Let b ≥ 2 be an integer and let E/ be a fixed elliptic curve. In this paper, we estimate the number of primes p ≤ x such that the number of points nE(p) on the reduction of E modulo p is a base b prime or pseudoprime. In particular, we improve previously known bounds which applied only to prime values of nE(p).

The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the G-representation naturally associated to the p∞-Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.

Let Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the boundfor any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds:We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, thenFinally we show that if g is a generator of F*p then for any M < p the number of solutions of the equationis less than . This implies that if p1/2 < M < p, then

In this paper, we study the linear structure of sets A ⊂ with doubling constant σ(A) < 2, where σ(A):=|A+A|/|A|. In particular, we show that A is contained in a small affine subspace. We also show that A can be covered by at most four shifts of some subspace V with |V| ≤ |A|. Finally, we classify all binary sets with small doubling constant.

Let andWe prove that there is an absolute constant c1 > 0 such thatfor every s ∈ (0, ∞) and n ≥ 9, where the supremum is taken for all f ∈ Gn withThis is what we call (an essentially sharp) Remez-type inequality for the class Gn. We also prove the right higher dimensional analog of the above result.

We study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

Let F(x) be the remainder term in the mean square formula of the error term Δ(t) in the Dirichlet divisor problem. We improve on the upper estimate of F(x) obtained by Preissmann around twenty years ago. The method is robust, which applies to the same problem for the error terms in the circle problem and the mean square formula of the Riemann zeta-function.

One of the fundamental problems in the study of moduli spaces is to give an intrinsic characterisation of representable functors of schemes, or of functors that are quotients of representable ones of some sort. Such questions are in general hard, leading naturally to geometry of algebraic stacks and spaces (see [1, 4]).

Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such as Coq and Agda (cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature.

Let ϕ(z) ∈ (z) be a rational function of degree d ≥ 2 with ϕ(0) = 0 and such that ϕ does not vanish to order d at 0. Let α ∈ have infinite orbit under iteration of ϕ and write ϕn(α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primitive divisor, i.e., there is a prime p such that p | An and p ∤ Ai for all i < n. More generally, we prove an analogous result when ϕ is defined over a number field and 0 is a preperiodic point for ϕ.

We prove a domination theorem for operators on Köthe function spaces by probability measures which includes both the Maurey–Rosenthal domination theorem and the Pisier domination theorem as special cases.

The relationship between posets that are cycle-free and graphs that have more than one end is considered. We show that any locally finite bipartite graph corresponding to a cycle-free partial order has more than one end, by showing a correspondence between the ends of the graph and those of the Hasse graph of its Dedekind–MacNeille completion. If, in addition, the cycle-free partial order is k-CS-transitive for some k ≥ 3 we show that the associated graph is end-transitive. Using this approach we go on to prove that, for infinite locally finite 3-CS-transitive posets with maximal chains of height 2, the properties of being crown-free and being cycle-free are equivalent. In contrast to this we show that the non-locally finite bipartite graphs arising from skeletal cycle-free partial orders each have only one end. We include a corrected proof of a result from an earlier paper on the axiomatizability of the class of cycle-free partial orders.

We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical bicategory, this being a bicategory enriched in the monoidal 2-category of pseudo double categories. Finally, we show that every sufficiently well-behaved locally cubical bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.

Given a1, . . ., ar ∈ ℚ \ {0, ±1}, the Schinzel–Wójcik problem is to determine whether there exist infinitely many primes p for which the order modulo p of each a1, . . ., ar coincides. We prove on the GRH that the primes with this property have a density and in the special case when each ai is a power of a fixed rational number, we show unconditionally that such a density is non zero. Finally, in the case when all the ai's are prime, we express the density it terms of an infinite product.

We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group.

A notion of good behavior is introduced for a definable subcategory of left R-modules. It is proved that every finitely presented left R-module has a pure projective left -approximation if and only if the associated torsion class of finite type in the functor category (mod-R, Ab) is coherent, i.e., the torsion subobject of every finitely presented object is finitely presented. This yields a bijective correspondence between such well-behaved definable subcategories and preenveloping subcategories of the category Add(R-mod) of pure projective left R-modules. An example is given of a preenveloping subcategory ⊆ Add(R-mod) that does not arise from a covariantly finite subcategory of finitely presented left R-modules. As a general example of this good behavior, it is shown that if R is a ring over which every left cotorsion R-module is pure injective, then the smallest definable subcategory (R-proj) containing every finitely generated projective module is well-behaved.

Let Sw+2(Γ0(N)) be the vector space of cusp forms of weight w + 2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2(Γ0(2)), and extend the Eichler–Shimura–Manin isomorphism theorem to Γ0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on Sw+2(Γ0(2)). As an application of main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of Sw+2(Γ0(2)).

We prove a theorem that implies:

Let G1=〈G1, ⋅, ≤〉 andG2=〈G2, ⋅, ≤〉 be ordered groups with subgroupsH1andH2, respectively. If ϕ : H1 ≅ H2is an order-preserving isomorphism, then the free product ofG1andG2withH1andH2amalgamated via ϕ is right orderable.

This solves Problem 15⋅34 from the Kourovka Notebook.

We extend this result to an arbitrary family of ordered groups with order-preserving-isomorphic subgroups amalgamated.

Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m > 2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directed m-string satellites of mutants share the same Homfly polynomial for m < 6 in general, and for all choices of m when the satellite is based on a cable knot pattern.

We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are different, by comparing their quantum sl(3) invariants.

The following theorem is proved. For any positive integer n there exists t depending only on n such that if the word w is a multilinear commutator, then the class of all groups G in which w(G) is locally finite and the product of any t values of w has order dividing n is a variety.