An extension of Stern's congruence

Eisenstein同余在tame族中的性质与一个metabelian扩张的类群

Eisenstein congruences in tame families and the class group of a metabelian extension

本文利用tame族中Eisenstein同余存在性的新结果，解释了一个特定metabelian数域中素数分裂行为的计算观察，延续了Ribet开创的范式。

An Extension of a Congruence by Tauraso

For a positive integer $n$ let $H_n=\sum_{k=1}^{n}1/n$ be the $n$th harmonic number. In this note we prove that for any prime $p\ge 7$, $$ \sum_{k=1}^{p-1}\frac{H_k}{k^2}\equiv \sum_{k=1}^{p-1}\frac{H_k^2}{k} \equiv\frac{3}{2p}\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^2}. $$ Notice that the first part of this congruence is recently proposed by R. Tauraso as a problem in Amer. Math. Monthly. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers and a combinatorial i…

An Extension of a Congruence by Kohnen

Let $p>3$ be a prime, and let $q_p(2)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base 2. Recently, Z. H. Sun proved that \sum_{k=1}^{p-1}\frac{1}{k\cdot 2^k}\equiv q_p(2)-\frac{p}{2}q_p(2)^2 \pmod{p^2} which is a generalization of a congruence due to W. Kohnen. In this note we give an elementary proof of the above congruence which is based on several combinatorial identities and congruences involving the Fermat quotient $q_p(2)$, harmonic or alternating harmonic sums.

Genus fields of abelian extensions of congruence rational function fields

In the published version of this paper [Finite Fields and Their Applications {\bf 20} (2013) 40--54], there is an error in the proof of Theorem 4.2 of the paper. Here we correct the error and give the right statments for Theorems 4.2, 4.5 and 5.2 We give a construction of genus fields for congruence function fields. First we consider the cyclotomic function field case following the ideas of Leopoldt and then the general case. As applications we give explicitly the genus fields of Kummer, Artin--Schreier and cyclic $p$--extensions. Kummer extensions were obtained previously by G. Peng and Artin…

V Tree -- Continued Fraction Expansion, Stern-Brocot Tree, Minkowski's $?(x)$ Function In Binary: Exponentially Faster

The Stern-Brocot tree and Minkowki's question mark function $?(x)$ (or Conway's box function) are related to the continued fraction expansion of numbers from Q with unary encoding of the partial denominators. We first define binary encodings $C_I, C_{II}$ of the natural numbers, adapted to the Gauß-Kuz'min measure for the distribution of partial denominators. We then define the V$_1$ tree as analogue to the Stern-Brocot tree, using the binary encondings $C_I, C_{II}$. We shall see that all numbers with denominator $q$ are present in the first $3.44\log_2(q)$ levels, instead of $1/q$ appearing …

Some congruences on prime factors of class number of finite algebraic extensions K/Q

This paper is a contribution to the description of some congruences on the odd prime factors of the class number of the number fields. An example of results obtained is: Let L/Q be a finite Galois solvable extension with [L:Q]=N, where N > 1 is odd. Let h(L) be the class number of L. Suppose that h(L) > 1. Let p be a prime dividing h(L). Let r be the rank of the p-class group of L. Then the product p\times (p^r-1)\times(p^{r-1}-1)\times ....\times (p-1) and N are not coprime. The proofs are elementary.

Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences

New congruences are found for Andrews' smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part alpha(24z) of a certain weak Maass form M(z) of weight 3/2. We show that a normalized form of the generating function for spt(n) is an eigenform modulo 72 for the Hecke operators T(p^2) for primes p > 3, and an eigenform modulo t for t = 5, 7 or 13 provided that (t, 6p) = 1. The result for the modulus 3 was observed earlier by the author and considered by Ono and Folsom. Similar congruences for higher powers of t (namely 5^6, 7^4 and 13^2) occur…

Special values of L-functions and false Tate curve extensions II

In this paper we show how one can combine the p-adic Rankin-Selberg product construction of Hida with freeness results of Hecke modules of Wiles to establish interesting congruences between special values of L-functions. These congruences is a part of some deep conjectural congruences that follow from the work of Kato on the non-commutative Iwasawa theory of the false Tate curve extension.

