Let D = p1 … pm, where p1, …, pm are distinct rational primes ≡ 1(mod 8), and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the Hecke L-function of the congruent elliptic curve ED2 : y2 = x3 − D2x at s = 1, divided by the period ω defined below, to be exactly divisible by 4m. As a corollary, we obtain a series of non-congruent numbers whose number of prime factors tends to infinity, and for which the corresponding elliptic curves have non-trivial 2-part of Tate–Shafarevich group, which greatly generalizes a result of Razar [8]. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer.

Let r1(n) denote the number of representations of the positive integer n as the sum of a square of a positive integer and the square of a positive prime number. We prove an asymptotic evaluation for [sum ]n[les ]xr1(n)2, as x → ∞, thereby improving upon a O-result of Rieger [7]. We further prove an asymptotic formula for the number of positive integers n [les ] x with r1(n) [ges ] 1, which answers a question stated at the end of [7]. Our result in particular shows that for almost all integers represented as the sum of a square and a square of a prime, the representation is unique.

Soit P un polynôme à coefficients entiers de degré 2 et Δ son discriminant. La quantité τP(n), définie par τP(n) := card {P(m) > 0: P(m)[mid ]n, m ∈ ℤ}, compte le nombre de diviseurs d de n qui s'écrivent sous la forme d = P(m). Lorsque P(X) = X(X + 1), le cardinal τX(X+1)(n) est égal au nombre de diviseurs consécutifs de n.

Let G be a finite group scheme over a field k, that is, an affine group scheme whose coordinate ring k[G] is finite dimensional. The dual algebra k[G]* ≡ Homk(k[G], k) is then a finite dimensional cocommutative Hopf algebra. Indeed, there is an equivalence of categories between finite group schemes and finite dimensional cocommutative Hopf algebras (cf. [19]). Further the representation theory of G is equivalent to that of k[G]*. Many familiar objects can be considered in this context. For example, any finite group G can be considered as a finite group scheme. In this case, the algebra k[G]* is simply the group algebra kG. Over a field of characteristic p > 0, the restricted enveloping algebra u([gfr ]) of a p-restricted Lie algebra [gfr ] is a finite dimensional cocommutative Hopf algebra. Also, the mod-p Steenrod algebra is graded cocommutative so that some finite dimensional Hopf subalgebras are such algebras.

Over the past thirty years, there has been extensive study of the modular representation theory (i.e. over a field of positive characteristic p > 0) of such algebras, particularly in regards to understanding cohomology and determining projectivity of modules. This paper is primarily interested in the following two questions:

Questions1·1. Let G be a finite group scheme G over a field k of characteristic p > 0, and let M be a rational G-module.

(a) Does there exist a family of subgroup schemes of G which detects whether M is projective?

(b) Does there exist a family of subgroup schemes of G which detects whether a cohomology class z ∈ ExtnG(M, M) (for M finite dimensional) is nilpotent?

Let I = [α, β] be a subinterval of [0, 1]. For each positive integer Q, we denote by [Fscr ]I(Q) the set of Farey fractions of order Q from I, that is

and order increasingly its elements γj = aj/qj as α [les ] γ1 < γ2 < … < γNI(Q) [les ] β. The number of elements of [Fscr ]I(Q) is

We simply let [Fscr ](Q) = [Fscr ][0,1](Q), N(Q) = N[0,1](Q).

Farey sequences have been studied for a long time, mainly because of their role in problems related to diophantine approximation. There is also a connection with the Riemann zeta function which has motivated their study. Farey sequences seem to be distributed as uniformly as possible along [0, 1]; a way to prove it is to show that

for all ε > 0, as Q → ∞. Yet this is a very strong statement, as Franel and Landau [3, 4] have shown that (1·3) is equivalent to the Riemann Hypothesis.

Our object here is to investigate the distribution of spacings between Farey points in subintervals of [0, 1]. Various results related to this problem have been obtained by [2, 3, 5–8, 10–13].

Let [afr ] be an ideal of a local ring (R, [mfr ]) and let N be a finitely generated R-module of dimension d: It is shown that Hd[afr ](N) ≃ Hd[mfr ](N)/[sum ]n∈ℕ〈[mfr ]〉 (0: Hd[mfr ](N)[afr ]n); where for an Artinian R-module X we put 〈[mfr ]〉X = ∩n∈ℕ[mfr ]nX. As a consequence several vanishing and connectedness results are deduced.

There are different definitions of ends in non-locally-finite graphs which are all equivalent in the locally finite case. We prove the compactness of the end-topology that is based on the principle of removing finite sets of vertices and give a proof of the compactness of the end-topology that is constructed by the principle of removing finite sets of edges. For the latter case there exists already a proof in [1], which only works on graphs with countably infinite vertex sets and in contrast to which we do not use the Theorem of Tychonoff. We also construct a new topology of ends that arises from the principle of removing sets of vertices with finite diameter and give applications that underline the advantages of this new definition.

