Freiman's Theorem in an arbitrary abelian group

MSTD sets and Freiman isomorphisms

An MSTD set is a finite set with more pairwise sums than differences. $(Υ,Φ)$-ismorphisms are generalizations of Freiman isomorphisms to arbitrary linear forms. These generalized isomorphisms are used to prove that every finite set of real numbers is Freiman isomorphic to a finite set of integers. This implies that there exists no MSTD set $A$ of real numbers with $|A| \leq 7$, and, up to Freiman isomorphism, there exists exactly one MSTD set $A$ of real numbers with $|A| = 8$.

整数和有限域中的多项式弗里曼-鲁兹萨（马尔顿）猜想

The Polynomial Freiman-Ruzsa (Marton) Conjecture in Integers and Finite Fields via Spectral Stability

本文通过谱稳定性二元性与弗里曼建模，解决了整数和有限域中的多项式弗里曼-鲁兹萨猜想，并扩展了其在有限域中的应用。

带有限挠率的阿贝尔群中的Marton猜想

Marton's Conjecture in abelian groups with bounded torsion

本研究证明了带有限挠率的任意阿贝尔群中具多项式界的Freiman--Ruzsa型定理，完全证实了Marton猜想，给出了满足和集增长条件的子集的子群陪集覆盖界，论证基于F₂ⁿ情形论证的变体。

Small doubling in ordered groups: generators and structure

We prove several new results on the structure of the subgroup generated by a small doubling subset of an ordered group, abelian or not. We obtain precise results generalizing Freiman's 3k-3 and 3k-2 theorems in the integers and several further generalizations.

一般四次范数的逆理论

General inverse theory for the $\mathsf{U}^4$ norm

本文提出了一种一般有限阿贝尔群中Gowers四次范数的逆理论，通过近立方多项式和结构结果，揭示了均匀性障碍与多项式扩展的联系，并解决了有限torsion群的逆理论问题。

On sets with small doubling

Let G be an arbitrary Abelian group and let A be a finite subset of G. A has small additive doubling if |A+A| < K|A| for some K>0. These sets were studied in papers of G.A. Freiman, Y. Bilu, I. Ruzsa, M.C.--Chang, B. Green and T.Tao. In the article we prove that if we have some minor restrictions on K then for any set with small doubling there exists a set Lambda, |Lambda| << K log |A| such that |A\cap Lambda| >> |A| / K^{1/2 + c}, where c > 0. In contrast to the previous results our theorem is nontrivial for large K. For example one can take K equals |A|^η, where η>0. We use an elementary met…

A probabilistic technique for finding almost-periods of convolutions

We introduce a new probabilistic technique for finding 'almost-periods' of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth's theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain-Green theorem on the existence of long arithmetic progressions in sumsets A+B that works with sparser subs…

Small doubling in groups with moderate torsion

We determine the structure of a finite subset $A$ of an abelian group given that $|2A|<3(1-ε)|A|$, $ε>0$; namely, we show that $A$ is contained either in a "small" one-dimensional coset progression, or in a union of fewer than $ε^{-1}$ cosets of a finite subgroup. The bounds $3(1-ε)|A|$ and $ε^{-1}$ are best possible in the sense that none of them can be relaxed without tightened another one, and the estimate obtained for the size of the coset progression containing $A$ is sharp. In the case where the underlying group is infinite cyclic, our result reduces to the well-known Freiman's $(3n-3)$-…

Sharp bound on the number of maximal sum-free subsets of integers

Cameron and Erdős asked whether the number of \emph{maximal} sum-free sets in $\{1, \dots , n\}$ is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of maximal sum-free sets. Here, we prove the following: For each $1\leq i \leq 4$, there is a constant $C_i$ such that, given any $n\equiv i \mod 4$, $\{1, \dots , n\}$ contains $(C_i+o(1)) 2^{n/4}$ maximal sum-free sets. Our proof makes use of container and removal lemmas of Green, a structural result of Deshouillers, Freiman, Sós and Temkin and a recent bound on…

On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field

We prove that for every field k and every positive integer n, there exists an absolutely simple n-dimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we let S(k,n) denote the fraction of the isogeny classes of n-dimensional abelian varieties over k that consist of absolutely simple ordinary abelian varieties. Then for every n, the quantity S(F_q,n) approaches 1 as q approaches infinity over the prime powers.

在任意有限域上给定维数 $g>1$ 的简单非循环阿贝尔簇的存在性

Existence of simple non-cyclic abelian varieties over arbitrary finite fields and of a given dimension $g>1$

本文证明对任意有限域，不存在所有维数大于1的简单阿贝尔簇的有理点群均为循环群的情况，推广了Vl\u0103du\u021b关于椭圆曲线的结果。

A purity theorem for abelian schemes

We prove a purity theorem for abelian schemes in arbitrary unramified mixed characteristic (0,p). The case p=2 is completely new and the case p>2 fixes several errors in the literature.

Arithmetic Sets in Groups

We define a notion of an arithmetic set in an arbitrary countable group and study properties of these sets in the cases of Abelian groups and non-abelian free groups.

Mass formula for supersingular abelian surfaces

We show a mass formula for arbitrary supersingular abelian surfaces in characteristic $p$.

