Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x + y = z, with x, y, z belonging to the set A. Let SF(G) denotes the set of all sum-free subets of $G$ and $σ(G)$ denotes the number $ n^{-1}(\log_2 |SF(G)|) $. In this article we shall improve the error term in the asymptotic formula of $σ(G)$ which was obtained recently by Ben Green and Ruzsa. The methods used are a slight refinement of methods developed by Ben Green and Ruzsa.
Let $a(n)$ denote the number of non-isomorphic abelian groups with $n$ elements. In 1991 Ivić proved an asymptotic formula of the sum $\sum_{n\leq x} a(n+ a(n)).$ In this paper, we will prove a sharper asymptotic formula for this sum.
For a positive integer $h$ and a subset $A$ of a given finite abelian group, we let $hA$, $h \hat{\;} A$, and $h_{\pm}A$ denote the $h$-fold sumset, restricted sumset, and signed sumset of $A$, respectively. Here we review some of what is known and not yet known about the minimum sizes of these three types of sumsets, as well as their corresponding critical numbers. In particular, we discuss several new open direct and inverse problems.
Let $G$ be an additive finite abelian group of order $n$, and let $S$ be a sequence of $n+k$ elements in $G$, where $k\geq 1$. Suppose that $S$ contains $t$ distinct elements. Let $\sum_n(S)$ denote the set that consists of all elements in $G$ which can be expressed as the sum over a subsequence of length $n$. In this paper we prove that, either $0\in \sum_n(S)$ or $|\sum_n(S)|\geq k+t-1.$ This confirms a conjecture by Y.O. Hamidoune in 2000.
As a consequence of their work, Bruce C. Berndt and Ronald J. Evans in 1977 and Larry Joel Goldstein and Michael Razar in 1976 obtained a formula for the square of the class number of an imaginary quadratic number field in terms of Dedekind sums. Here we focus only on this formula and give a new short and simple proof of it. We finally express the relative class numbers of imaginary abelian number fields in terms of Dedekind sums.
Using combinatorial techniques, we prove that the weighted sum of the inverse number of automorphisms of all finite abelian $p$-groups $\sum_G |G|^{-u} |\text{Aut}(G)|^{-1}$ is equal to $\prod_{j=u+1}^\infty\left(1-1/p^j\right)^{-1}$, where $u$ is a non-negative integer. This result was originally obtained by H. Cohen and H. W. Lenstra, Jr. In this paper we give a new elementary proof of their result.
A finite abelian group $G$ of cardinality $n$ is said to be of type III if every prime divisor of $n$ is congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of largest possible cardinality in a finite abelian group $G$ of type III. This theorem, when taken together with known results, gives a complete characterisation of sum-free subsets of the largest cardinality in any finite abelian group $G$. We supplement this result with a theorem on the structure of sum-free subsets of cardinality "close" to the largest possible in a type III abelian group $G$. We then give …
Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x + y = z, with x, y, z belonging to the set A. In this paper we shall characterise the largest possible sum-free subsets of G in case the order of G is only divisible by primes which are congruent to 1 modulo 3. The result is based on a recent result of Ben Green and Imre Ruzsa.