Some congruences on prime factors of class number of algebraic extensions K/Q

This paper is an updated version of ANT-0372 (2002 dec 4) with the same title. Several errors are corrected in this version. An example of the kind of results obtained is: Let K/\Q be an abelian extension with N = [K:\Q] > 1, N odd. Let h(K) be the class number of K. Suppose that h(K) > 1. Let p be a prime dividing h(K). Let r_p be the rank of the p-class group of K. Then p \times (p^{r_p}-1) and N are not coprime. The paper is at elementary level and contains a lot of numerical examples.

On Euler numbers modulo powers of two

To determine Euler numbers modulo powers of two seems to be a difficult task. In this paper we achieve this and apply the explicit congruence to give a new proof of a classical result due to M. A. Stern.

Genus Fields of Congruence Function Fields

Let $k$ be a rational congruence function field and consider an arbitrary finite separable extension $K/k$. If for each prime in $k$ ramified in $K$ we have that at least one ramification index is not divided by the characteristic of $K$, we find the genus field $\g K$, except for constants, of the extension $K/k$. In general, we describe the genus field of a global function field.

Euler Characteristics and their Congruences in the Positive Rank Setting

The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible $p$-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves.

Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)

In 1862 Wolstenholme proved that for any prime $p\ge 5$ the numerator of the fraction $$ 1+\frac 12 +\frac 13+...+\frac{1}{p-1} $$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of the fraction $$ 1+\frac{1}{2^2} +\frac{1}{3^2}+...+\frac{1}{(p-1)^2} $$ written in reduced form is divisible by $p$. The first of the above congruences, the so called {\it Wolstenholme's theorem}, is a fundamental congruence in combinatorial number theory. In this article, consisting of 11 sections, we provide a historical survey of Wolstenholme's type congruences and related problems. Namely…

Congruences between abelian pseudomeasures, II

We extend the main result of [Math. Res. Lett. 15 (2008), 715-725] to Galois extensions L/K of totally real number fields of arbitrary odd prime power degree, thereby offering support for the validity of the 'main conjecture' of equivariant Iwasawa theory.

用于素性判定的二项式系数同余的充分集合

The Sufficient Set of Congruences of Binomial Coefficients for Primality

该研究证明整数p的素性等价于二项式系数模p的⌈(1/2)log₂p⌉个同余式，并将结果推广到其他类似同余式，为素性判定提供了新的充分条件。

A family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial

We prove a two-parameter family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Crucial ingredients in our proof are George Andrews' multiseries extension of the Watson transformation, and a Karlsson--Minton type summation for very-well-poised basic hypergeometric series due to George Gasper. The new family of $q$-congruences is then used to prove two conjectures posed earlier by the authors.

Gessel-Lucas congruences for sporadic sequences

For each of the $15$ known sporadic Apéry-like sequences, we prove congruences modulo $p^2$ that are natural extensions of the Lucas congruences modulo $p$. This extends a result of Gessel for the numbers used by Apéry in his proof of the irrationality of $ζ(3)$. Moreover, we show that each of these sequences satisfies two-term supercongruences modulo $p^{2r}$. Using special constant term representations recently discovered by Gorodetsky, we prove these supercongruences in the two cases that remained previously open.

Apéry数的Gauss同余的推广

An extension of Gauss congruences for Apéry numbers

本文通过Bernoulli数建立了包含多种Apéry数及Apéry-like数为特例的数列A_n^{(r,s,t)}的Gauss同余的深一层同余推广。

Polynomial extension of Fleck's congruence

Let $p$ be a prime, and let $f(x)$ be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the $p$-adic order of the sum $$\sum_{k=r(mod p^β)}\binom{n}{k}(-1)^k f([(k-r)/p^α]),$$ where $α\geβ\ge 0$, $n\ge p^{α-1}$ and $r\in Z$. This polynomial extension of Fleck's congruence has various backgrounds and several consequences such as $$\sum_{k=r(mod p^α)}\binom{n}{k} a^k\equiv 0 (mod p^{[(n-p^{α-1})/ϕ(p^α)]})$$ provided that $α>1$ and $a\equiv-1(mod p)$.

eta-quotient新形式的例外同余

Exceptional Congruences for Eta-quotient newforms

本文研究了eta-quotient新形式系数的I型和II型同余，利用模形式理论证明了这些同余及对素数幂的扩展。

Lucas' theorem: its generalizations, extensions and applications (1878--2014)

