Piotr Miska
Donnay, Garcia and Rouse classified nonsingular quadratic forms $Q$ with integral coefficients and prime numbers $p$ such that the set of quotients of values of $Q$ attained for integer arguments is dense in the field of $p$-adic numbers. The aim of this note is to give another proof of this classification.
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The ratio set of a set of positive integers $A$ is defined as $R(A) := \{a / b : a, b \in A\}$. The study of the denseness of $R(A)$ in the set of positive real numbers is a classical topic and, more recently, the denseness in the set of $p$-adic numbers $\mathbb{Q}_p$ has also been investigated. Let $A_1, \ldots, A_k$ be a partition of $\mathbb{N}$ into $k$ sets. We prove that for all prime numbers $p$ but at most $\lfloor \log_2 k \rfloor$ exceptions at least one of $R(A_1), \ldots, R(A_k)$ is dense in $\mathbb{Q}_p$. Moreover, we show that for all prime numbers $p$ but at most $k - 1$ excep…
The quotient set, or ratio set, of a set of integers $A$ is defined as $R(A) := \left\{a/b : a,b \in A,\; b \neq 0\right\}$. We consider the case in which $A$ is the image of $\mathbb{Z}^+$ under a polynomial $f \in \mathbb{Z}[X]$, and we give some conditions under which $R(A)$ is dense in $\mathbb{Q}_p$. Then, we apply these results to determine when $R(S_m^n)$ is dense in $\mathbb{Q}_p$, where $S_m^n$ is the set of numbers of the form $x_1^n + \cdots + x_m^n$, with $x_1, \dots, x_m \geq 0$ integers. This allows us to answer a question posed in [Garcia et al., $p$-adic quotient sets, Acta Ari…
On $p$-adic denseness of quotients of values of integral forms
研究整形式值的商集在\(p\)进数中的密度问题，通过考虑由形式定义的簇并借助代数几何工具，证明某些条件下的结论，给出反例，猜想素数次形式的情况，还构造了特定的整形式。
For a set of integers $A$, we consider $R(A)=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values attained by an integral form. This problem has been answered for quadratic forms. Very recently, Antony and Barman have studied this problem for the diagonal binary cubic forms $ax^3+by^3$, where $a$ and $b$ are integers. In this article, we study this problem for diagonal forms. We extend the results of Antony and Barman to the diagonal binary forms $ax^n+by^n$ for all $n\geq 3$. We also study $p$-adic d…
For $A\subseteq \{1, 2, \ldots\}$, we consider $R(A)=\{a/b: a, b\in A\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values assumed by a cubic form. We study this problem for the cubic forms $ax^3+by^3$, where $a$ and $b$ are integers. We also prove that if $A$ is the set of nonzero values assumed by a non-degenerate, integral and primitive cubic form with more than 9 variables, then $R(A)$ is dense in $\mathbb{Q}_p$.
For $A \subseteq \mathbb{N}$, the question of when $R(A) = \{a/a' : a, a' \in A\}$ is dense in the positive real numbers $\mathbb{R}_+$ has been examined by many authors over the years. In contrast, the $p$-adic setting is largely unexplored. We investigate conditions under which $R(A)$ is dense in the $p$-adic numbers. Techniques from elementary, algebraic, and analytic number theory are employed in this endeavor. We also pose many open questions that should be of general interest.
For $A \subseteq \{1,2,\ldots\}$, we consider $R(A) = \{a/a' : a,a' \in A\}$. If $A$ is the set of nonzero values assumed by a quadratic form, when is $R(A)$ dense in the $p$-adic numbers? We show that for a binary quadratic form $Q$, $R(A)$ is dense in $\mathbb{Q}_{p}$ if and only if the discriminant of $Q$ is a nonzero square in $\mathbb{Q}_{p}$, and for a quadratic form in at least three variables, $R(A)$ is always dense in $\mathbb{Q}_{p}$. This answers a question posed by several authors in 2017.
A resolution of Kellner's conjectures on Wilson quotients
本研究针对凯尔纳关于威尔逊商的猜想，利用\(p\)进对数和指数函数给出同余式的\(p\)进证明，还通过完全贝尔多项式给出多项式\(\psi_\nu\)的显式生成函数，从而解决了凯尔纳的两个猜想。
Let $(x_n)_{n\geq0}$ be a linear recurrence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In [`The quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_p$', \emph{Bull. Aust. Math. Soc.} \textbf{96} (2017), 24-29], Sanna posed an open question to classify primes $p$ for which the quotient set of $(x_n)_{n\geq0}$ is dense in $\mathbb{Q}_p$. In this article, we find a sufficient condition for denseness of the quotient set of the $k$th-order linea…
We prove hypergeometric type summation identities for a function defined in terms of quotients of the $p$-adic gamma function by counting points on certain families of hyperelliptic curves over $\mathbb{F}_{q}$. We also find certain special values of that function.
We prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$ hypergeometric series and the Fourier coefficients of certain weight four modular form.
For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of $p$-adic Gamma functions that are stronger than those one can expect to prove using commutative formal group laws. We prove a number of such supercongruences by using classical hypergeometric transformation formulae. These formulae (see the appendix), most of which are decades or centuries old, allow us to write the terminating series as the ratio of products of of $Γ$-values. At this point sums have become quotients. Writing these $Γ$-quotients as $Γ_p$-quotients, we are …
The arithmetic of continued fractions in the field of $p$-adic numbers
本文研究了p进数域中连分数的算术，提出了一种计算莫比乌斯变换和双线性分数变换的完整方法，并证明了某些情况下部分商无法通过初始连分数恢复，但此类元素集合在p进数域上具有零测度。
For a prime $p$ and a matrix $A \in \mathbb{Z}^{n \times n}$, write $A$ as $A = p (A \,\mathrm{quo}\, p) + (A \,\mathrm{rem}\, p)$ where the remainder and quotient operations are applied element-wise. Write the $p$-adic expansion of $A$ as $A = A^{[0]} + p A^{[1]} + p^2 A^{[2]} + \cdots$ where each $A^{[i]} \in \mathbb{Z}^{n \times n}$ has entries between $[0, p-1]$. Upper bounds are proven for the $\mathbb{Z}$-ranks of $A \,\mathrm{rem}\, p$, and $A \,\mathrm{quo}\, p$. Also, upper bounds are proven for the $\mathbb{Z}/p\mathbb{Z}$-rank of $A^{[i]}$ for all $i \ge 0$ when $p = 2$, and a conje…
- Let K be a totally imaginary number field. Denote by G ur K (2) the Galois group of the maximal unramified pro-2 extension of K. By comparing cup-products in {é}tale cohomology of SpecO K and cohomology of uniform pro-2 groups, we obtain situations where G ur K (2) has no non-trivial uniform analytic quotient, proving some new special cases of the unramified Fontaine-Mazur conjecture. For example, in the family of imaginary quadratic fields K for which the 2-rank of the class group is equal to 5, we obtain that for at least 33.12% of such K, the group G ur K (2) has no non-trivial uniform an…
Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the field of the $p$--adic numbers $\mathbb Q_p$. First, we introduce them from a formal point of view, i.e., without considering a specific algorithm that produces the partial quotients of a MCF, and we perform a general study about their convergence in $\mathbb Q_p$. In particular, we derive some conditions about their convergence and we prove that convergent MCFs always strongly co…
$p$-Adic quotient sets: linear recurrence sequences with reducible characteristic polynomials
本文针对特征多项式在有理数域上可约的一类线性递推序列，解答了Sanna提出的判断其$p$-进商集在$p$-进数域中是否稠密的公开问题，补充了此前不可约情形下的相关研究结论。
Transcendence of $p$-adic continued fractions and a quantitative $p$-adic Roth theorem
本文研究了p进数列的超越性，证明回文和拟周期性p进数列收敛于超越数或二次无理数，并提供了一个量化版的p进Roth定理。
We describe an algorithm for computing certain quaternionic quotients of the Bruhat-Tits tree for GL2(Qp). As an application, we describe an algorithm to obtain (conjectural) equations for the canonical embedding of Shimura curves.
Previous works have shown that certain weight $2$ newforms are $p$-adic limits of weakly holomorphic modular forms under repeated application of the $U$-operator. The proofs of these theorems originally relied on the theory of harmonic Maass forms. Ahlgren and Samart obtained strengthened versions of these results using the theory of holomorphic modular forms. Here, we use such techniques to express all weight $2$ CM newforms which are eta quotients as $p$-adic limits. In particular, we show that these forms are $p$-adic limits of the derivatives of the Weierstrass mock modular forms associate…
On $\mathfrak{P}$-adic continued fractions with extraneous denominators: some explicit finiteness results
本文提出了一种新的算法方法，用于构造数域中的除法链，通过允许有限个分母来定义满足有限性性质的$\mathfrak P$-adic连续分数。
Serre obtained the p-adic limit of the integral Fourier coefficient of modular forms on $SL_2(\mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on $Γ_{0}(4N)$ for $N=1,2,4$. A proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications we obtain congruences of Borcherds exponents, congruences of quotient of Eisentein series and congruences of values of $L$-functions at a certain point are also studied. Furthermore, the congruences of the Fourier coeff…
Let $X$ be a quotient of the modular curve $X_0(N)$ whose Jacobian $J_X$ is a simple factor of $J_0(N)^{new}$ over $\mathbb{Q}$. Let $f$ be the newform of level $N$ and weight 2 associated with $J_X$; assume $f$ has analytic rank 1. We give analytic methods for determining the rational points of $X$ using quadratic Chabauty by computing two $p$-adic Gross-Zagier formulas for $f$. Quadratic Chabauty requires a supply of rational points on the curve or its Jacobian; this new method eliminates this requirement. To achieve this, we give an algorithm to compute the special value of the anticyclotom…
Quotient branching law for $p$-adic $(\mathrm{GL}_{n+1}, \mathrm{GL}_n)$ I: generalized Gan-Gross-Prasad relevant pairs
本文针对非阿基米德局部域上的GL_{n+1}与GL_n，给出并证明了二者不可约光滑表示间同态非零的充要条件，解决了商分支律问题，还将结果推广到A型仿射Hecke代数并细化了分支律。
By analyzing the coefficients of the power series defining the Kubota--Leopoldt $p$-adic $L$-function associated to the non-trivial character of a real quadratic field, we prove a congruence of Ankeny--Artin--Chowla-type for prime power modulus. Additionally, we show how some classical congruences relating Bernoulli numbers and Wilson quotients fit naturally into the theory of the $p$-adic Riemann zeta function.
