 Research Associate Professor (特聘副研究员)

My research interests lie in number theory, especially elliptic curves, class groups and character sums.

## Experience

 2021/12 - present Research Associate Professor, HFUT 2018/04 - 2021/11 Postdoctoral Fellow, Mentor Yi Ouyang, USTC 2016/03 - 2018/03 Postdoctoral Fellow, Mentor Ye Tian, AMSS CAS 2010/09 - 2015/11 Ph.D. in Mathematics, Advisor Yi Ouyang, USTC 2006/09 - 2010/06 B.S. in Mathematics, USTC

## Research

1. On a comparison of Cassels pairings of different elliptic curves
S. Zhang. (2022), preprint.
PDF | 幻灯片 | Abs
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.
2. On the quadratic twist of elliptic curves with full $2$-torsion
Z. Wang, S. Zhang. (2022), preprint.
PDF | Abs
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.
3. 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 {Zhang2022,
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},
}
4. On linearity of the periods of subtraction games
S. Zhang. (2021), preprint.
PDF | Abs
The subtraction game is an impartial combinatorial games involving a finite set $S$ of positive integers. The nim-sequence $\mathcal{G}_S$ associated to 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.
5. The $3$-class groups of $\mathbb Q(\sqrtp)$ and its normal closure
J. Li, S. Zhang. Math Z. 300 (2022), no. 1, 209-215.
PDF | PubPDF | Abs | Bib
We determine the $3$-class groups of $\mathbb{Q}(\sqrt{p})$ and $K = \mathbb{Q}(\sqrtp, \sqrt{-3})$ when $p\equiv 4, 7 \bmod 9$ 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 {LiZhang2022,
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},
}
6. $\ell$-Class groups of fields in Kummer towers
J. Li, Y. Ouyang, Y. Xu, S. Zhang. Publ. Mat. 66(2022), no. 1, 235-267.
PDF | PubPDF | Abs | Bib
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 $(n, m)\in\mathbb{Z}^2_{\ge0}$ in the case $p=2$ or $p\equiv 3,5 \bmod 8$, and for $(n, m) = (n, 0), (n, 1)$ or $(1, m)$ in the case $p\equiv 7 \bmod 16$, 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 {LiOuyangXuZhang2022,
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},
}
7. The distinctness and generating fields of twisted Kloosterman sums
S. Zhang. (2021), preprint.
PDF | Slide & 幻灯片 | Abs
We use the Kloosterman sheaves constructed by Fisher to show when two Kloosterman sums differ 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.
8. The generating fields of two twisted Kloosterman sums
S. Zhang. J. Univ. Sci. Technol. China 51 (2021), no. 12, 879-888.
PDF | PubPDF | Abs
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 pis large with respect to the order dof 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.
9. 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.
10. Birch's lemma over global function fields
Y. Ouyang, S. Zhang. Proc. Amer. Math. Soc. 145 (2017), no. 2, 577-584.
PDF | PubPDF | Abs | Bib
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 {OuyangZhang2017,
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},
}
11. 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.
PDF | PubPDF | Abs | Bib
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 $\log_pq$. The main tools are Dwork’s trace formula and Zhu’s rigid transform theorem.
@article {OuyangZhang2016,
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},
}
12. On second 2-descent and non-congruent numbers
Y. Ouyang, S. Zhang. Acta Arith. 170 (2015), no. 4, 343-360.
PDF | PubPDF & Errata | Abs | Bib
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 {OuyangZhang2015,
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},
}
13. On non-congruent numbers with 1 modulo 4 prime factors
Y. Ouyang, S. Zhang. Sci. China Math. 57 (2014), no. 3, 649-658.
PDF | PubPDF | Abs | Bib
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 {OuyangZhang2014,
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 notes.

1. The curve and $p$-adic Hodge theory
Laurent Fargus, 2019/11/01 - 2020/01/10, MCM Beijing
PDF | Abs
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.
2. $p$-adic abelian integrals
Pierre Colmez, 2016/09/14 - 2016/10/26, BICMR Beijing
PDF | Abs
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.

## Teaching

### 教学中

#### 2022年秋 复变函数与积分变换(1400261B)

• 教材：西安交通大学高等数学教研室编《复变函数》，张元林《积分变换》
• 日期：2022/08/30-2022/11/03
• 时间： 009班(电信工) 周二 8:10-10:00，周四 10:20-12:10
010班(光信息，智感工) 周二 10:20-12:10，周四 8:10-10:00
• 地点：翡翠湖校区 翠五教103
• 课件：📖1, 2, 3, 4, 5, 6

### 历史记录

034Y01

2022年春 📖课件 1, 2, 3, 4
📝试卷
🎥课程精讲 1, 2, 3, 4

2021年秋(助教) 📖课件 5, 6

2022年夏(助教)
2021年夏(助教)
📝作业和试卷

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

2016年春(助教) 📝试卷
📏习题课

001548.05

2020年秋 📝试卷

MA05109

2020年春 📖讲义
📝试卷
🎥课程视频

001356.01

2014年秋(助教)
2012年秋(助教)
📝试卷
📏习题课

001704.01

2013年春(助教) 📝试卷

001010.01

2011年秋(助教) 📝试卷

001012.02

2011年春(助教) 📝试卷

 📗 丘成桐大学生数学竞赛历年笔试题和口试题 📗 10000 个科学难题·数学卷 📘 一份不太简短的 $\LaTeX2\varepsilon$ 介绍 6.02版 推荐使用 TeX Live 或 MiKTeX 自带的编辑器 TeXworks 📘 XYpic中文简介 更详细的中文介绍请参考《LaTeX入门与提高》§12.4 📘 李文威: 数学写作漫谈 📘 李文威: 教学实践：经验、反思与构想 📘 How to Give a Good Colloquium - John E. McCarthy
 Yi Ouyang (欧阳毅) Ye Tian (田野) Xinyi Yuan (袁新意) Wen-Wei Li (李文威) Yiwen Ding (丁一文) Yang Jinbang (杨金榜) Yue Xu (许跃) Jianing Li (李加宁) Xin Wan (万昕) Xu Shen (申旭) Yongquan Hu (胡永泉) Weizhe Zheng (郑维喆) Shou-Wu Zhang (张寿武) Wei Zhang (张伟) Tonghai Yang (杨同海) Daqing Wan (万大庆) Liang Xiao (肖梁) Ruochuan Liu (刘若川) Chenyang Xu (许晨阳) Lei Zhang (张磊) Yifeng Liu (刘一峰) Zhiwei Yun (恽之玮) Chao Li (李超) Yihang Zhu (朱艺航) Jie Shu (舒杰) Yanbing Zhao (赵彦冰) Guohuan Qiu (邱国寰) Yijun Yuan (袁轶君) Henri Darmon Laurent Fargues Vincent Lafforgue NUMDAM DigiZ Z-Library MathSciNet Zentralblatt MATH MSC2020 LetPub LMFDB Math Genealogy MathOverflow Gitee GitHub 原版模组更多的合成 原版模组入门教程