Byeong Moon Kim, Dayoon Park
In this article, we consider the rank of universal $m$-gonal forms for all sufficiently large $m$. Especially, we determine the minimal rank of universal $m$-gonal form and the maximal rank of kinds of proper universal $m$-gonal form.
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In this paper, we study the set of positive integers that characterize the universality of $m$-gonal form.
In this article, we consider the representation of $m$-gonal forms over $\mathbb N_0$. We show that any $m$-gonal forms over $\mathbb N_0$ of rank $\ge 5$ is almost regular and ponder the sufficiently large integers which are indeed represented over $\mathbb N_0$ among the integers which are locally represented. And as its consequential result, we prove a finiteness theorem for universal (original polygonal number version's) $m$-gonal forms over $\mathbb N_0$.
In this paper, we show that for any $m$-gonal form $F_m(\mathbf x)$ with $m \ge 12$ which represents every positive integer up to $m-4$, by putting together only unary $m$-gonal form, we may complete an universal form.
It is known that any $m$-gonal form of rank $n \ge 5$ is almost regular. On the other words, any $m$-gonal form of rank $n \ge 5$ represents every sufficiently large integer which is locally represented. In this article, we study the sufficiently large integers which are represented by (almost regular) $m$-gonal forms of rank $n \ge 5$.
For a positive integer $n$, the set of all integers greater than or equal to $n$ is denoted by $\mathcal T(n)$. A sum of generalized $m$-gonal numbers $g$ is called tight $\mathcal T(n)$-universal if the set of all nonzero integers represented by $g$ is equal to $\mathcal T(n)$. In this article, we prove the existence of a minimal tight $\mathcal T(n)$-universality criterion set for a sum of generalized $m$-gonal numbers for any pair $(m,n)$. To achieve this, we introduce an algorithm giving all candidates for tight $\mathcal T(n)$-universal sums of generalized $m$-gonal numbers for any given …
In this paper we study universal quadratic polynomials which arise as sums of polygonal numbers. Specifically, we determine an asymptotic upper bound (as a function of $m$) on the size of the set $S_m\subset\mathbb{N}$ such that if a sum of $m$-gonal numbers represents $S_m$, then it represents $\mathbb{N}$.
The maximal $n$ which admits a non-almost regular $m$-gonal form of rank $n$ is $4$. In this article, we consider the $m$-gonal forms of rank $4$ which is not almost regular.
It is known that any $m$-gonal form of $\rank n \ge 5$ is almost regular. In this article, we study the sufficiently large integers which are represented by (almost regular) $m$-gonal forms of $\rank n \ge 6$.
In this paper, we classify the $(a_1,a_2,a_3,a_4,a_5)$ for which the universality of an $m$-gonal form $F_m(\mathbf x)$ having its first five coefficients as $(a_1,a_2,a_3,a_4,a_5)$ is characterized as the representability of positive integers up to $m-4$ and see its applications.
In this paper, we show that for a fixed rank $n$, there are only finitely many $m$ for which there is a regular $m$-gonal form of rank $n$ and determine every type of the (generalized) regular $m$-gonal form for every sufficiently large $m$.
An integer of the form $P_m(x)= \frac{(m-2)x^2-(m-4)x}{2}$ for an integer $x$, is called a generalized $m$-gonal number. For positive integers $α_1,\dots,α_u$ and $β_1,\dots,β_v$, a mixed sum $Φ=α_1P_4(x_1)+\cdots+α_uP_4(x_u)+β_1P_8(y_1)+\cdots+β_vP_8(y_v)$ of generalized $4$- and $8$-gonal numbers is called universal if $Φ=N$ has an integer solution for any nonnegative integer $N$. In this article, we prove that there are exactly 1271 proper universal mixed sums of generalized $4$- and $8$-gonal numbers. Furthermore, the "$61$-theorem" is proved, which states that an arbitrary mixed sum of ge…
Universal quaternary mixed sums involving generalized 3-, 4-, 5- and 8-gonal numbers via products of Ramanujan's theta functions
本文利用Ramanujan theta函数恒等式理论，系统确定了由广义3-、4-、5-和8-角数构成的四元混合和何时能表示所有非负整数，即其通用性。
Tight universality of $m$-gonal forms with minimal criterion sets
本文确定了紧致 $\mathcal{T}(n)$-普适 $m$ 角型的最小判定集，给出了 $n\geq8$ 时除两个例外情形的完整刻画，并证明 $m$ 的上界最优。
Consider a smooth projective curve $\overline{C}$ over a finite field $\mathbb{F}_q$, equipped with a simply branched morphism $\overline{C} \to \mathbb{P}^1$ of degree $d \leq 5$. Assume char$\, \mathbb{F}_q > 2$ if $d \leq 4$, and char$\, \mathbb{F}_q > 3$ if $d=5$. In this paper we describe how to efficiently compute a lift of $\overline{C}$ to characteristic zero, such that it can be fed as input to Tuitman's algorithm for computing the Hasse-Weil zeta function of $\overline{C} / \mathbb{F}_q$. Our method relies on the parametrizations of low rank rings due to Delone-Faddeev and Bhargava.