Abelian varieties over finite fields as basic abelian varieties

In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.

The characteristic polynomials of abelian varieties of higher dimension over finite fields

The characteristic polynomials of abelian varieties over the finite field $\mathbb{F}_q$ with $q=p^n$ elements have a lot of arithmetic and geometric information. They have been explicitly described for abelian varieties up to dimension 4, but little is known in higher dimension. In this paper, among other things, we obtain the following three results on the characteristic polynomial of abelian varieties. First, we prove a relation between $n$ and $e$, where $e$ is a certain multiplicity associated with a simple abelian variety of arbitrary dimension over $\mathbb{F}_q$. Second, we explicitly …

Uniformity of stably integral points on principally polarized abelian surfaces

We prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a number field. This gives a uniform version, in the spirit of a result of Caporaso-Harris-Mazur, of an unconditional theorem of Faltings. We utilize recent results of Alexeev and Nakamura on complete moduli for quasi-abelian varieties. We expect that a thorough understanding of current work of Alexeev should give a more general result for abelian varieties of an …

Corestriction Principle in non-abelian Galois cohomology

This is a revision of a McMaster University preprint, with extension. In this paper we prove that over local or global fields of characteristic 0, the Corestriction Principle holds for kernel and image of all maps which are connecting maps in group cohomology and the groups of $R$-equivalences. Some related questions over arbitrary fields of characteristic 0 are also discussed. AMS Mathematics Subject Classification (1991): Primary 11E72, Secondary 18G50, 20G10

任意域中小MSTD集合的分类

A Classification of Small MSTD Sets in Arbitrary Fields

该研究证明加法阿贝尔群无大小为5的MSTD集，借助计算机程序完成任意域中6-9大小MSTD集的分类，并探究Z/pZ乘法子群中MSTD集的最小基数。

On the abelian fivefolds attached to cubic surfaces

To a family of smooth projective cubic surfaces one can canonically associate a family of abelian fivefolds. In characteristic zero, we calculate the Hodge groups of the abelian varieties which arise in this way. In arbitrary characteristic we calculate the monodromy group of the universal family of abelian varieties, and thus show that the Galois group of the 27 lines on a suitably general cubic surface in positive characteristic is as large as possible.

Reciprocity law for compatible systems of abelian mod $p$ Galois representations

The main result of the paper is a reciprocity law which proves that compatible systems of semisimple, abelian mod $p$ representations (of arbitrary dimension) of absolute Galois groups of number fields, arise from Hecke characters. In the last section of the paper analogs for Galois groups of function fields of these results are explored, and a question is raised whose answer will require developments in transcendence theory in characteristic $p$.

Characteristic Polynomials of Simple Ordinary Abelian Varieties over Finite Fields

We provide an easy method for the construction of characteristic polynomials of simple ordinary abelian varieties ${\mathcal A}$ of dimension $g$ over a finite field ${\mathbb F}_q$, when $q\ge 4$ and $2g=ρ^{b-1}(ρ-1)$ for some prime $ρ\ge 5$, with $b\ge 1$. Moreover, we show that ${\mathcal A}$ is absolutely simple if $b=1$ and $ρ=5$, but ${\mathcal A}$ is not absolutely simple for arbitrary $ρ$ with $b>1$.

Which quaternion algebras act on a modular abelian variety?

Let A be a modular abelian variety over \Q of arbitrary even dimension. We establish criteria to prevent a given quaternion algebra over a totally real number field to be the endomorphism algebra of A over \bar\Q. We accomplish this by analyzing the representation of Gal(\bar\Q /\Q) on the points of N-torsion of A at primes N which ramify in B.

On the subgroups of finite Abelian groups of rank three

We describe the subgroups of the group $\Z_m \times \Z_n \times \Z_r$ and derive a simple formula for the total number $s(m,n,r)$ of the subgroups, where $m,n,r$ are arbitrary positive integers. An asymptotic formula for the function $n\mapsto s(n,n,n)$ is also deduced.

Theta functions of arbitrary order and their derivatives

In this paper we establish the relationships between theta functions of arbitrary order and their derivatives. We generalize our previous work math.AG/0310085 and prove that for any n>1 the map sending an abelian variety to the set of Gauss images of its points of order 2n is an embedding into an appropriate Grassmannian (note that for n=1, i.e. points of order 2, we only get generic injectivity). We further discuss the generalizations of Jacobi's derivative formula for any dimension and any order. The series of embedding of the level covers of moduli spaces of principally polarized abelian va…

Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion

In this paper we prove the Geyer-Jarden conjecture on the torsion part of the Mordell-Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on l-torsion points, for almost all primes l, contains the full symplectic group.

Endomorphism fields of abelian varieties

We give a sharp divisibility bound, in terms of g, for the degree of the field extension required to realize the endomorphisms of an abelian variety of dimension g over an arbitrary number field; this refines a result of Silverberg. This follows from a stronger result giving the same bound for the order of the component group of the Sato-Tate group of the abelian variety, which had been proved for abelian surfaces by Fite-Kedlaya-Rotger-Sutherland. The proof uses Minkowski's reduction method, but with some care required in the extremal cases when p equals 2 or a Fermat prime.