About
me
My research interests lie in number theory, especially elliptic curves, class groups and character sums.
- Default: Hefei / 默认:合肥
- 04-04 ~ 04-06, Tongling / 铜陵
- 04-08 ~ 04-13, Nanjing / 南京


Experience
Research
- The generating fields of twisted Kloosterman sums
S. Zhang. Int. J. Number Theory (2025), to appear.
PDF | Slide & 幻灯片 | Abs
We use the Kloosterman sheaves constructed by Fisher to show when two Kloosterman sums differ a factor of a $(q-1)$-th root of unity, and use $p$-adic analysis to prove the non-vanishing of the Kloosterman sums. Then we can determine the generating fields by these results. - On the linearity of the periods of subtraction games
S. Zhang. Theor. Comput. Sci. 985 (2024).
PDF | PubPDF | 幻灯片 | Abs | Bib
A subtraction game is an impartial combinatorial game involving a finite set $S$ of positive integers. The nim-sequence $\mathcal G_S$ associated with this game is ultimately periodic. In this paper, we study the nim-sequence $\mathcal G_S\cup\{c\}$ where $S$ is fixed and $c$ varies. We conjecture that there is a multiplier $q$ of the period of $\mathcal G_S$, such that for sufficiently large $c$, the pre-period and period of $\mathcal G_S\cup\{c\}$ are linear on $c$, if $c$ modulo $q$ is fixed. We prove it in several cases.
We also give new examples with period $2$ inspired by this conjecture.@article {Zhang2024tcs, AUTHOR = {Zhang, Shenxing}, TITLE = {On the linearity of the periods of subtraction games}, JOURNAL = {Theor. Comput. Sci.}, FJOURNAL = {Theoretical Computer Science}, VOLUME = {985}, YEAR = {2024}, PAGES = {114350}, ISSN = {0304-3975}, MRCLASS = {91A46, 91A05}, DOI = {10.1016/j.tcs.2023.114350}, URL = {https://doi.org/10.1016/j.tcs.2023.114350}, }
- On a comparison of Cassels pairings of different elliptic curves
S. Zhang. Acta Arith. 211 (2023), no. 1, 1-23.
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Let $e_1, e_2, e_3$ be nonzero integers satisfying $e_1+e_2+e_3=0$. Let $(a,b,c)$ be a primitive triple of odd integers satisfying $e_1a^2+e_2b^2+e_3c^2=0$. Denote by $E: y^2=x(x-e_1)(x+e_2)$ and $\mathcal E: y^2=x(x-e_1a^2)(x+e_2b^2)$. Assume that the $2$-Selmer groups of $E$ and $\mathcal E$ are minimal. Let $n$ be a positive square-free odd integer, where the prime factors of $n$ are nonzero quadratic residues modulo each odd prime factor of $e_1e_2e_3abc$. Then under certain conditions, the $2$-Selmer group and the Cassels pairing of the quadratic twist $E^{(n)}$ coincide with those of $\mathcal E^{(n)}$. As a corollary, $E^{(n)}$ has Mordell-Weil rank zero without order $4$ element in its Shafarevich-Tate group, if and only if these holds for $\mathcal E^{(n)}$. We also give some applications for the congruent elliptic curve.@article {Zhang2023aa, AUTHOR = {Zhang, Shenxing}, TITLE = {On a comparison of {C}assels pairings of different elliptic curves}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {211}, YEAR = {2023}, NUMBER = {1}, PAGES = {1-23}, ISSN = {0065-1036}, MRCLASS = {11G05 (11R11 11R29)}, DOI = {10.4064/aa220709-15-7}, URL = {https://doi.org/10.4064/aa220709-15-7}, }
- On the Newton polygons of twisted $L$-functions of binomials
S. Zhang. Finite Fields Appl. 80 (2022).
PDF | PubPDF | 幻灯片 | Abs | Bib
Let $\chi$ be an order $c$ multiplicative character of a finite field and $f(x) = x^d+\lambda x^e$ a binomial with $(d,e)=1$. We study the twisted classical and $T$-adic Newton polygons of $f$. When $p>(d-e)(2d-1)$, we give a lower bound of Newton polygons and show that they coincide if $p$ does not divide a certain integral constant depending on $p\bmod cd$.