Let C be a germ of curve singularity embedded in (kn, 0). It is well known that the blowing-up of C centred on its closed ring, Bl(C), is a finite union of curve singularities. If C is reduced we can iterate this process and, after a finite number of steps, we find only non-singular curves. This is the desingularization process. The main idea of this paper is to linearize the blowing-up of curve singularities Bl(C) → C. We perform this by studying the structure of [Oscr ]Bl(C)/[Oscr ]C as W-module, where W is a discrete valuation ring contained in [Oscr ]C. Since [Oscr ]Bl(C)/[Oscr ]C is a torsion W-module, its structure is determined by the invariant factors of [Oscr ]C in [Oscr ]Bl(C). The set of invariant factors is called in this paper as the set of micro-invariants of C (see Definition 1·2).

In the first section we relate the micro-invariants of C to the Hilbert function of C (Proposition 1·3), and we show how to compute them from the Hilbert function of some quotient of [Oscr ]C (see Proposition 1·4).

The main result of this paper is Theorem 3·3 where we give upper bounds of the micro-invariants in terms of the regularity, multiplicity and embedding dimension. As a corollary we improve and we recover some results of [6]. These bounds can be established as a consequence of the study of the Hilbert function of a filtration of ideals g = {g[r,i+1]}i [ges ] 0 of the tangent cone of [Oscr ]C (see Section 2). The main property of g is that the ideals g[r,i+1] have initial degree bigger than the Castelnuovo–Mumford regularity of the tangent cone of [Oscr ]C.

Section 4 is devoted to computation the micro-invariants of branches; we show how to compute them from the semigroup of values of C and Bl(C) (Proposition 4·3). The case of monomial curve singularities is especially studied; we end Section 4 with some explicit computations.

In the last section we study some geometric properties of C that can be deduced from special values of the micro-invariants, and we specially study the relationship of the micro-invariants with the Hilbert function of [Oscr ]Bl(C). We end the paper studying the natural equisingularity criteria that can be defined from the micro-invariants and its relationship with some of the known equisingularity criteria.

1·1. Background. Throughout this note Q stands for a finitely generated multiplicative Abelian group of torsion-free rank n, R for a commutative ring with 1, and M for a finitely generated RQ-module. The geometric invariant ΣM of M was introduced in [BS1, BS2]. It can be viewed as a subset of the ℝ-vector space of all (additive) characters of Q, Q* = Hom(Q, ℝ) ≅ ℝn, as follows: for every character χ: Q → ℝ one considers the submonoid Qχ = {q ∈ Q [mid ] χ(q) [ges ] 0} of Q and puts

Note that 0 ∈ ΣM. It is often convenient to work with the complement ΣcM of ΣM in Q*.

The geometric invariant ΣM has been investigated for two reasons. Firstly, if R is a Dedekind domain, then ΣM turns out to be a polyhedral (i.e. a finite union of finite intersection of (open) vector half spaces) subset of Q*. This rather subtle fact was conjectured, for R a field, by Bergman [B] and established by Bieri and Groves in [BG2]; it opens the possibility for computations and imposes arithmetic restrictions on automorphisms of M. Secondly, for R = ℤ, ΣM contains interesting information on the (metabelian) groups G which are extensions of M by Q (i.e. G fits into a short exact sequence M [rarrtl ] G [Rarr ] Q). In [BS1] it is proved that G has a finite presentation if and only if ΣM ∪ −ΣM = Q*. A number of attempts have been undertaken to extend this result to a characterization of the higher dimensional finiteness property that G is of typeA group of G is of type FPm if the trivial G-module ℤ admits a free resolution F [Rarr ] ℤ with finitely generated m-skeleton. For metabelian groups G it is known, by [BS1], that FP2 is equivalent to finite presentability.FPm for m > 2, and they all revolve around the following:

FPm-Conjecture: G is of type FPmif and only if 0 ∈ Q* is not in the convex hull of m points of ΣcM.

The conjecture appeared in [BG1].

Throughout the ground field is always supposed to be algebraically closed of characteristic zero. Let X be a smooth projective threefold of general type, denote by ϕm the m-canonical map of X which is nothing but the rational map naturally associated with the complete linear system [mid ]mKX[mid ]. Since, once given such a 3-fold X, ϕm is birational whenever m [Gt ] 0, quite an interesting thing to find is the optimal bound for such an m. This bound is important because it is not only crucial to the classification theory, but also strongly related to other problems. For example, it can be applied to determine the order of the birational automorphism group of X [21, remark in section 1]. To fix the terminology we say that ϕm is stably birational if ϕt is birational onto its image for all t [ges ] m. It is well known that the parallel problem in the surface case was solved by Bombieri [1] and others. In the 3-dimensional case, many authors have studied the problem, in quite different ways. Because, in this paper, we are interested in the results obtained by Hanamura [7], we do not plan to mention more references here. According to 3-dimensional MMP, X has a minimal model which is a normal projective 3-fold with only ℚ-factorial terminal singularities. Though X may have many minimal models, the singularity index (namely the canonical index) of any of its minimal models is uniquely determined by X. Denote by r the canonical index of minimal models of X. When r = 1 we know that ϕ6 is stably birational by virtue of [3, 6, 13 and 14]. When r [ges ] 2, Hanamura proved the following theorem.