For a natural number $m$, generalized $m$-gonal numbers are those numbers of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this paper we establish conditions on $m$ for which the ternary sum $p_m(x)+p_m(y)+p_m(z)$ is almost universal.
For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary quadratic forms to determine completely when the mixed sum $ax^2+by^2+cP_{p^k+2}(z)$ represents all but finitely many positive integers.
For a natural number $m$, generalized $m$-gonal numbers are defined by the formula $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this paper, we determine a criterion on $a,b,c,m$ for which the weighted ternary sum $P_{a,b,c,m}:=ap_m(x)+bp_m(y)+cp_m(z)$ is almost universal. We also prove for some $a,b,c,m$ that the form $P_{a,b,c,m}$ is not almost universal, while it represents all possible congruence classes.
For a subset $S$ of nonnegative integers and a vector $\mathbf{a}=(a_1,\dots,a_k)$ of positive integers, let $V'_S(\mathbf{a})=\{ a_1s_1+\cdots+a_ks_k : s_i\in S\} \setminus \{0\}$. For a positive integer $n$, let $\mathcal T(n)$ be the set of integers greater than or equal to $n$. In this paper, we consider the problem of finding all vectors $\mathbf{a}$ satisfying $V'_S(\mathbf{a})=\mathcal T(n)$, when $S$ is the set of (generalized) $m$-gonal numbers and $n$ is a positive integer. In particular, we completely resolve the case when $S$ is the set of triangular numbers.
Number fields without universal quadratic forms of small rank exist in most degrees
该研究证明，凡次数可被2或3整除时，均有无穷多个全实数域使其通用二次型的秩必须任意大。
In this paper, we will prove that if $d$ is sufficiently large square-free positive rational integer, then there is no integral septenary universal quadratic lattice over $\mathcal{O}_F$ where $F=\mathbb Q(\sqrt{d})$.
We begin a systematic investigation of universal norms for $p$-adic representations in higher rank Iwasawa theory. After establishing the basic properties of the module of higher rank universal norms we construct an Iwasawa-theoretic pairing that is relevant to this setting. This allows us, for example, to refine the classical Iwasawa Main Conjecture for cyclotomic fields, and also to give applications to various well-known conjectures in arithmetic concerning Iwasawa invariants and leading terms of $L$-functions.
Minimal rank of universal lattices and number of indecomposable elements in real multiquadratic fields
本文通过给出实多重二次域中泛二次格最小秩及不可分解元素数量的上界，证明了相应域的密度为零，并研究了实双二次域中不可分解元素的结构与具体计算。
Towards a finite-slope universal Rankin-Selberg p-adic L-function
本文提出在通用变形家族中研究有限斜率兰利-塞尔伯格 $p$-adic $L$-函数的框架，并探讨了残余表示与变形家族之间的关系。
Lower bound estimates for the rank of universal quadratic forms in some families of real cubic fields with density one
本文在密度为一的实三次域族中，利用不可分解整数和短向量性质，给出了泛二次型秩的明确下界估计，并推广到表示给定整数所有倍数的情形。
We show that if $K$ is a monogenic, primitive, totally real number field, that contains units of every signature, then there exists a lower bound for the rank of integer universal quadratic forms defined over $K$. In particular, we extend the work of Blomer and Kala, to show that there exist infinitely many totally real cubic number fields that do not have a universal quadratic form of a given rank defined over them. For the real quadratic number fields with a unit of negative norm, we show that the minimal rank of a universal quadratic form goes to infinity as the discriminant of the number f…
Real quadratic fields with a universal quadratic form of given rank have density zero
该研究通过二次格短向量与系数有界周期连分数数量估计，证明给定秩泛在二次型的实二次域数量有显式上界、密度为零，并推广至全正定二次格情形。
Minimal rank of primitively $n$-universal integral quadratic forms over local rings
本文完全确定了在任意正整数 $n$ 及满足 $2$ 为单位或素元的任意局部环 $R$ 上，本原 $n$-泛整二次型的极小秩，推广了先前关于 $p$-adic 整数环上本原 $1$-泛情形的结果。
Extending bounds on minimal ranks of universal quadratic lattices to larger number fields
本研究改进Kala的数域推广技术，通过伽罗瓦理论将数域复合域子域结构转化为群论问题，证明若d次全实数域存在泛二次格最小秩≥r的数域，则所有k≥3的kd次全实数域中也存在，拓展了泛二次格最小秩相关结论的适用数域范围。
We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field $\mathbb Q(\sqrt D)$ and obtain lower and upper bounds for it in terms of certain sums of coefficients of the associated continued fraction. We also estimate such sums in terms of $D$ and establish a link between continued fraction expansions and special values of $L$-functions in the spirit of Kronecker's limit formula.
