Research in Number Theory最新文献

We investigate generalizations along the lines of the Mordell-Lang conjecture of the author's p-adic formal Manin-Mumford results for n-dimensional p-divisible formal groups $F$ . In particular, given a finitely generated subgroup $\Gamma$ of $F\left({\overline{Q}}_p\right)$ and a closed subscheme $X\hookrightarrow F$ , we show under suitable assumptions that for any points $P\in X\left(C_p\right)$ satisfying $nP\in \Gamma$ for some $n\in N$ , the minimal such orders n are uniformly bounded whenever X does not contain a formal subgroup translate of positive dimension. In contrast, we then provide counter-examples to a full p-adic formal Mordell-Lang result. Finally, we outline some consequences for the study of the Zariski-density of sets of automorphic objects in p-adic deformations. Specifically, we do so in the context of the nearly ordinary p-adic families of cuspidal cohomological automorphic forms for the general linear group constructed by Hida.

Let $\overline{p}\left(n\right)$ denote the overpartition function. In this paper, our primary goal is to study the asymptotic behavior of the finite differences of the logarithm of the overpartition function, i.e., ${(-1)}^{r-1}{\Delta }^{r}log\overline{p}\left(n\right)$ , by studying the inequality of the following form $\begin{array}{cc}log(1+\frac{C\left(r\right)}{n^{r-1/2}}-\frac{1+C_1\left(r\right)}{n^r})& <{(-1)}^{r-1}{\Delta }^{r}log\overline{p}\left(n\right)\\ & <log(1+\frac{C\left(r\right)}{n^{r-1/2}})\text{for}n\ge N\left(r\right),\end{array}$ where $C\left(r\right),C_1\left(r\right),\text{and}N\left(r\right)$ are computable constants depending on the positive integer r, determined explicitly. This solves a problem posed by Wang, Xie and Zhang in the context of searching for a better lower bound of ${(-1)}^{r-1}{\Delta }^{r}log\overline{p}\left(n\right)$ than 0. By settling the problem, we are able to show that

Recently Ono, Saad and the second author [21] initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the p-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the p-adic setting and obtain that their limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of pth Hecke operators acting on the spaces of cusp forms of even weight $k\ge 4$ and levels 4 and 8 in terms of p-adic hypergeometric function which is of independent interest. These results can be viewed as p-adic analogous of some trace formulas of [1, 2, 6].

We present a new elementary algorithm that takes $\text{time}{O}_{ϵ}\left(x^{\frac{3}{5}},{(logx)}^{\frac{8}{5}+ϵ}\right)\text{and}\text{space}O\left(x^{\frac{3}{10}},{(logx)}^{\frac{13}{10}}\right)$ (measured bitwise) for computing $M\left(x\right)={\sum }_{n\le x}\mu \left(n\right),$ where $\mu \left(n\right)$ is the Möbius function. This is the first improvement in the exponent of x for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to $O\left(x^{1/5}{(logx)}^{5/3}\right)$ by the use of (Helfgott in: Math Comput 89:333-350, 2020), at the cost of letting time rise to the order of $x^{3/5}{(logx)}^{2}loglogx$ .

The geometric sieve for densities is a very convenient tool proposed by Poonen and Stoll (and independently by Ekedahl) to compute the density of a given subset of the integers. In this paper we provide an effective criterion to find all higher moments of the density (e.g. the mean, the variance) of a subset of a finite dimensional free module over the ring of algebraic integers of a number field. More precisely, we provide a geometric sieve that allows the computation of all higher moments corresponding to the density, over a general number field K. This work advances the understanding of geometric sieve for density computations in two ways: on one hand, it extends a result of Bright, Browning and Loughran, where they provide the geometric sieve for densities over number fields; on the other hand, it extends the recent result on a geometric sieve for expected values over the integers to both the ring of algebraic integers and to moments higher than the expected value. To show how effective and applicable our method is, we compute the density, mean and variance of Eisenstein polynomials and shifted Eisenstein polynomials over number fields. This extends (and fully covers) results in the literature that were obtained with ad-hoc methods.

We postulate axioms for a chiral half of a nonarchimedean 2-dimensional bosonic conformal field theory, that is, a vertex operator algebra in which a p-adic Banach space replaces the traditional Hilbert space. We study some consequences of our axioms leading to the construction of various examples, including p-adic commutative Banach rings and p-adic versions of the Virasoro, Heisenberg, and the Moonshine module vertex operator algebras. Serre p-adic modular forms occur naturally in some of these examples as limits of classical 1-point functions.