Let a ∈ ℕ [setmn ] {0, 1} and let bn be a sequence of rational integers satisfying bn = O(η−2n) for every η ∈]0, 1[. We prove that the number S = [sum ]+∞n=0 1/(a2n + bn) is transcendental by using a special form of Mahler's transcendence method.

Let p be an odd prime. The results in this paper concern the units of the infinite extension of Qp generated by all p-power roots of unity. Let

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where μpn+1 denote the pn+1th roots of 1. Let [pscr ]n be the maximal ideal of the ring of integers of Φn and let Un be the units congruent to 1 modulo [pscr ]n.

Let ζn be a fixed primitive pn+1th root of unity such that ζpn = ζn − 1, ∀n [ges ] 1. Put πn = ζn − 1. Thus πn is a local parameter for Φn. Let

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Kummer already exploited the obvious fact that every u0 ∈ U0 can be written in the form

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where f0(T) is some power series in Zp[[T]]. Here of course, the power series f0(T) is not uniquely determined. Let

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the inverse limit being with respect to norm maps. Coates and Wiles (see [3]) discovered that any unit u = (un) ∈ U∞ has a unique power series fu(T) in Zp[[T]] with fu(πn) = un. The uniqueness of such a power series is obvious by Weierstrass Preparation Theorem, but the existence is in no way obvious. They worked with the formal group of height one attached to an elliptic curve with complex multiplication at an ordinary prime, but their ideas apply to any Lubin–Tate group defined over Zp. Almost immediately, Coleman [4] gave a totally different proof of the existence of the fu(T), which holds for all Lubin–Tate groups. We refer to such a power series as a Coleman power series. In this paper we adopt the same approach as [3]. We first prove the following result which is stronger than the original one in [3].

Let E be a modular elliptic curve over ℚ and let L(E, s) denote the associated L-function. The Birch and Swinnerton-Dyer conjecture then predicts that L(E, s) has a zero at s = 1 of order precisely equal to the rank of the Mordell–Weil group E(ℚ). According to Waldspurger's theorem [26] we know that there exists a real quadratic character χ such that the twisted L-function L(E, χ, s) does not vanish at s = 1. Recently Kolyvagin [16] has proved that E(ℚ) is finite provided L(E, 1) ≠ 0 and there exists a suitable real quadratic character χt such that L(E, χt, s) has a simple zero at s = 1. The latter condition was proved to be true for infinitely many t [4, 19]. Iwaniec [15] and Perelli and Pomykała [20] have proved quantitative results on this condition; Pomykała [21] has generalized it to the nth derivative of L(E, χ, s).

Variants and generalizations are possible. For instance, Friedberg and Hoffstein [10] established a nonvanishing theorem for quadratic twists of the L-series of an arbitrary cuspidal automorphic form on GL(2) over any number field. In some cases such a result can be applied to the construction of l-adic representations associated to modular forms over imaginary quadratic fields [25]. Further examples as well as some perspective concerning the higher rank case is discussed in an excellent survey article [5].

We should also mention that there are examples of irreducible cuspidal automorphic representations of GL(2) over a number field such that the corresponding twisted L-function vanishes at the centre of the critical strip for all quadratic characters [22].

Here is our contribution to the above picture.

Let M be a pure motive over ℚ, and let L(M, s) denote the corresponding L-function. In this paper we prove, under certain assumptions, (a quantitative version of) a nonvanishing theorem for n-th derivatives of quadratic twists of L(M, s) at the centre of the critical strip (Theorem 3). Also, assuming the Birch and Swinnerton-Dyer conjecture for abelian varieties over ℚ, we obtain a bound for the order of the (twisted) Shafarevich–Tate group (Theorems 4 and 5).

All the results of this paper except Section 5 could be stated (and proved) without using any motivic language. Almost all we need to know about motives concerns the conjectural description of their L-functions (see Section 1).

Let A ⊂ B be a homogeneous inclusion of standard graded algebras with A0 = B0. To relate properties of A and B we intermediate with another algebra, the associated graded ring G = grA1B(B). We give criteria as to when the extension A ⊂ B is integral or birational in terms of the codimension of certain modules associated to G. We also introduce a series of multiplicities associated to the extension A ⊂ B. There are applications to the extension of two Rees algebras of modules and to estimating the (ordinary) multiplicity of A in terms of that of B and of related rings. Many earlier results by several authors are recovered quickly.

The geometry of a desingularization Ym of an arbitrary subvariety of a generic hypersurface Xn in an ambient variety W (e.g. W = ℙn+1) has received much attention over the past decade or so. Clemens [5] has proved that for m = 1, n = 2, W = ℙ3 and X of degree d, Y has genus g [ges ] 1 + d(d − 5)/2; Xu [13, 14] improved this to g [ges ] d(d −3)/2 − 2 for d [ges ] 5 and showed that if equality holds for d [ges ] 6 then Y is planar; he also gave some lower bounds on the geometric genus pg(Y) in case m = n − 1. Voisin [10, 11] proved that, for X of degree d in ℙn+1, n [ges ] 3, m [les ] n − 2 then then pg(Y) > 0 if d [ges ] 2n + 1 − m and KY separates generic points if d [ges ] 2n + 2 − m (see also [1, 4]). For Xn a generic complete intersection of type (d1, …, dk) in any smooth polarized (n + k)-fold M, Ein [6, 7] proved that pg(Y) > 0 if d1 + … + dk [ges ] 2n + k − m + 1 and Y is of general type if d1 + … + dk [ges ] 2n + k − m + 2.