In 1878 É. Lucas proved a remarkable result which provides a simple way to compute the binomial coefficient ${n\choose m}$ modulo a prime $p$ in terms of the binomial coefficients of the base-$p$ digits of $n$ and $m$: {\it If $p$ is a prime, $n=n_0+n_1p+\cdots +n_sp^s$ and $m=m_0+m_1p+\cdots +m_sp^s$ are the $p$-adic expansions of nonnegative integers $n$ and $m$, then \begin{equation*} {n\choose m}\equiv \prod_{i=0}^{s}{n_i\choose m_i}\pmod{p}. \end{equation*}} The above congruence, the so-called {\it Lucas' theorem} (or {\it Theorem of Lucas}), plays an important role in Number Theory and C…

A New Super Congruence Involving Multiple Harmonic Sums

Let ${\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p\ge 5$ and $r\ge 2$, we prove that \begin{equation} \sum\limits_{\begin{smallmatrix} {{l}_{1}}+{{l}_{2}}+\cdots +{{l}_{5}}={{p}^{r}} {{l}_{1}},\cdots ,{{l}_{5}}\in {\mathcal{P}_{p}} \end{smallmatrix}}{\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}}}\equiv -\frac{5!}{6}{{B}_{p-5}}{{p}^{r-1}} \pmod{{{p}^{r}}}. \end{equation} This gives an extension of a family of super congruences found by Wang, Cai and Zhao.

Artin--Schreier and Cyclotomic Extensions

In this paper we prove that any Artin--Schreier extension of a congruence rational function field is contained in the composite of a cyclotomic function field and a constant field extension that are explicitly prescribed.

Théorème de Chebotarev et Congruences de suites récurrentes linéaires, liens avec les algorithmes de factorisations sur $\mathbb{F}_p$

The classical congruences satisfied by the Fibonacci and Lucas sequences are reflected with the decomposition of primes in the ring generated by the gold number. This generalizes to establish a correspondence that we hope will be new between Chebotarev's theorem and the congruences satisfied by linear sequences. This link is done into the context of number field extensions. In particular we characterize primes ideals totally decomposed by simple congruences on the terms of linear recurrent sequences. Our results are illustrated by numerous examples, including Padovan sequencesof group $S_3$ an…

Modularity of residual Galois extensions and the Eisenstein ideal

For a totally real field $F$, a finite extension $\mathbf{F}$ of $\mathbf{F}_p$ and a Galois character $χ: G_F \to \mathbf{F}^{\times}$ unramified away from a finite set of places $Σ\supset \{\mathfrak{p} \mid p\}$ consider the Bloch-Kato Selmer group $H:=H^1_Σ(F, χ^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\overlineρ$ giving rise to lines in $H$ which are modular by some $ρ_f$ (also unramified outside $Σ$) satisfies $d \geq n:= \dim_{\mathbf{F}} H$. This was proved under the assu…

一阶同余syntomic上同调群

Congruences of first syntomic cohomology groups

研究一阶syntomic上同调群的同余性质，通过比较其模p^n约简与附带Breuil-Kisin模和Nygaard滤子的模p^{2n}约简的同构性，揭示局部Bloch-Kato Selmer群的细化结构。

Congruences for critical values of higher derivatives of twisted Hasse-Weil L-functions

Let A be an abelian variety over a number field k and F a finite cyclic extension of k of p-power degree for an odd prime p. Under certain technical hypotheses, we obtain a reinterpretation of the equivariant Tamagawa number conjecture (eTNC) for A, F/k and p as an explicit family of p-adic congru- ences involving values of derivatives of the Hasse-Weil L-functions of twists of A, normalised by completely explicit twisted regulators. This reinterpretation makes the eTNC amenable to numerical verification and furthermore leads to explicit predictions which refine well-known conjectures of Mazur…

Euler Characteristics and their Congruences for Multi-signed Selmer Groups

The notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the cohomology groups are not finite. Let $p$ be an odd prime, $E_1$ and $E_2$ be elliptic curves over a number field $F$ with semistable reduction at all primes $v|p$ such that the $\operatorname{Gal}(\bar{F}/F)$-modules $E_1[p]$ and $E_2[p]$ are irreducible and isomorphic. We compare the Iwasawa invariants of certain imprimitive multisigned Selmer groups of $E_1$ and $E_2$. Leveraging these results, congruence relations for the truncated Euler charac…