This paper is devoted to the complete calculation of the additive structure of the 2-torsion of the integral cohomology of the infinite special linear group SL(ℤ) over the ring of integers ℤ. This enables us to determine the best upper bound for the order of the Chern classes of all integral and rational representations of discrete groups.

In this paper, we will study the local cohomology modules HiI(S) of a polynomial ring S = k[x1, …, xn] with supports in a (radical) monomial ideal I. When S/I is a Cohen–Macaulay ring of dimension d (more generally, if Extn−d(S/I, [wfr ]S) is Cohen–Macaulay), we can ‘visualize’ a ℤn-graded minimal injective resolution of Hn−dI(S) using Stanley–Reisner's simplicial complex of I.

Let I be an ideal of a commutative ring R and M an R-module. It is well known that the I-adic completion functor ΛI defined by ΛI(M) = lim←tM/ItM is an additive exact covariant functor on the category of finitely generated R-modules, provided R is Noetherian. Unfortunately, even if R is Noetherian, ΛI is neither left nor right exact on the category of all R-modules. Nevertheless, we can consider the sequence of left derived functors {LIi} of ΛI, in which LI0 is right exact, but in general LI0 ≠ ΛI. Therefore the computation of these functors is in general very difficult. For the case that R is a local Noetherian ring with the maximal ideal [mfr ] and I is generated by a R-regular sequence, Matlis proved in [9, 10] that

where D(−) = HomR(−; E(R/[mfr ])) is the Matlis dual functor, and that

In [18, 5] A.-M. Simon shows that LIo(M) = M and LIi(M) = 0 for i > 0, provided that M is complete with respect to the I-adic topology.

Later, Greenlees and May [3] using the homotopy colimit, or telescope, of the cochain of Koszul complexes to define so-called local homology groups of a module M (over a commutative ring R) by

where x is a finitely generated system of I. Then they showed, under some conditions on x which are satisfied when R is Noetherian, that HI[bull ](M) ≅ LI[bull ](M). Recently, Tarrío, López and Lipman [1] have presented a sheafified derived-category generalization of Greenlees–May results for a quasi-compact separated scheme. The purpose of this paper is to study, with elementary methods of homological and commutative algebra, local homology modules for the category of Artinian modules over Noetherian rings.

In an earlier paper we studied the impact of equivalences between derived categories of group rings on their cohomology rings. Especially the group of auto-equivalences TrPic(RG) of the derived category of a group ring RG as introduced by Raphaël Rouquier and the author defines an action on the cohomology ring of this group. We study this action with respect to the restriction map, transfer, conjugation and the local structure of the group G.

We show that there is a well-defined cap-product structure on the Fintushel–Stern spectral sequence and the induced cap-product structure on the ℤ8-graded instanton Floer homology. The cap-product structure provides an essentially new property of the instanton Floer homology, from a topological point of view, which multiplies a finite-dimensional cohomlogy class by an infinite-dimensional homology class (Floer cycles) to get another infinite-dimensional homology class.

An invariant of a three-manifold with boundary is derived from topological quantum field theory. This invariant is then used as an obstruction to embedding one 3-manifold into another.

According to the result by Dyer and Formanek [4], the automorphism group of a finitely generated free two-step nilpotent group is complete except in the case when this group is a one- or three-generator (the three-generator groups have automorphism tower of height 2). The purpose of this paper is to prove that the automorphism group of an infinitely generated free two-step nilpotent group is also complete. (Recall that a group G is said to be complete if G is centreless and every automorphism is inner.)

The paper may be considered as a contribution to the study of automorphism towers of relatively free groups.

Let a cyclic group of odd prime order p act on a ℤ(p)-Poincaré duality space X. We prove a relation between the Witt classes associated to the [ ]p-cohomology rings of the fixed point set of this action and of X. This is applied to show a similar result for actions of finite p-groups on ℤ(p)-homology manifolds.

We prove an analogue of Hardy's Theorem for Fourier transform pairs in ℝ for arbitrary simply connected nilpotent Lie groups, thus extending earlier work on ℝn and the Heisenberg groups ℍn.

Fix a prime integer p. We show that a closed orientable 3-manifold M admits an action of Zp with the fixed point set S1 if and only if M can be obtained as the result of surgery on a p-periodic framed link L and Zp acts freely on the components of L. We prove a similar theorem for free Zp-actions. As an interesting application, we prove the following, rather unexpected result: for any M as above and for any odd prime p, H1(M, Zp) ≠ Zp. We also prove a similar criterion of 2-periodicity for rational homology 3-spheres.