Let $G$ be an abelian group of finite order $n$, and let $h$ be a positive integer. A subset $A$ of $G$ is called {\em weakly $h$-incomplete}, if not every element of $G$ can be written as the sum of $h$ distinct elements of $A$; in particular, if $A$ does not contain $h$ distinct elements that add to zero, then $A$ is called {\em weakly $h$-zero-sum-free}. We investigate the maximum size of weakly $h$-incomplete and weakly $h$-zero-sum-free sets in $G$, denoted by $C_h(G)$ and $Z_h(G)$, respectively. Among our results are the following: (i) If $G$ is of odd order and $(n-1)/2 \leq h \leq n-2$…
We study the maximal cross number $\mathsf{K}(G)$ of a minimal zero-sum sequence and the maximal cross number $\mathsf{k}(G)$ of a zero-sum free sequence over a finite abelian group $G$, defined by Krause and Zahlten. In the first part of this paper, we extend a previous result by X. He to prove that the value of $\mathsf{k}(G)$ conjectured by Krause and Zahlten hold for $G \bigoplus C_{p^a} \bigoplus C_{p^b}$ when it holds for $G$, provided that $p$ and the exponent of $G$ are related in a specific sense. In the second part, we describe a new method for proving that the conjectured value of $…
We call a subset $A$ of the (additive) abelian group $G$ {\it $t$-independent} if for all non-negative integers $h$ and $k$ with $h+k \leq t$, the sum of $h$ (not necessarily distinct) elements of $A$ does not equal the sum of $k$ (not necessarily distinct) elements of $A$ unless $h=k$ and the two sums contain the same terms in some order. A {\it weakly $t$-independent} set satisfies this property for sums of distinct terms. We give some exact values and asymptotic bounds for the size of a largest $t$-independent set and weakly $t$-independent set in abelian groups, particularly in the cyclic …
For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$ρ(G, m, h) = \min \{|hA| \; : \; A \subseteq G, |A|=m\}$$ and $$ρ_{\pm} (G, m, h) = \min \{|h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and $h_{\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $ρ(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. In previous work we provided an upper bound for $ρ_{\pm} (G, m, h)$ that we believe is exact, and proved that $ρ_{\pm} (G, m, h)$ agrees with $ρ(G, m, h)$ when $G$ is cyclic. Here we s…
In this note, we give the explicit formula for the number of multisubsets of a finite abelian group $G$ with any given size such that the sum is equal to a given element $g\in G$. This also gives the number of partitions of $g$ into a given number of parts over a finite abelian group. An inclusion-exclusion formula for the number of multisubsets of a subset of $G$ with a given size and a given sum is also obtained.
Suppose $G$ is a finite abelian group and $S=g_{1}\cdots g_{l}$ is a sequence of elements in $G$. For any element $g$ of $G$ and $A\subseteq\mathbb{Z}\backslash\left\{ 0\right\} $, let $N_{A,g}(S)$ denote the number of subsequences $T=\prod_{i\in I}g_{i}$ of $S$ such that $\sum_{i\in I}a_{i}g_{i}=g$ , where $I\subseteq\left\{ 1,\ldots,l\right\}$ and $a_{i}\in A$. The purpose of this paper is to investigate the lower bound for $N_{A,0}(S)$. In particular, we prove that $N_{A,0}(S)\geq2^{|S|-D_{A}(G)+1}$, where $D_{A}(G)$ is the smallest positive integer $l$ such that every sequence over $G$ of …
Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in S}|. A simple application of the pigeonhole principle shows that |A|+|B|>|G|+L_S implies A\wedge^S B=G. We then prove that if |A|+|B|=|G|+L_S then |A\wedge^S B|>= |G|-2|S|. We also characterize the triples of sets (A,B,S) such that |A|+|B|=|G|+L_S and |A\wedge^S B|= |G|-2|S|. Moreover, in this case, we also provide the structure of the set G\setminus (A\wedge^…
Let $G$ be a finite abelian group. For any positive integers $d$ and $m$, let $φ_G(d)$ be the number of elements in $G$ of order $d$ and $\mathsf M(G,m)$ be the set of all zero-sum sequences of length $m$. In this paper, for any finite abelian group $H$, we prove that $$|\mathsf M(G,|H|)|=|\mathsf M(H,|G|)|$$ if and only if $φ_G(d)=φ_H(d)$ for any $d|(|G|,|H|)$. We also consider an extension of this result to non-abelian groups in terms of invariant theory.
Let S be an abelian semigroup, and A a finite subset of S. The sumset hA consists of all sums of h elements of A, with repetitions allowed. Let |hA| denote the cardinality of hA. Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial p(t) such that |hA| = p(h) for all sufficiently large h. Lattice point counting is also used to prove that sumsets of the form h_1A_1 + >... + h_rA_r have multivariate polynomial growth.
The purpose of the article is to provide an unified way to formulate zero-sum invariants. Let $G$ be a finite additive abelian group. Let $B(G)$ denote the set consisting of all nonempty zero-sum sequences over G. For $Ω\subset B(G$), let $d_Ω(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has a subsequence in $Ω$.We provide some first results and open problems on $d_Ω(G)$.
Let $G$ be a finite abelian group with exponent $n$. Let $η(G)$ denote the smallest integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a zero-sum subsequence of length at most $n$. We determine the precise value of $η(G)$ when $G$ is a $p$-group whose Davenport constant is at most $2n-1$. This confirms one of the equalities in a conjecture by Schmid and Zhuang from 2010.
Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a constant depending on G and lying in the interval [2/7,1/2]. We also estimate the number of sum-free subsets of G. It turns out that log_2 of this number is c(G)|G| + o(|G|), which is tight up to the o-term. For certain abelian groups, those whose order is divisible by a small prime of the form 3k + 2, we can obtain an asymptotic for the number of sum-free s…
In this paper we study sum-free sets of order $m$ in finite Abelian groups. We prove a general theorem on 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting. As a consequence, we determine the typical structure and asymptotic number of sum-free sets of order $m$ in Abelian groups $G$ whose order is divisible by a prime $q$ with $q \equiv 2 \pmod 3$, for every $m \ge C(q) \sqrt{n \log n}$, thus extending and refining a theorem of Green and Ruzsa. In particular, we prove that almost all sum-free subsets of size $m…
In this expository paper, we review the formula of Chowla and Selberg for the periods of elliptic curves with complex multiplication, and discuss two methods of proof. One uses Kronecker's limit formula and the other uses the geometry of a family of abelian varieties. We discuss a generalization of this formula, which was proposed by Colmez, as well as some explicit Hodge cycles which appear in the geometric proof.
A subset $A$ of a given finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ does not equal the sum of $l$ (not necessarily distinct) elements of $A$. We are interested in finding the maximum size $λ_{k,l}(G)$ of a $(k,l)$-sum-free subset in $G$. A $(2,1)$-sum-free set is simply called a sum-free set. The maximum size of a sum-free set in the cyclic group $\mathbb{Z}_n$ was found almost forty years ago by Diamanda and Yap; the general case for arbitrary finite abelian groups was recently settled by Green and Ruzsa. Here we find the va…
Suppose that $A$, $B$ and $S$ are non-empty subsets of a finite abelian group $G$. Then the generalized restricted sumset $$ A\stackrel{S}+B:=\{a+b:\,a\in A,\ b\in B,\ a-b\not\in S\} $$ contains at least $$ \min\{|A|+|B|-3|S|,p(G)\} $$ elements, where $p(G)$ is the least prime factor of $|G|$. Further, we also have $$ |A\stackrel{S}+B|\geq \min\{|A|+|B|-|S|-2,p(G)\}, $$ provided that both $|A|$ and $|B|$ are large with respect to $|S|$.
Let $F$ be a totally real field with ring of integers $O_F$, and $D$ be a totally definite quaternion algebra over $F$. A well-known formula established by Eichler and then extended by Körner computes the class number of any $O_F$-order in $D$. In this paper we generalize the Eichler class number formula so that it works for arbitrary $\mathbb{Z}$-orders in $D$. The motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a prime finite field $\mathbb{F}_p$. We give explicit formulas for the number of these isomorphism classes for all prim…
In this paper, we study the minimal number of elements of maximal order within a zero-sumfree sequence in a finite Abelian p-group. For this purpose, in the general context of finite Abelian groups, we introduce a new number, for which lower and upper bounds are proved in the case of finite Abelian p-groups. Among other consequences, the method that we use here enables us to show that, if we denote by exp(G) the exponent of the finite Abelian p-group G which is considered, then a zero-sumfree sequence S with maximal possible length in G must contain at least exp(G)-1 elements of maximal order,…
Consider a quartic $q$-Weil polynomial $f$. Motivated by equidistribution considerations we define, for each prime $\ell$, a local factor which measures the relative frequency with which $f\bmod \ell$ occurs as the characteristic polynomial of a symplectic similitude over $\mathbb{F}_\ell$. For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over $\mathbb{F}_q$ with Weil polynomial $f$.
Consider a $q$-Weil polynomial $f$ of degree $2g$. Using an equidistribution assumption that is too strong to be true, we define and compute a product of local relative densities of matrices in $\rm{GSp}_{2g}(\mathbb{F}_\ell)$ with characteristic polynomial $f\mod\ell$ when $g$ is an odd prime. This infinite product is closely related to a ratio of class numbers. When $g=3$ we conjecture that the product gives the size of an isogeny class of principally polarized abelian threefolds.
In this paper, we present a reciprocity on finite abelian groups involving zero-sum sequences. Let $G$ and $H$ be finite abelian groups with $(|G|,|H|)=1$. For any positive integer $m$, let $\mathsf M(G,m)$ denote the set of all zero-sum sequences over $G$ of length $m$. We have the following reciprocity $$|\mathsf M(G,|H|)|=|\mathsf M(H,|G|)|.$$ Moreover, we provide a combinatorial interpretation of the above reciprocity using ideas from rational Catalan combinatorics. We also present and explain some other symmetric relationships on finite abelian groups with methods from invariant theory. A…
Let G be a finite abelian group. For g in G and i an integer we define N(i,g) to be the number of subsets of G of size i which sum up to g. We will give a short proof, using character theory, of a formula for these N(i,g) due to Li and Wan. We also give a formula for N(i,g)*, the number of subsets of G not containing 0 of size i which sum up to g. This generalizes another result of Wan.