In [2] the first two authors applied the classical method of focal loci of families as in Segre [8] and Ciliberto–Sernesi [3] (really normal bundle considerations) to give a new proof of one of the main results of Xu in [13, 14] and to extend the lower bounds for the genus in the cases of general surfaces in a component of the Noether–Lefschetz locus in ℙ3 and of general projectively Cohen–Macaulay surfaces in ℙ4.

In this paper we introduce some notions of ‘filling’ subvarieties and use them to prove two new results which inter alia give another perspective on the above results, especially the genus bounds. The general philosophy is as usual that as Y moves with X, sections of KY (or some twist) can be produced through differentiation; but here this is implemented by exploiting and extending an elementary but perhaps surprising technique in the spirit of classical projective geometry which goes back to [2] and which gives a useful lower bound, depending on the dimension of the projective span of Y, on the number of independent sections of KY produced by the differentiation process.

As to our results, in Theorem 1 below we extend and refine in the above sense most of the above-quoted results (not including Voisin's), by giving a lower bound on the number of sections of a certain twist of KY involving KW (for a recent extension, including the proof of a conjecture of Clemens, see [15]). Our second result (Theorem 2), based on the notion of ‘r-filling’ subvariety, indicates an apparently new and unexpected direction as it deals with some higher-order tensors on Y, manifested in the form of an effective divisor on a Cartesian product Yr having certain vanishing order on a diagonal locus as well as on a ‘double point’ locus associated to the map Y → X. As one application, we conclude a lower bound on the number of quadrics (and higher-degree hypersurfaces) containing certain ‘adjoint-type’ projective images of Y (and even on the dimension of the kernel of certain ‘symmetric Gaussian’ maps) (Corollary 2·1 below). Our feeling, however, is that the latter is only the tip of an iceberg and we hope to explore further in this direction in the future.

Let S (respectively, S′) be a finite subset of a compact connected Riemann surface X (respectively, X′) of genus at least two. Let [Mscr ] (respectively, [Mscr ]′) denote a moduli space of parabolic stable bundles of rank 2 over X (respectively, X′) with fixed determinant of degree 1, having a nontrivial quasi-parabolic structure over each point of S (respectively, S′) and of parabolic degree less than 2. It is proved that [Mscr ] is isomorphic to [Mscr ]′ if and only if there is an isomorphism of X with X′ taking S to S′.

Let N ⊆ M be a pair of closed smooth manifolds and L an algebraic model for the submanifold N. In this paper, we will give an obstruction to finding an algebraic model X of M so that the submanifold N corresponds in X to an algebraic subvariety isomorphic to L.

We prove that if k, F are fields of different characteristics then the permutation module kFn for the affine group AGL(n, F) has simple augmentation submodule. This proves a conjecture of A. R. Camina.

The main result of the paper is that the real characters of a group G of type FPm (Fm respectively) that do not vanish on a normal locally polycyclic-by-finite subgroup represent elements of the geometric invariant [sum ]m(G, ℤ) ([sum ]m(G) respectively). In the case m = 2 a stronger result is proved. Some consequences of the main result are considered.

In this paper, various Euler characteristic formulas for simplicial maps are obtained, which generalize the Izumiya–Marar formula [14], the Banchoff triple point formula [3] and the formula due to Szücs for maps of surfaces into 3-space [27]. Moreover, we obtain new results about the Euler characteristics of the multiple point sets and the images of generic smooth maps and the numbers of their singularities.

In this note we first show, using results of McDuff, that for a Cr-manifold S, diffeomorphic to the interior of a compact manifold with boundary, the class of all Cr-diffeomorphisms lying in a one parameter group of S generates the connected component of 1S in Diffr (S, S). Then we use this result to obtain two extension theorems for stratified maps defined on some strata of a stratified space [Xscr ]. Our extension theorems hold for Mather's abstract stratified sets, for Whitney (b)-regular, Bekka (c)-regular, Verdier (w)-regular and Lipschitz-regular stratified spaces.

We determine first the structure of certain exponential reduced subsemigroups of the Lie group ℝ × G, where G is a compact and connected Lie group. Next, we construct the Bohr compactification of these subsemigroups. The main result is a structure theorem of this Bohr compactification. In the final section we make some links to the theory of one-parameter semigroups of subsets of Lie groups (due to Rådström).

We generalize the Carathéodory version of the Schwarz reflection principle for pseudo-metrics

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where w is analytic in some domain of the complex plane, [mid ] w [mid ] < 1 and use a unit disk version of the new result to study certain homeomorphisms of the unit ball in [Hscr ]([ ]) onto itself.

The random record distribution ν associated with a probability distribution μ can be written as a convolution series, ν = [sum ]∞n=1n−1 (n + 1)−1μ*n. Various authors have obtained results on the behaviour of the tails ν((x, ∞)) as x → ∞, using Laplace transforms and the associated Abelian and Tauberian theorems. Here we use Gelfand transforms and the Wiener–Lévy–Gelfand Theorem to obtain expansions of the tails under moment conditions on μ. The results differ notably from those known for other convolution series.