We conjecture that this condition holds if $p$ is large enough with respect to $c,d$ by combining all known results and the conjecture given by Zhang-Niu. As an example, we show that it holds for $e=d-1$.@article {Zhang2022ffa, AUTHOR = {Zhang, Shenxing}, TITLE = {On the {N}ewton polygons of twisted {$L$}-functions of binomials}, JOURNAL = {Finite Fields Appl.}, FJOURNAL = {Finite Fields and their Applications}, VOLUME = {80}, YEAR = {2022}, PAGES = {Paper No. 102026, 20}, ISSN = {1071-5797}, MRCLASS = {11S40 (11T23)}, MRNUMBER = {4388988}, DOI = {10.1016/j.ffa.2022.102026}, URL = {https://doi.org/10.1016/j.ffa.2022.102026}, }
- The $3$-class groups of $\mathbb Q(\sqrt[3]p)$ and its normal closure
J. Li, S. Zhang. Math Z. 300 (2022), no. 1, 209-215.
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We determine the $3$-class groups of $\mathbb Q(\sqrt[3]p)$ and $K=\mathbb Q(\sqrt[3]p,\sqrt{-3})$ when $p\equiv4,7\bmod9$ is a prime and $3$ is a cube modulo $p$. This confirms a conjecture made by Barrucand-Cohn, and proves the last remaining case of a conjecture of Lemmermeyer on the $3$-class group of $K$.@article {LiZhang2022mz, AUTHOR = {Li, Jianing and Zhang, Shenxing}, TITLE = {The 3-class groups of {$\mathbb{Q}(\root3p)$} and its normal closure}, JOURNAL = {Math. Z.}, FJOURNAL = {Mathematische Zeitschrift}, VOLUME = {300}, YEAR = {2022}, NUMBER = {1}, PAGES = {209--215}, ISSN = {0025-5874}, MRCLASS = {11R29 (11R16)}, DOI = {10.1007/s00209-021-02797-5}, URL = {https://doi.org/10.1007/s00209-021-02797-5}, }
- $\ell$-Class groups of fields in Kummer towers
J. Li, Y. Ouyang, Y. Xu, S. Zhang. Publ. Mat. 66 (2022), no. 1, 235-267.
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Let $\ell$ and $p$ be prime numbers and $K_{n,m}=\mathbb Q(p^{1/\ell^n},\zeta_{2\ell^m})$. We study the $\ell$-class group of $K_{n,m}$ in this paper. When $\ell=2$, we determine the structure of the $2$-class group of $K_{n,m}$ for all $(n,m)\in\mathbb Z^2_{\ge 0}$ in the case $p=2$ or $p\equiv3,5\bmod 8$, and for $(n,m)=(n,0),(n,1)$ or $(1,m)$ in the case $p\equiv7\bmod{16}$, generalizing the results of Parry about the $2$-divisibility of the class number of $K_2,0$. We also obtain results about the $\ell$-class group of $K_{n,m}$ when $\ell$ is odd and in particular $\ell=3$. The main tools we use are class field theory, including Chevalley’s ambiguous class number formula and its generalization by Gras, and a stationary result about the $\ell$-class groups in the $2$-dimensional Kummer tower $\{K_{n,m}\}$.@article {LiOuyangXuZhang2022publmat, AUTHOR = {Li, Jianing and Ouyang, Yi and Xu, Yue and Zhang, Shenxing}, TITLE = {{$\ell$}-class groups of fields in {K}ummer towers}, JOURNAL = {Publ. Mat.}, FJOURNAL = {Publicacions Matem\`atiques}, VOLUME = {66}, YEAR = {2022}, NUMBER = {1}, PAGES = {235--267}, ISSN = {0214-1493}, DOI = {10.5565/publmat6612210}, URL = {https://doi.org/10.5565/publmat6612210}, }