We define a pro-$p$ Abelian sheaf on a modular curve of a fixed level $N \geq 5$ divisible by a prime number $p \neq 2$. Every $p$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ associated to an eigenform is obtained as a quotient of its étale cohomology. For any compact $\mathbb{Z}_p[[1 + N \mathbb{Z}_p]]$-algebra $Λ_1$ satisfying certain suitable conditions, we construct a representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ over $Λ_1$ associated to a $Λ_1$-adic cuspidal eigenform of finite slope as a scalar extension of a quotient of the étale cohomology.
Quadratic Chabauty for Atkin-Lehner quotients of modular curves via weakly holomorphic modular forms: Hodge Filtrations
本研究针对素级亏格≥2的Atkin-Lehner商模曲线，提出用弱全纯模形式计算二次Chabauty方法中关键的Hodge滤过的算法，提升了计算效率，并在$X_0^+(193)$上发现了权2尖点形式积分的同余关系。
The formulas for local root numbers of abelian varieties of dimension one are known. In this paper we treat the simplest unknown case in dimension two by considering a curve of genus 2 defined over a $5$-adic field such that the inertia acts on the first $\ell$-adic cohomology group through the largest possible finite quotient, isomorphic to $C_5\rtimes C_8$. We give a few criteria to identify such curves and prove a formula for their local root numbers in terms of invariants associated to a Weierstrass equation.
Convergence, Finiteness and Periodicity of Several New Algorithms of p-adic Continued Fractions
本文提出若干改进的$p$进连分数展开算法，证明有理数部分商长度有界且小素数下二次无理数均可周期展开，并给出对任意素数构造此类算法的方法。
On the pro-étale cohomology of quotient stacks of Drinfeld spaces
该研究聚焦p进域上Drinfeld空间商栈的pro-étale上同调问题，借助Lubin-Tate塔与Drinfeld塔的同构及已有Drinfeld空间p进上同调结果，给出两类商栈ℓ进与p进pro-étale上同调的显式描述，并应用于计算GLₙ(ℚₚ)的连续群上同调。
Multiple Zeta Values in Positive Characteristic: A Kaneko-Zagier-Type Conjecture
本文在正特征中构造了∞-进与有限多重zeta值之间的满射比较映射，通过除以ζ_A(q-1)确立Kaneko-Zagier型猜想映射，并证明其与Carlitz多重多对数相容，最终验证了Shi维数猜想的上界。
Let k be a totally real number field ant let k$\infty$ be its cyclotomic Zp-extension for a prime p\textgreater{}2. We give (Theorem 3.2) a sufficient condition of nullity of the Iwasawa invariants lambda, mu, when p totally splits in k, and we obtain important tables of quadratic fields and p for which we can conclude that lambda = mu=0.We show that the number of ambiguous p-classes of kn (nth stage in k$\infty$) is equal to the order of the torsion group T, of the Galois group of the maximal Abelian p-ramified pro-p-extension of k (Theorem 4.2), for all n \textgreater{}\textgreater{} e, wher…
In this paper we introduce a new method for finding Galois groups by computer. This is particularly effective in the case of Galois groups of p-extensions ramified at finitely many primes but unramified at the primes above p. Such Galois groups have been regarded as amongst the most mysterious objects in number theory. Very little has hitherto been discovered regarding them despite their importance in studying p-adic Galois representations unramified at p. The conjectures of Fontaine-Mazur say that they should have no p-adic analytic quotients (equivalently, the images of the Galois representa…
This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline representation of the Galois group of Qp with nonnegative Hodge-Tate weights. In this paper, we study these Coleman maps using Perrin-Riou's p-adic regulator L_V. Denote by H(Γ) the algebra of Qp-valued distributions on Γ= Gal(Qp(μ(p^\infty) / Qp). Our first result determines the H(Γ)-elementary divisors of the quotient of D_{cris}(V) \otimes H(Γ) by the H(Γ)-submodule generated by (ϕ* N(V))^{ψ= 0}, where N(V) is the Wac…
We shall make a slight improvement to a result of p-adic logarithms which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient x^{p-1}-1.