A subset $A$ of a finite abelian group is called $(k,\ell)$-sum-free if $kA \cap \ell A=\emptyset.$ In this paper, we extend this concept to compact abelian groups and study the question of how large a measurable $(k,\ell)$-sum-free set can be. For integers $1 \leq k <\ell$ and a compact abelian group $G$, let $$λ_{k,\ell}(G)=\sup\{ μ(A): kA \cap \ell A =\emptyset \}$$ be the maximum possible size of a $(k,\ell)$-sum-free subset of $G$. We prove that if $G=\mathbb{I} \times M$, where $\mathbb{I}$ is the identity component of $G$, then $$λ_{k, \ell}(G)=\max \left\{ λ_{k, \ell}(M), λ_{k, \ell}(\…
Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this paper is to investigate the lower bound for $N_g(S)$. In particular, we prove that either $N_g(S)=0$ or $N_g(S) \ge 2^{|S|-D(G)+1}$, where $D(G)$ is the smallest positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a nonempty zero-sum subsequence. We also characterize the structures of the extremal sequences for which the equality holds for some groups.
Let $K$ be a field finitely generated over ${\Q}$, and $A$ an Abelian variety defined over $K$. Then by the Mordell-Weil Theorem, the set of rational points $A(K)$ is a finitely-generated Abelian group. In this paper, assuming Tate's Conjecture on algebraic cycles, we prove a limit formula for the Mordell-Weil rank of an arbitrary family of Abelian varieties $A$ over a number field $k$; this is the Abelian fibration analogue of the Nagao formula for elliptic surfaces $E$, originally conjectured by Nagao, and proven by Rosen and Silverman to be equivalent to Tate's Conjecture for $E$. We also g…
We give an explicit formula for the Mordell-Weil rank of an abelian fibered variety and some of its applications for an abelian fibered hyperkähler manifold. As a byproduct, we also give an explicit example of an abelian fibered variety in which the Picard number of the generic fiber in the sense of scheme is different from the Picard number of generic closed fibers.
This paper gives a natural extension of Frobenius-Stickelberger formula and Kiepert formula to Abelian functions for "Purely Trigonal Curves", especially, of degree four. A description on the theory of Abelian functions for general trigonal curves of degree four is also included.
Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60], the author claimed some surprising connections among these seemingly unrelated fascinating areas. In this paper we establish further connections between zero-sum problems for abelian p-groups and covers of the integers. For example, we extend the famous Erdos-Ginzburg-Ziv theorem in the follo…
Let $A$ be a finite, nonempty subset of an abelian group. We show that if every element of $A$ is a sum of two other elements, then $A$ has a nonempty zero-sum subset. That is, a (finite, nonempty) sum-full subset of an abelian group is not zero-sum-free.
We express the Frobenius-Hecke traces on the compactly supported cohomology of a Shimura variety of abelian type in terms of elliptic parts of stable Arthur-Selberg trace formulas for the endoscopic groups. This confirms predictions of Langlands and Kottwitz at primes where the level is hyperspecial.
Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is best possible, and we obtain the stronger (exact best possible) bound in almost all cases. We prove similar results in the case |G| is even. Our proof requires us to extend a theorem of Olson on the number of subset sums of anti-symmetric subsets S from the case of Z_p to the case of a general finite abelian group. To do so, we adapt Olson's method using a g…
Using the theory of polarised flag type quotients, we determine mass formulae for all principally polarised supersingular abelian threefolds defined over an algebraically closed field $k$ of characteristic $p$. We combine these results with computations of the automorphism groups to study Oort's conjecture; we prove that every generic three-dimensional principally polarised supersingular abelian variety over $k$ of characteristic $\neq 2$ has automorphism group $\{ \pm 1 \}$.
We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and $A$-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and $A$-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.