Numerical evidence toward a 2-adic equivariant "main conjecture"

Recently Ritter and Weiss introduced an equivariant "main conjecture" than generalizes and refines the Main Conjecture of Iwasawa theory. In this paper, we show that, for the prime 2 and a dihedral extension of order 8 over Q, this conjecture is equivalent to a congruence condition on the coefficients of a power series with 2-adic integral coefficients constructed using the 2-adic L-series associated to the extension. We then verify that this congruence condition holds for the first coefficients in a large number of examples.

Mazur 增长数猜想与同余

Mazur's growth number conjecture and congruences

本文研究虚二次域上 Mazur 关于 Selmer 秩增长的猜想在 Galois 表示 $p$-同余下的保持性，证明了对特定三元组及模形式 Greenberg Selmer 群的类似结果，并得出非反圆扩张中 Mordell-Weil 秩的有界性。

Asymptotics of class number and genus for abelian extensions of an algebraic function field

Among abelian extensions of a congruence function field, an asymptotic relation of class number and genus is established. The proof is classical, employing well-known results from congruence function field theory.

Some Extensions to Touchard's Theorem on Odd Perfect Numbers

The multiplicative structure of an odd perfect number $n$, if any, is $n=π^αM^2$, where $π$ is prime, $\gcd(π,M)=1$ and $π\equiv α\equiv1\pmod{4}$. An additive structure of $n$, established by Touchard, is that "$\bigl(n\equiv 9\pmod{36}\bigr )$ OR $\bigl (n\equiv1\pmod{12}\bigr )$". A first extension of Touchard's result is that the proposition "$\bigl(n\equiv x^2\pmod{4 x^2}\bigr )$ OR $\bigl (n\equiv π\equiv1\pmod{4 x}\bigr )$" holds for $x=3$ (the extension is due to the fact that the second congruence contains also $π$). We further extend the proof to $x=α+2$, $α+2$ prime, with the restri…

On Mordell-Weil groups and congruences between derivatives of twisted Hasse-Weil L-functions

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair ( h^1(A/F)(1), Z[Gal(F/k)] ). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mor…

Congruences for critical values of higher derivatives of twisted Hasse-Weil L-functions, III

Let $A$ be an abelian variety defined over a number field $k$, let $p$ be an odd prime number and let $F/k$ be a cyclic extension of $p$-power degree. Under not-too-stringent hypotheses we give an interpretation of the $p$-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of $A$ of the ${\rm Gal}(F/k)$-valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in…

关于具有 $\ell$-正则非划线部分的超分拆函数的一些新同余式

Some New Congruences For Overpartition Function With $\ell$-Regular Non-Overlined Parts

本文推广了 Alanzi 和 Sellers 的结果，证明了 $\ell=4,5k,6,8$ 时 $\overline R_\ell^\ast(n)$ 的无穷族及特定同余式，并建立了与其他分拆函数的联系。

S-算术子群的同余子群性质：当S具有正狄利克雷密度时

The congruence subgroup property for $S$-arithmetic subgroups of simple algebraic groups when $S$ has positive Dirichlet density

本文研究了当S具有正狄利克雷密度时，简单代数群的S-算术子群的同余子群性质，证明了同余核为平凡群，为塞雷的同余子群猜想提供了证据。

Extensions of Wilson's lemma and the Ax-Katz theorem

A classical result of A. Fleck states that if p is a prime, and n>0 and r are integers, then $$\sum_{k=r(mod p)}\binom {n}{k}(-1)^k=0 (mod p^{[(n-1)/(p-1)]}).$$ Recently R. M. Wilson used Fleck's congruence and Weisman's extension to present a useful lemma on polynomials modulo prime powers, and applied this lemma to reprove the Ax-Katz theorem on solutions of congruences modulo p and deduce various results on codewords in p-ary linear codes with weights. In light of the recent generalizations of Fleck's congruence given by D. Wan, and D. M. Davis and Z. W. Sun, we obtain new extensions of Wil…

