A finiteness theorem for universal $m$-gonal forms

Universal mixed sums of generalized $4$- and $8$-gonal numbers

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…

通过Ramanujan theta函数乘积研究涉及广义3-、4-、5-和8-角数的通用四元混合和

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。

具有最小判定集的 $m$ 角型的紧致普适性

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。

关于广义多角数之和的有限性定理

On finiteness theorems for sums of generalized polygonal numbers

本文研究广义多角数混合和的普适性，通过限定满足特定最小公倍数条件的子类，获得了仅需验证有限正整数子集即可判定其表示所有正整数能力的有限性条件及常数渐近增长界。

Finiteness theorems for universal sums of squares of almost primes

In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.

Almost universal ternary sums of polygonal numbers

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.

Almost universal mixed sums of squares and polygonal numbers

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.

Almost Universal Weighted Ternary Sums of Polygonal Numbers

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.

Finiteness of unramified deformation rings

We prove that the universal unramified deformation ring $R^{\mathrm{unr}}$ of a continuous Galois representation $\overlineρ: G_{F^{+}} \rightarrow \mathrm{GL}_n(k)$ (for a totally real field $F^{+}$ and finite field $k$) is finite over $\mathcal{O} = W(k)$ in many cases. We also prove (under similar hypotheses) that the universal deformation ring $R^{\mathrm{univ}}$ is finite over the local deformation ring $R^{\mathrm{loc}}$.

Tight universal triangular forms

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.

亏格至少为2的模曲线的正交性结果

Finiteness results for modular curves of genus at least 2

有理数域Q上的曲线X是模曲线，如果对某个N，X被X_1（N）支配;另外，如果它在J_1（N）中的雅可比像包含在J_1（N）的新子簇中，则X称为新的模曲线。证明了对任意不小于2的整数g，亏格为g的Q上的新模曲线的集合是有限的且可计算的.对于可计算性的结果，我们证明了一个算法版本的de Franchis-Severi定理。对于有界性的新模曲线，对于雅可比矩阵是J_0（N）的新部分与N可被指定素数整除的商的新模曲线，以及对于水平在限制集中的模曲线（新的或非新的），证明了类似的有限性结果.我们详细研究了新的模超椭圆曲线。特别是，我们发现所有新的模曲线的亏格2明确，并构建可能是所有新的模超椭圆曲线的所有属的完整列表。最后证明了对特征为零的域k和至少为2的整数g，k上由Fermat曲线支配的亏格g曲线集是有限的且可计算的。

Ternary universal sums of generalized pentagonal numbers

For any $m\ge3$, every integer of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}2$ with $x \in \z$ is said to be a generalized $m$-gonal number. Let $a\le b\le c$ be positive integers. For every non negative integer $n$, if there are integers $x,y,z$ such that $n=ap_k(x)+bp_k(y)+cp_k(z)$, then the quadruple $(k;a,b,c)$ is said to be {\it universal}. Sun gave in \cite{s1} all possible quadruple candidates that are universal and proved some quadruples to be universal (see also \cite{gs}). He remains the following quadruples $(5,1,1,k)$ for $k=6,8,9,10$, $(5,1,2,8)$, and $(5,1,3,s)$ for $7 \le s \le 8$ …

通用的几乎质数长度的广义多边形数之和

Universal sums of generalized polygonal numbers of almost prime length

本文研究了广义多边形数的通用和，证明了在输入素因子数量受限的情况下，通用和的有限性定理，并给出了不同条件下的最优界。

A Manin-Mumford theorem for the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves

We study the intersection of an algebraic variety with the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves. For this intersection we show a Manin-Mumford type statement. This answers some questions posed by Corvaja-Masser-Zannier which arose in connection with their investigation of the intersection of a curve with real analytic subgroups of various algebraic groups. They prove finiteness in the situation of a single elliptic curve. Using Khovanskii's zero-estimates combined with a stratification result of Gabrielov-Vorobjov and recent work of the au…

On universal and periodic $β$-expansions, and the Hausdorff dimension of the set of all expansions

In this paper we study the topology of a set naturally arising from the study of $β$-expansions. After proving several elementary results for this set we study the case when our base is Pisot. In this case we give necessary and sufficient conditions for this set to be finite. This finiteness property will allow us to generalise a theorem due to Schmidt and will provide the motivation for sufficient conditions under which the growth rate and Hausdorff dimension of the set of $β$-expansions are equal and explicitly calculable.

An effective Arakelov-theoretic version of the hyperbolic isogeny theorem

For an integer $e$ and hyperbolic curve $X$ over $\overline{\mathbb Q}$, Mochizuki showed that there are only finitely many isomorphism classes of hyperbolic curves $Y$ of Euler characteristic $e$ with the same universal cover as $X$. We use Arakelov theory to prove an effective version of this finiteness statement.

实二次域上的 escalations 与判别准则

Escalations and criteria over real quadratic fields

针对全实代数数域上通用二次型的判别准则仅在Q(√5)上明确已知的问题，本文开发数域上的escalate方法，证明判别准则有限性并提供计算工具，对Q(√2)、Q(√3)进行计算得到猜想，还发展对角二次型的escalation理论并给出对应结果。