- The generating fields of two twisted Kloosterman sums
S. Zhang. J. Univ. Sci. Technol. China 51 (2021), no. 12, 879-888.
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In this paper, we study the generating fields of the twisted Kloosterman sums $\mathrm{Kl}(q,a,\chi)$ and the partial Gauss sums $g(q,a,\chi)$. We require that the characteristic $p$ is large with respect to the order $d$ of the character $\chi$ and the trace of the coefficient $a$ is nonzero. When $p\equiv\pm1\bmod d$, we can characterize the generating fields of these character sums. For general $p$, when $a$ lies in the prime field, we propose a combinatorial condition on $(p,d)$ to ensure one can determine the generating fields.@article {Zhang2021just, AUTHOR = {Zhang, Shenxing}, TITLE = {The generating fields of two twisted {K}loosterman sums}, JOURNAL = {J. Univ. Sci. Technol. China}, FJOURNAL = {Journal of University of Science and Technology of China}, VOLUME = {51}, YEAR = {2021}, NUMBER = {12}, PAGES = {879--888}, ISSN = {0253-2778}, DOI = {10.52396/JUST-2021-0077}, URL = {https://doi.org/10.52396/JUST-2021-0077}, }
- Birch's lemma over global function fields
Y. Ouyang, S. Zhang. Proc. Amer. Math. Soc. 145 (2017), no. 2, 577-584.
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We obtain a function field version of Birch’s Lemma, which reveals non-torsion points in quadratic twists of an elliptic curve over a global function field, where the quadratic twists have many prime factors. The proof is based on Brown’s Euler system for Heegner points of function fields and Vigni’s result.@article {OuyangZhang2017pams, AUTHOR = {Ouyang, Yi and Zhang, Shenxing}, TITLE = {Birch's lemma over global function fields}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {145}, YEAR = {2017}, NUMBER = {2}, PAGES = {577--584}, ISSN = {0002-9939}, MRCLASS = {11G05 (11D25 11G40 14H52)}, DOI = {10.1090/proc/13265}, URL = {https://doi.org/10.1090/proc/13265}, }
- Newton polygons of $L$-functions of polynomials $x^d+ax^{d-1}$ with $p\equiv-1\bmod d$
Y. Ouyang, S. Zhang. Finite Fields Appl. 37 (2016), 285-294.
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For prime $p\equiv-1\bmod d$ and $q$ a power of $p$, we obtain the slopes of the $q$-adic Newton polygons of $L$-functions of $x^d+ax^{d-1}\in \mathbb F_q[x]$ with respect to finite characters $\chi$ when $p$ is larger than an explicit bound depending only on $d$ and $\log_p q$. The main tools are Dwork’s trace formula and Zhu’s rigid transform theorem.@article {OuyangZhang2016ffa, AUTHOR = {Ouyang, Yi and Zhang, Shenxing}, TITLE = {Newton polygons of {$L$}-functions of polynomials {$x^d+ax^{d-1}$} with {$p\equiv-1\mod d$}}, JOURNAL = {Finite Fields Appl.}, FJOURNAL = {Finite Fields and their Applications}, VOLUME = {37}, YEAR = {2016}, PAGES = {285--294}, ISSN = {1071-5797}, MRCLASS = {11T06}, DOI = {10.1016/j.ffa.2015.10.003}, URL = {https://doi.org/10.1016/j.ffa.2015.10.003}, }
- On second 2-descent and non-congruent numbers
Y. Ouyang, S. Zhang. Acta Arith. 170 (2015), no. 4, 343-360.
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We use the so-called second $2$-decent method to find several series of non-congruent numbers. We consider three different $2$-isogenies of the congruent elliptic curves and their duals, and find a necessary condition to estimate the size of the images of the $2$-Selmer groups in the Selmer groups of the isogeny.@article {OuyangZhang2015aa, AUTHOR = {Ouyang, Yi and Zhang, Shenxing}, TITLE = {On second 2-descent and non-congruent numbers}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {170}, YEAR = {2015}, NUMBER = {4}, PAGES = {343--360}, ISSN = {0065-1036}, MRCLASS = {11G05 (11D25)}, DOI = {10.4064/aa170-4-3}, URL = {https://doi.org/10.4064/aa170-4-3}, }
