- On a comparison of rank zero quadratic twists of different elliptic curves
S. Zhang.
(2022), preprint.
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Let $E: y^2=x(x-e_1)(x+e_2)$ be an elliptic curve, where $e_1,e_2,e_3=-e_1-e_2$ are integers. Let $(a,b,c)$ be a primitive triple of odd integers satisfying $e_1a^2+e_2b^2+e_3c^2=0$. Denote by $\mathcal E: y^2=x(x-e_1a^2)(x+e_2b^2)$. Assume that the $2$-Selmer groups of $E$ and $\mathcal E$ are minimal. Let $n$ be a positive 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 quadratic twist $\mathcal E^{(n)}$ has Mordell-Weil rank zero and there is no order $4$ element in its Shafarevich-Tate group, if and only if these holds for $E^{(n)}$. We also give some applications for the congruent elliptic curve.
- 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.
- On the Newton polygons of twisted $L$-functions of binomials
S. Zhang.
Finite Fields Appl. 80 (2022).
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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 $e=d-1$.
- On linearity of the periods of subtraction games
S. Zhang.
(2021), preprint.
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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.
- The generating fields of two twisted Kloosterman sums
S. Zhang.
J. Univ. Sci. Technol. China
(2022), to appear.
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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 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.
- 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 $p\equiv 4, 7 \bmod 9$ 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},
}
- $\ell$-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 $(n, m)\in\mathbb{Z}^2_{\ge0}$ 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},
}
- The distinctness and generating fields of twisted Kloosterman sums
S. Zhang.
(2021), preprint.
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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.
- The virtual periods of linear recurrence sequences in cyclotomic fields
S. Zhang.
arXiv: 2010.08342 (2020).
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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.
- 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 {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},
}
- 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 $\log_pq$. 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},
}
- 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 {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},
}
- 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 {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.