of our category is an "eigen-module" with respect to these functors. In order to prove this, we use the fusion product of modules over the affine Kac-Moody algebra.
TL;DR: In this article, the authors derived a rigorous derivation of the classical Debye $T^3$ law on the specific heat at low temperatures using a special quantization of crystal lattices.
Abstract: We discuss, from a geometric standpoint, the specific heat of a solid. This is a classical subject in solid state physics which dates back to a pioneering work by Einstein (1907) and its refinement by Debye (1912). Using a special quantization of crystal lattices and calculating the asymptotic of the integrated density of states at the bottom of the spectrum, we obtain a rigorous derivation of the classical Debye $T^3$ law on the specific heat at low temperatures. The idea and method are taken from discrete geometric analysis which has been recently developed for the spectral geometry of crystal lattices.
TL;DR: In this article, a family of subvarieties of the flag variety defined by certain linear conditions, called Hessenberg varieties, are studied and compared to Schubert varieties, and it is shown that Hessenberg types are not always pure dimensional.
Abstract: We study a family of subvarieties of the flag variety defined by certain linear conditions, called Hessenberg varieties. We compare them to Schubert varieties. We prove that some Schubert varieties can be realized as Hessenberg varieties and vice versa. Our proof explicitly identifies these Schubert varieties by their permutation and computes their dimension.
We use this to answer an open question by proving that Hessenberg varieties are not always pure dimensional. We give examples that neither semisimple nor nilpotent Hessenberg varieties need be pure; the latter are connected, non-pure-dimensional Hessenberg varieties. Our methods require us to generalize the definition of Hessenberg varieties.
TL;DR: In this article, the authors prove an explicit identity between period integrals of automorphic forms and special values of automomorphic L-series and show that certain Shimura subvarieties on Hilbert modular varieties are equidistributed.
Abstract: The objective of this paper is to prove an explicit identity between period integrals of automorphic forms and special values of automorphic L-series. As an application, we will show that certain Shimura subvarieties on Hilbert modular varieties are equidistributed. In the following we describe the main results.
TL;DR: It is proved that for every n×n orthogonal matrix U there is a non-commutative convex combination A of permutation matrices which approximates U entry-wise within an error of cn− 1 2 lnn and in the Frobenius norm within anerror of c lnn.
Abstract: Motivated in part by a problem of combinatorial optimization and in part by analogies with quantum computations, we consider approximations of orthogonal matrices U by “non-commutative convex combinations”A of permutation matrices of the type A = ∑ Aσσ, where σ are permutation matrices and Aσ are positive semidefinite n× n matrices summing up to the identity matrix. We prove that for every n×n orthogonal matrix U there is a non-commutative convex combination A of permutation matrices which approximates U entry-wise within an error of cn− 1 2 lnn and in the Frobenius norm within an error of c lnn. The proof uses a certain procedure of randomized rounding of an orthogonal matrix to a permutation matrix.
TL;DR: In this paper, an analogue of the Novikov conjecture for algebraic (or Kahler) manifolds is presented. But it does not consider non-birational mappings.
Abstract: This paper attempts to provide an analogue of the Novikov conjecture for algebraic (or Kahler) manifolds. Inter alia, we prove a conjecture of Rosenberg’s on the birational invariance of higher Todd genera. We argue that in the algebraic geometric setting the Novikov philosophy naturally includes non-birational mappings.
TL;DR: In this paper, the authors study morphisms φ : AM → AN of affine space whose extension φ̄ : PM → PN need not be a morphism.
Abstract: h ( φ(P ) ) = d · h(P ) + O(1) for all P ∈ P (K̄) combined with the fact that there are only finitely many K-rational points of bounded height leads immediately to a proof of Northcott’s Theorem [18] stating that φ has only finitely many K-rational preperiodic points. The situation is more complicated if φ : PN → PN is only required to be a rational map. An initial difficulty arises because there may be orbits Oφ(P ) that “terminate” because some iterate φn(P ) arrives at a point where φ is not defined. In this paper we study morphisms φ : AM → AN of affine space whose extension φ̄ : PM → PN need not be a morphism. (For the application to dynamics, we take M = N .) Example 1. The simplest automorphisms of A2 with interesting dynamics are the Hénon maps φ : A → A, φ(x, y) = (y, ax + f(y)), with a ∈ C∗ and f(y) ∈ C[y].
TL;DR: Goresky and MacPherson as mentioned in this paper showed that the Lefschetz hyperplane theorem for intersection homology of complex singular spaces is equivalent to the result of the Stratified Morse Theory for complex analytic spaces.
