学术报告
报告题目1：Class numbers and $p$-ranks in $/mathbb{Z}^d_p$-towers
报 告 人：万大庆 教授 （美国加州大学欧文分校）
报告时间：2017年12月29日 （星期五）下午13:30-14:30
报告题目2：Multiple reciprocal sums and multiple reciprocal star sums of polynomials are almost never integers
报 告 人： 洪绍方 教授 （四川大学）
报告时间：2017年12月29日 （星期五）下午15:00-16:00
邀 请 人：胡甦 副教授
报告地点：四号楼4318室
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数学学院
2017年12月28日
报告1摘要：To extend Iwasawa's classical theorem from $/mathbb{Z}_p$-towers to $/mathbb{Z}^d_p$-towers, Greenberg conjectured that the exponent of $p$ in the $n$-th class number in a $/mathbb{Z}^d_p$-tower of a global field $K$ ramified at finitely many primes is given by a polynomial in $pn$ and $n$ of total degree at most d for sufficiently large $n$. This conjecture remains open for $d/ge 2$. In this talk, we prove that this conjecture is true in the function field case. Further, we propose a series of general conjectures on $p$-adic stability of zeta functions in a $p$-adic Lie tower of function fields.
报告2摘要： Let $n$ and $k$ be integers such that $1/le k/le n$ and $f(x)$ be a nonzero polynomial of integer coefficients. For any $n$-tuple $/vec{s}=(s_1, ..., s_n)$ of positive integers, we define $$H_{k,f}(/vec{s}, n):=/sum/limits_{1/leq i_{1}</cdots<i_{k}/le n}/prod/limits_{j=1}^{k}/frac{1}{f(i_{j})^{s_j}}$$and$$H_{k,f}^*(/vec{s}, n):=/sum/limits_{1/leq i_{1}/leq /cdots/leq i_{k}/leq n}/prod/limits_{j=1}^{k}/frac{1}{f(i_{j})^{s_j}}.$$If all $s_j$ are 1, then let $H_{k,f}(/vec{s}, n):=H_{k,f}(n)$ and $H_{k,f}^*(/vec{s}, n):=H_{k,f}^*(n)$. Hong and Wang refined the results of Erd/"{o}s and Niven, and of Chen and Tang by showing that $H_{k,f}(n)$ is not an integer if $n/geq 4$ and $f(x)=ax+b$ with $a$ and $b$ being positive integers. Meanwhile, Luo, Hong, Qian and Wang established the similar result when $f(x)$ is of nonnegative integer coefficients and of degree no less than two. For any $n$-tuple $/vec{s}=(s_1, ..., s_n)$ of positive integers, Pilehrood, Pilehrood and Tauraso proved that $H_{k,f}(/vec{s},n)$ and $H_{k,f}^*(/vec{s},n)$ are nearly never integers if $f(x)=x$. In this talk, we show that if $f(x)$ is a nonzero polynomial of nonnegative integer coefficients such that either $/deg f(x)/ge 2$ or $f(x)$ is linear and $s_j/ge 2$ for all integers $j$ with $1/le j/le n$, then $H_{k,f}(/vec{s}, n)$ and $H_{k,f}^*(/vec{s}, n)$ are not integers except for the case $f(x)=x^{m}$ with $m/geq1$ being an integer and $n=k=1$, in which case, both of $H_{k,f}(/vec{s}, n)$ and $H_{k,f}^*(/vec{s}, n)$ are integers. Furthermore, we prove that if $f(x)=2x-1$, then both $H_{k,f}(/vec{s}, n)$ and $H_{k,f}^*(/vec{s}, n)$ are not integers except when $n=1$, in which case $H_{k,f}(/vec{s}, n)$ and $H_{k,f}^*(/vec{s}, n)$ are integers. The method of the proofs is analytic and $p$-adic.
报告人简介：
万大庆，美国加州大学尔湾分校（University of California, Irvine）教授。中科院数学院研究所海外杰出访问教授，清华大学高研中心海外访问教授，国家杰岀青年基金（B类) 获得者，教育部海外杰出青年，入选中科院百人计划，获得国际华人数学家晨兴（Morningside）数学银奖。已在数学顶尖杂志Annals of Mathematic、Inventiones Mathematicae、Journal of American Mathematical Society 等发表了多篇文章。现为国际著名数论杂志《Journal of Number Theory》、《Finite Fields and Their Applications》编委，在数论、算术几何、编码、密码和计算复杂性领域都有很高的研究成就。
洪绍方，四川大学数学学院教授，博士生导师，研究领域为数论，算术几何和编码理论，入选教育部2006年度新世纪优秀人才计划，已发表学术论文90多篇，SCI收录论文60篇。