Finite Fourier Duality, Radial Values, and Arithmetic of a Mod Eight False Theta Quotient
研究模八假 theta 矩商式的径向行为，建立有限傅里叶闭包，证明周期狄利克雷值非零，并给出系数整除性、二进插值及欧拉变换正性等算术结果。
In this note, we investigate the p-adic behavior of Weil numbers in the cyclotomic $\mathbb Z\_p$-extension of $\mathbb Q(ζ\_p).$ We determlne the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in Iwasawa Theory.
Let $|| \cdot ||$ denote the distance to the nearest integer and, for a prime number $p$, let $| \cdot |_p$ denote the $p$-adic absolute value. In 2004, de Mathan and Teulié asked whether $\inf_{q \ge 1} \, q \cdot || q α|| \cdot | q |_p = 0$ holds for every badly approximable real number $α$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number $α$ grows too rapidly or too slowly, then their conjecture is true for the pair $(α, p)$ with $p$ an arbitrary prime.
Let K/Q be Galois and let eta in K* be such that the multiplicative Z[G]-module generated by eta is of Z-rank n.We define the local theta-regulators Delta\_p^theta(eta) in F\_p for the Q\_p-irreducible characters theta of G=Gal(K/Q). Let V\_theta be the theta-irreducible representation. A linear representation L^theta=delta.V\_theta is associated withDelta\_p^theta(eta) whose nullity is equivalent to delta$\ge$1 (Theorem 3.9). Each Delta\_p^theta(eta) yields Reg\_p^theta(eta) modulo p in the factorization $\prod$\_theta (Reg\_p^theta(eta))^phi(1) of Reg\_p^G(eta) := Reg\_p(eta)/p^[K : Q] (norm…
In previous work we described when a single geometric representation, valued in a linear algebraic group, of the Galois group of a number field lifts through a central torus quotient to a geometric representation. In this paper we prove a much sharper result for systems of l-adic representations, such as the l-adic realizations of a motive, having common "good reduction" properties. Namely, such systems admit geometric lifts with good reduction outside a common finite set of primes. The method yields new proofs of theorems of Tate (the original result on lifting projective representations over…
The Euler quotient modulo an odd-prime power $p^r~(r>1)$ can be uniquely decomposed as a $p$-adic number of the form $$ \frac{u^{(p-1)p^{r-1}} -1}{p^r}\equiv a_0(u)+a_1(u)p+\ldots+a_{r-1}(u)p^{r-1} \pmod {p^r},~ \gcd(u,p)=1, $$ where $0\le a_j(u)<p$ for $0\le j\le r-1$ and we set all $a_j(u)=0$ if $\gcd(u,p)>1$. We firstly study certain arithmetic properties of the level sequences $(a_j(u))_{u\ge 0}$ over $\mathbb{F}_p$ via introducing a new quotient. Then we determine the exact values of linear complexity of $(a_j(u))_{u\ge 0}$ and values of $k$-error linear complexity for binary sequences de…
We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL(2,\mathbb{Q}_p)$ for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigenfunctions on covers of a fixed regular graph or along nested sequences of regular graphs. Our results are the first of their kind on any p-adic arithmetic quotient. They may be understood as analogues of Lindenstrauss's theorem on the equidistribution of Maass forms on a compact arithmetic surface. The new ingredients here include the introduction of a repre…
Let p be a prime, and let n>0 and r be integers. In 1913 Fleck showed that $$F_p(n,r)=(-p)^{-[(n-1)/(p-1)]}\sum_{k=r(mod p)}\binom{n}{k}(-1)^k\in\Z.$$ Nowadays this result plays important roles in many aspects. Recently Sun and Wan investigated $F_p(n,r)$ mod p in [SW2]. In this paper, using p-adic methods we determine $(F_p(m,r)-F_p(n,r))/(m-n)$ modulo p in terms of Bernoulli numbers, where m>0 is an integer with $m\not=n$ and $m=n (mod p(p-1))$. Consequently, $F_p(n,r)$ mod $p^{ord_p(n)+1}$ is determined; for example, if $n=n_*(mod p-1)$ with $0<n_*<p-2$ then $$\frac{F_p(pn,0)}{pn}=\frac{n_*…
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