In this article, we construct zeta morphisms for the universal deformations of odd absolutely irreducible two dimensional mod p Galois representations satisfying some mild assumptions, and prove that our zeta morphisms interpolate Kato's zeta morphisms for Galois representations associated to Hecke eigen cusp newforms. The existence of such morphisms was predicted by Kato's generalized Iwasawa main conjecture. Based on Kato's original construction, we construct our zeta morphisms using many deep results in the theory of p-adic (local and global) Langlands correspondence for GL_{2/Q}. As an app…
Primitively $2$-universal senary integral quadratic forms
本文证明了本原 $2$-全泛二次型的最小秩为六，并确定了本原 $2$-全泛六元二次型恰有 $201$ 个等价类。
Sails for universal quadratic forms
本文建立广义连分数中的帆与全实数域算术的新联系，并用于双二次域，刻画通用二次型秩和不可分解整数数量随判别式的幂次或对数增长。
In this short note, we answer a question raised by M. Papikian on a universal upper bound for the degree of the extension of $K_\infty$ given by adjoining the periods of a Drinfeld module of rank 2. We show that contrary to the rank 1 case such a universal upper bound does not exist, and the proof generalises to higher rank. Moreover, we give an upper and lower bound for the extension degree depending on the valuations of the defining coefficients of the Drinfeld module. In particular, the lower bound shows the non-existence of a universal upper bound.
Asymptotics of $n$-universal lattices over number fields
本研究针对全实数域上的$n$-泛性格，推导了其最小秩对数的显式渐近公式，并证明对$n≥3$，秩或准则集规模不超过给定常数$C$的全实域仅有有限个且均可有效计算。
Using continued fractions with prescribed period for universal quadratic forms
本文通过分析连分数周期与二次型判别式的同余条件，改进了实二次域上通用二次型的大秩结果。
A (positive definite and integral) quadratic form is said to be $\textit{prime-universal}$ if it represents all primes. Recently, Doyle and Williams in [2] classified all prime-universal diagonal ternary quadratic forms, and all prime-universal diagonal quaternary quadratic forms under two conjectures proposed by themselves. In this article, we classify all prime-universal diagonal quadratic forms regardless of ranks. Furthermore, we prove, so called, $67$-Theorem for a diagonal quadratic form to be prime-universal.
For each positive integer $r$, we construct a nowhere-vanishing, single-cuspidal Drinfeld modular form for $\GL_r(\FF_q[θ])$, necessarily of least possible weight, via determinants using rigid analytic trivializations of the universal Drinfeld module of rank $r$ and deformations of vectorial Eisenstein series. Along the way, we deduce that the cycle class map from de Rham cohomology to Betti cohomology is an isomorphism for Drinfeld modules of all ranks over $\FF_q[θ]$.
We obtain two-variable Hecke-Rogers identities for three universal mock theta functions. This implies that many of Ramanujan's mock theta functions, including all the third order functions, have a Hecke-Rogers-type double sum representation. We find new generating function identities for the Dyson rank function, the overpartition rank function, the M2-rank function and related spt-crank functions. Results are proved using the theory of basic hypergeometric functions.
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