For a finite abelian group $G$ with subsets $A$ and $B$, the sumset $AB$ is $\{ab \mid a\in A, b \in B\}$. A fundamental problem in additive combinatorics is to find a lower bound for the cardinality of $AB$ in terms of the cardinalities of $A$ and $B$. This article addresses the analogous problem for sequences in abelian groups and flags in field extensions. For a positive integer $n$, let $[n]$ denote the set $\{0,\dots,n-1\}$. To a finite abelian group $G$ of cardinality $n$ and an ordering $G = \{1=v_0,\dots,v_{n-1}\}$, associate the function $T \colon [n] \times [n] \rightarrow [n]$ defin…
Given a finite subset A of an abelian group G, we study the set k \wedge A of all sums of k distinct elements of A. In this paper, we prove that |k \wedge A| >= |A| for all k in {2,...,|A|-2}, unless k is in {2,|A|-2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite subsets A of G for which |k \wedge A| = |A| for some k in {2,...,|A|-2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.
In this article, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds true for finite cyclic groups, finite Abelian p-groups and for finite Abelian groups of rank two. Also, the results obtained here enable us to improve, via the resolution of a linear integer program, a result of W. Gao and A. Geroldinger concerning the minimal number of elements with maximal order in a long zero-sumfree sequence of a finite Abelian group of r…
Colmez conjectured a product formula for periods of abelian varieties with complex multiplication by a field K, analogous to the standard product formula in algebraic number theory. He proved this conjecture up to a rational power of 2 for K/Q abelian. In this paper, we complete the proof of Colmez for K/Q abelian by eliminating this power of 2. Our proof relies on analyzing the Galois action on the De Rham cohomology of Fermat curves in mixed characteristic (0, 2), which in turn relies on understanding the stable reduction of Z/2^n-covers of the projective line, branched at three points.
Let S be an abelian semigroup, written additively. Let A be a finite subset of S. We denote the cardinality of A by |A|. For any positive integer h, the sumset hA is the set of all sums of h not necessarily distinct elements of A. We define 0A = {0}. If A_1,...,A_r, and B are finite sumsets of A and h_1,...,h_r are nonnegative integers, the sumset h_1A + ... + h_rA_r + B is the set of all elements of S that can be represented in the form u_1 + ... + u_r + b, where u_i \in h_iA_i and b \in B. The growth function of this sumset is γ(h_1,...,h_r) = |h_1A + ... + h_rA_r + B|. Applying the Hilbert …
We study a zero-sum problem dealing with minimal zero-sum sequences of maximal length over finite abelian groups. A positive answer to this problem yields a structural description of sets of lengths with maximal elasticity in transfer Krull monoids over finite abelian groups.
We prove formulas for the p-adic logarithm of quaternionic Darmon points on p-adic tori and modular abelian varieties over Q having purely multiplicative reduction at p. These formulas are amenable to explicit computations and are the first to treat Stark-Heegner type points on higher-dimensional abelian varieties.
Let $G$ be a finite, non-trivial abelian group of exponent $m$, and suppose that $B_1, ..., B_k$ are generating subsets of $G$. We prove that if $k>2m \ln \log_2 |G|$, then the multiset union $B_1\cup...\cup B_k$ forms an additive basis of $G$; that is, for every $g\in G$ there exist $A_1\subset B_1, ..., A_k\subset B_k$ such that $g=\sum_{i=1}^k\sum_{a\in A_i} a$. This generalizes a result of Alon, Linial, and Meshulam on the additive bases conjecture. As another step towards proving the conjecture, in the case where $B_1, ..., B_k$ are finite subsets of a vector space we obtain lower-bound e…
We give a formula for the Eisenstein cohomology of local systems on the partial compactification of the moduli of principally polarized abelian varieties given by rank 1 degenerations. For genus 2 we give a formula for the full Eisenstein cohomology.
DICATAM, Università degli Studi di Brescia（布雷西亚大学 DICATAM）；DII, Università degli Studi di Brescia（布雷西亚大学 DII）；Department of Engineering and Sciences, Universitas Mercatorum（Mercatorum大学工程与科学系）；Department of Mathematics, Universitat Politècnica de Catalunya（加泰罗尼亚理工大学数学系）
Let $A$ be an abelian variety over the function field of a smooth projective curve $C$ over an algebraically closed field $k$. We compute the $l$-adic cohomology groups of $C$ with coefficients in the locally constant sheaf associated to $H^1(\bar A,\mathbf{Q}_l)$ in terms of arithmetico-geometric invariants of $A$. We apply this, when $k$ is the algebraic closure of a finite field, to a motivic computation of the $L$-function of $A$.