Abstract: In 1974, Mark Goresky and Robert MacPherson began their development of intersection homology theory (see [24] in these volumes), and their first paper on this topic appeared in 1980; see [12]. At that time, they were missing a fundamental tool which was available for the study of smooth manifolds; they had no Morse Theory for stratified spaces. Goresky and MacPherson wished to have a Stratified Morse Theory to allow them to prove a Lefschetz hyperplane theorem for the intersection homology of complex singular spaces, just as ordinary Morse Theory yields the Lefschetz Hyperplane Theorem for ordinary homology of complex manifolds ([34], §7). The time was ripe for a stratified version of Morse Theory. In 1970, Mather had given a rigorous proof of Thom’s first isotopy lemma [33]; this result says that proper, stratified, submersions are locally-trivial fibrations. In 1973, Morse functions on singular spaces had been defined by Lazzeri in [25], and the density and stability of Morse functions under perturbations had been proved in [37]. We shall recall these definitions and results in Section 2. What was missing was the analog of the fundamental result of Morse Theory, a theorem describing how the topology of a space is related to the critical points of a proper Morse function. In [16], Goresky and MacPherson proved such a theorem for stratified spaces. Suppose that M is a smooth manifold, that X is a Whitney stratified subset ofM , and that f : X → R is a proper function which is the restriction of a smooth function on M . For all v ∈ R, let X≤v := f−1((−∞, v]). Suppose that a, b ∈ R, a < b, and f−1([a, b]) contains a single (stratified) critical point, p, which is non-degenerate (see Definition 2.3) and contained in the open set f−1((a, b)). Let S be the stratum containing p. Then, the Main Theorem of Stratified Morse Theory (see Theorem 2.16) says that the topological space X≤b is obtained from the space X≤a by attaching a space A to X≤a along a subspace B ⊆ A, where the pair (A,B), the Morse data, is the product of the tangential Morse data of f at p and the normal Morse data of f at p. This result is especially powerful in the complex analytic case, where the normal Morse data depends on the stratum S, but not on the point p or on the particular Morse function f . Detailed proofs of these results appeared in the 1988 book Stratified Morse Theory [16]; we present a summary of a number of these results in Section 2. Even before the appearance of [16], Goresky and MacPherson published two papers, Stratified Morse Theory [15] and Morse Theory and Intersection Homology Theory [14], which contained announcements of many of the fundamental definitions and results of Stratified Morse Theory. In addition, these two papers showed that Stratified Morse Theory has a number of important applications to complex analytic spaces, including homotopy results, the desired Lefschetz Hyperplane Theorem for intersection homology, and the first proof that the (shifted) nearby cycles of a perverse sheaf are again perverse. We shall discuss these results and others in Section 3.
TL;DR: In this article, the authors study descent and co-descent for the Iwasawa modules B ∞ = lim −→n and Y ∞= lim ←−Bn, and derive the kernels and cokernels of the natural maps Bn → (B∞)n and (Y ∞)Γn → Bn.
Abstract: Abstract: Let F be a totally real abelian number field, F∞ = ∪n>0Fn its cyclotomic Zp-extension (p 6= 2), Γn = Gal(F∞/Fn). Let Bn be the ppart of the quotient Un/Cn of the units of Fn modulo circular units. We study descent and co-descent for the Iwasawa modules B∞ = lim −→n and Y∞ = lim ←−Bn; more precisely, using the Main Conjecture, we determine the kernels and cokernels of the natural maps Bn → (B∞)n and (Y∞)Γn → Bn in terms of some (module theoretic) invariants attached to F∞/F . We derive some consequences related to Greenberg’s conjecture.
TL;DR: In this article, a variant of this degeneration, due essentially to Samuel, Rees, and Nagata, is described, in which Y flatly degenerates to the balanced normal cone CXY.
Abstract: Let X be a subscheme of a reduced scheme Y. Then Y has a flat degeneration to the normal cone CXY of X, and this degeneration plays a key step in Fulton and MacPherson’s “basic construction” in intersection theory. The intersection product has a canonical refinement as a sum over the components of CXY, for X and Y depending on the given intersection problem. The cone CXY is usually not reduced, which leads to the appearance of multiplicities in intersection formulae. We describe a variant of this degeneration, due essentially to Samuel, Rees, and Nagata, in which Y flatly degenerates to the “balanced” normal cone CXY. This space is reduced, and has a natural map onto the reduction (CXY)red of CXY. The multiplicity of a component now appears as the degree of this map. Hence intersection theory can be studied using only reduced schemes. Moreover, since the map CXY → (CXY)red may wrap several components of CXY around one component of CXY, writing the intersection product as a sum over the components of CXY gives a further canonical refinement. In the case that X is a Cartier divisor in a projective scheme Y, we describe the balanced normal cone in homotopy-theoretic terms, and prove a useful upper bound on the Hilbert function of CXY.