Some Observations on Modulo 5 Congruences for 2-Color Partitions

The 2-color partitions may be considered as an extension of regular partitions of a natural number $n$, with $p_{k}(n)$ defined as the number of 2-colored partitions of $n$ where one of the 2 colors appears only in parts that are multiples of $k$. In this paper, we record the complete characterization of the modulo 5 congruence relation $p_{k}(25n + 24 - k) \equiv 0 \pmod{5}$ for $k \in \{1, 2, \ldots, 24\}$, in connection with the 2-color partition function $p_k(n)$, providing references to existing results for $k \in \{1, 2, 3, 4, 7, 8, 17\}$, simple proofs for $k \in \{5, 10, 15, 20\}$ for …

Radical Extensions for the Carlitz--Hayes Module

Let $L/K$ be a finite extension of congruence function fields. We say that $L/K$ is a {\it radical extension} if $L$ is generated by roots of polynomials $u^{M}-α\in K[u]$, where $u^{M}$ is the action of Carlitz-Hayes. We study a special class of these extensions, the {\it radical cyclotomic} extensions. We prove that any radical cyclotomic extension has order a power of the characteristic of $K$. We also give bounds for the Carlitz-Hayes torsion of these extensions.

Generalizations of Iwasawa's 'Riemann-Hurwitz' Formula for Cyclic p-Extensions of Number Fields

We produce generalizations of Iwasawa's `Riemann-Hurwitz' formula for number fields. These generalizations apply to cyclic extensions of number fields of degree p^n for any positive integer n. We first deduce some congruences and inequalities and then use these formulas to establish a vanishing criterion for Iwasawa λ-invariants which generalizes a result of Takashi Fukuda et. al. for totally real number fields.

Congruences between modular forms and related modules

We fix $\ell$ a prime and let $M$ be an integer such that $\ell\not|M$; let $f\in S_2(Γ_1(M\ell^2))$ be a newform supercuspidal of fixed type related to the nebentypus, at $\ell$ and special at a finite set of primes. Let $\TT^ψ$ be the local quaternionic Hecke algebra associated to $f$. The algebra $\TT^ψ$ acts on a module $\mathcal M^ψ_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $\TT^ψ$ is the universal deformation ring of a global Galois deformation problem associated to $\orho_f$. Moreover $\mathcal M^ψ_f$ is free of …

A congruence modulo $n^3$ involving two consecutive sums of powers and its applications

For various positive integers $k$, the sums of $k$th powers of the first $n$ positive integers, $S_k(n+1)=1^k+2^k+...+n^k$, have got to be some of the most popular sums in all of mathematics. In this note we prove that for each $k\ge 2 $$ 2S_{2k+1}(n)- (2k+1)nS_{2k}(n)\equiv \{{array}{ll} 0\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\,even\,\,or}\,\, n\,\, {\rm is\,\, odd} & {\rm or} \,\, n\equiv 0\,(\bmod{\,4}) \frac{n^3}{2}\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\, odd} &,\,{\rm and}\,\, n\equiv 2\,(\bmod{\,4}). {array}.$$ The above congruence allows us to state an equivalent formulati…

Lucas定理的一个推广

An extension of Lucas Theorem

本文给出计算模素数二项式系数的同余判据的初等证明，并据此推广Lucas定理，应用于推导二项式系数和的恒等式与同余式。

Congruences for Fishburn numbers modulo prime powers

The Fishburn numbers $ξ(n)$ are defined by the formal power series \[ \sum_{n \geq 0} ξ(n) q^n = \sum_{n \geq 0} \prod_{j = 1}^n (1 - (1 - q)^j). \] Recently, G. Andrews and J. Sellers discovered congruences of the form $ξ(p m + j) \equiv 0$ modulo $p$, valid for all $m \geq 0$. These congruences have then been complemented and generalized to the case of $r$-Fishburn numbers by F. Garvan. In this note, we answer a question of Andrews and Sellers regarding an extension of these congruences to the case of prime powers. We show that, under a certain condition, all these congruences indeed extend …

Congruences for sums of binomial coefficients

Let q>1 and m>0 be relatively prime integers. We find an explicit period $ν_m(q)$ such that for any integers n>0 and r we have $[n+ν_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q), or a=1 (mod q) and 2|m, where $[n,r]_m(a)=\sum_{k=r(mod m)}\binom{n}{k}a^k$. This is a further extension of a congruence of Glaisher.