- On non-congruent numbers with 1 modulo 4 prime factors
Y. Ouyang, S. Zhang. Sci. China Math. 57 (2014), no. 3, 649-658.
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In this paper, we use the $2$-decent method to find a series of odd non-congruent numbers $\equiv1\pmod 8$ whose prime factors are $\equiv1\pmod4$ such that the congruent elliptic curves have second lowest Selmer groups, which includes Li and Tian’s result [LT00] as special cases.@article {OuyangZhang2014scm, AUTHOR = {Ouyang, Yi and Zhang, ShenXing}, TITLE = {On non-congruent numbers with 1 modulo 4 prime factors}, JOURNAL = {Sci. China Math.}, FJOURNAL = {Science China. Mathematics}, VOLUME = {57}, YEAR = {2014}, NUMBER = {3}, PAGES = {649--658}, ISSN = {1674-7283}, MRCLASS = {11G05 (11D25)}, DOI = {10.1007/s11425-013-4705-y}, URL = {https://doi.org/10.1007/s11425-013-4705-y}, }
Here are some preprint.
- On the quadratic twist of elliptic curves with full $2$-torsion
Z. Wang, S. Zhang. (2022), preprint.
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Let $E: y^2=x(x-a^2)(x+b^2)$ be an elliptic curve with full $2$-torsion group, where $a$ and $b$ are coprime integers and $2(a^2+b^2)$ is a square. Assume that the $2$-Selmer group of $E$ has rank two. We characterize all quadratic twists of $E$ with Mordell-Weil rank zero and $2$-primary Shafarevich-Tate groups $(\mathbb Z/2\mathbb Z)^2$, under certain conditions. We also obtain a distribution result of these elliptic curves. - The virtual periods of linear recurrence sequences in cyclotomic fields
S. Zhang. arXiv: 2010.08342 (2020).
PDF | arXivPDF | Abs
A linear recurrence sequence in a cyclotomic field produces a sequence of the generating fields of each term. We show that the later sequence is periodic after removing the first finite terms, and give a bound of its period. This can be applied to exponential sums.
Here are some notes.
- The curve and $p$-adic Hodge theory
Laurent Fargus, 2019-11-01 ~ 2020-01-10, MCM Beijing
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The main theme of this course will be to understand and give a meaning to the notion of a $p$-adic Hodge structure. Starting with the work of Fontaine, who introduced many of the basic notions in the domain, it took many years to understand the exact definition of a $p$-adic Hodge structure. We now have the right definition: this involves the fundamental curve of $p$-adic Hodge theory and vector bundles on it. In the course I will explain the construction and basic properties of the curve. I will moreover explain the proof of the classification of vector bundles theorem on the curve. As an application I will explain the proof of weakly admissible implies admissible. In the meanwhile I will review many objects that show up in $p$-adic Hodge theory like $p$-divisible groups and their moduli spaces, Hodge-Tate and de Rham period morphisms, and filtered $\phi$-modules. - $p$-adic abelian integrals
Pierre Colmez, 2016-09-14 ~ 2016-10-26, BICMR Beijing
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The study of complex abelian integrals, i.e., integrals of algebraic functions of one complex variable, was a major incentive to develop complex algebraic geometry (some 150 years ago). After briefly explaining the complex theory, I will study its analog in the $p$-adic world: this provides a concrete introduction to $p$-adic Hodge theory, a theory that was originated by Tate some 50 years ago and was turned into one of most powerful tools of number theory.
教学
教学中
历史记录
课程 | 时间 | 资料 |
---|---|---|
复变函数与积分变换 1400261B 合肥工业大学 |
2024年秋 2023年秋 2022年秋 |
课件 复习课 试卷 |
线性代数 1400071B 合肥工业大学 |
2024年秋 | 课件 复习课 |
数学(下) 034Y01 合肥工业大学 |
2023年春(代课) 2022年春 |
课件
1,
2.1-2.2 课件PPT 2, 3, 4 试卷 课程精讲 1, 2, 3, 4 |
线性代数 清华丘成桐中学生数学夏令营 |
2023年夏(助教) 2022年夏(助教) 2021年夏(助教) |
作业和试卷 |
概率论与数理统计 合肥工业大学 |
2021年秋(助课) | 课件PPT 5-6 |
复变函数B 001548.05 中国科学技术大学 |
2020年秋 | 试卷 |
代数数论 MA05109 中国科学技术大学 |
2020年春 | 讲义 试卷 课程视频 |
代数与数论 中科院暑期学校 |
2019年夏(助教) | 代数数论I讲义 代数数论II讲义 群表示论讲义 代数几何讲义 代数数论作业 |
初等数论 中国科学院大学 |
2016年春(助教) | 试卷 习题课 |
代数学基础 001356.01 中国科学技术大学 |
2014年秋(助教) 2012年秋(助教) |
试卷 习题课 |
近世代数(H) 001704.01 中国科学技术大学 |
2013年春(助教) | 试卷 |
近世代数 001010.01 中国科学技术大学 |
2011年秋(助教) | 试卷 |
复变函数 001012.02 中国科学技术大学 |
2011年春(助教) | 试卷 |
教研论文
- 袁志杰, 张神星. 复变函数在不同圆环域内洛朗展开的探究[J]. 南阳师范学院学报, 2024, 23(3): 38-41.