Local discriminants, kummerian extensions, and elliptic curves

Some thoughts on the congruence D=0,1(mod 4) for the absolute discriminant D of a number field

Combinatorial congruences and Stirling numbers

In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence $\sum_{j=0}^l{l\choose j}S(j,m)a^{l-j}(l=m,m+1,...)$ modulo a prime $p$, where $a$ and $m>0$ are integers, and those $S(j,m)$ are Stirling numbers of the second kind. We also give a new extension of Glaisher's congruence by showing that $(p-1)p^{[\log_p m]}$ is a period of the sequence $\sum_{j=r(mod p-1)}{l\choose j}S(j,m)(l=m,m+1,...)$ modulo $p$.

Extensions of CM elliptic curves and orbit counting on the projective line

There are several formulas for the number of orbits of the projective line under the action of subgroups of $GL_2$. We give an interpretation of two such formulas in terms of the geometry of elliptic curves, and prove a more general formula for a large class of congruence subgroups of Bianchi groups. Our formula involves the number of walks on a certain graph called an isogeny volcano. Underlying our results is a complete description of the group of extensions of a pair of CM elliptic curves, and of a pair of lattices in a quadratic field.

A maximal extension of the Bloom-Maynard bound for sets with no square differences

We show that if $h\in\mathbb{Z}[x]$ is a polynomial of degree $k$ such that the congruence $h(x)\equiv0\pmod{q}$ has a solution for every positive integer $q$, then any subset of $\{1,2,\ldots,N\}$ with no two distinct elements with difference of the form $h(n)$, with $n$ positive integer, has density at most $(\log N)^{-c\log\log\log N}$, for some constant $c$ that depends only on $k$. This improves on the best bound in the literature, due to Rice, and generalizes a recent result of Bloom and Maynard.

Proof of the Main Conjecture of Noncommutative Iwasawa Theory for Totally Real Number Fields in Certain Cases

Fix an odd prime $p$. Let $G$ be a compact $p$-adic Lie group containing a closed, normal, pro-$p$ subgroup $H$ which is abelian and such that $G/H$ is isomorphic to the additive group of $p$-adic integers $\mathbbZ_p$ . First we assume that $H$ is finite and compute the Whitehead group of the Iwasawa algebra, $Λ(G)$, of $G$. We also prove some results about certain localisation of $Λ(G)$ needed in Iwasawa theory. Let $F$ be a totally real number field and let $F_{\infty}$ be an admissible $p$-adic Lie extension of $F$ with Galois group $G$. The computation of the Whitehead groups are used to …

Extensions of abelian varieties defined over a number field

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.

Linear extension numbers of $n$-element posets

We address the following natural but hitherto unstudied question: what are the possible linear extension numbers of an $n$-element poset? Let $\mathbf{LE}(n)$ denote the set of all positive integers that arise as the number of linear extensions of some $n$-element poset. We show that $\mathbf{LE}(n)$ skews towards the "small" end of the interval $[1,n!]$. More specifically, $\mathbf{LE}(n)$ contains all of the positive integers up to $\exp\left(c\frac{n}{\log n}\right)$ for some absolute constant $c$, and $|\mathbf{LE}(n) \cap ((n-1)!,n!]|<(n-3)!$. The proof of the former statement involves so…

Inductive construction of the p-adic zeta functions for non-commutative p-extensions of totally real fields with exponent p

We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a pgroup with exponent p. We first calculate the Whitehead groups of the Iwasawa algebra and its canonical Ore localisation by using Oliver-Taylor's theory upon integral logarithms. This calculation reduces the existence of the non-commutative p-adic zeta function to certain congruence conditions among abelian p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne-Ribet'…

Invariance of the parity conjecture for p-Selmer groups of elliptic curves in a $D_{2p^n}$-extension

In section 2, we show a $p$-parity result in a $D_{2p^{n}}$-extension of number fields $L/K$ ($p\geq 5$) for the twist $1\oplus η\oplus τ$: W(E/K,1\oplus η\oplus τ)=(-1)^{< 1\oplusη\oplus τ, X_{p}(E/L)>}, where $E$ is an elliptic curve over $K,$ $η$ and $τ$ are respectively the quadratic character and an irreductible representation of degree 2 of $Gal(L/K)=D_{2p^{n}},$ and $X_{p}(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between $% ε_{0}$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction abov…