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针对解析函数在同圆心的不同圆环域内洛朗级数,给出了这些洛朗级数的系数之差与函数在两个圆环之间的极点的关系。特别地,由有理函数在某一圆环内的洛朗展开可以直接得到它在同圆心的任一圆环域内的洛朗级数。 - 欧阳毅, 张神星. 善用初等变换展开线性代数理论教学[J]. 大学数学, 2023, 39(5): 68-75.
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初等变换是线性代数的基本变换,在线性代数课程中常常被用来计算,例如求解线性方程组、计算方阵的行列式、矩阵的求逆以及更一般的矩阵方程 $AX=B$ 的求解、计算整数矩阵和域上多项式矩阵的 Smith 标准形、以及计算对称阵的相合标准形等。本文说明如何灵活利用初等变换,给出线性代数课程中一些重要理论结果的系统而又简洁的证明。
本科生毕业论文
- 邱修煜(2024):椭圆曲线的同源与密码学应用的研究
- 刘旭鸿(2022):二次域类群的结构以及算术应用
大学生创新创业训练计划
- 朱思雯、姚佳婧、王悦妍、邓颖、罗天曦(X202210359482):新型冠状病毒传染模型
链接
数学链接
LATEX 链接
本页面所有内容的TEX源代码可在此处找到。
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常用LaTeX代码以及所有入门知识与资料
LaTeX 工作室整理的常见参考资料,该网站也提供了在线问答社区。 -
一份不太简短的 LaTeX2ε 介绍
LaTeX 入门必看资料,先看附录A安装TeX Live 或 MiKTeX,然后可使用编辑器 TeXworks。也可以使用 VSCode 并安装扩展 LaTeX Workshop. -
LaTeX 常见符号表
页数有点多,建议用在线识别LaTeX符号。 -
宏包:合肥工业大学期末试卷模板 v1.9
使用前请查看说明文档。点此下载zip压缩包。 -
宏包:合肥工业大学学位论文模板 v1.0.4
点击下载说明文档,使用样例。 - 基金委基金申请书正文TEX模板,已尽量与Word版模板文字位置保持一致。本人不对该模板正确性负责。
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宏包:TikZ & PGF
功能强大的绘图宏包,但手册有上千页,不便于入门。建议看简略版可视化TikZ使用手册,也可看非官方简短入门手册。 -
宏包:Beamer
常用的LaTeX幻灯片宏包。点击下载中译版用户手册,英文版手册。 -
宏包:XY-pic
中文简介, 更详细的中文介绍请参考《LaTeX入门与提高》§12.4,点击下载英文版用户手册。
其它链接
蔡 立 | 丁一文 | 范洋宇 | 方江学 | 胡永泉 | 李加宁 | 李 超 | 李宋宋 | 李文威 | 李永雄 | 刘若川 | 刘 余 | 吕恒飞 | 欧阳毅 | 邱国寰 | 任汝飞 | 申 旭 | 舒 杰 | 田一超 | 田 野 | 万大庆 | 万 昕 | 王浩然 | 肖 梁 | 许 跃 | 许晨阳 | 杨金榜 | 杨同海 | 袁新意 | 袁轶君 | 恽之玮 | 张 磊 | 张寿武 | 张 伟 | 赵彦冰 | 郑维喆 | Henri Darmon | Laurent Fargues | Vincent Lafforgue
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