A Clean Approach to Rational Cubic Residues

On mod 3 triple Milnor invariants and triple cubic residue symbols in the Eisenstein number field

We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $\mathbb{Q}(\sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and Rédei's triple symbol, and describes the decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over $\mathbb{Q}(\sqrt{-3})$ with restricted ramification, which we construct concretely in the form similar to Rédei's dihedral extension over $\mathbb{Q}$. We also give a cohomological interpretation of our symbols by…

论有限域与高次互反律

On Finite Fields and Higher Reciprocity

本文系统阐述有限域、乘法特征及 Gauss/Jacobi 和，并在此基础上建立三次互反律、讨论素数2的三次剩余条件，最后综述四次互反律。

Characterizations of the $d$th-power residue matrices over finite fields

In a recent paper of the author with D. Dummit and H. Kisilevsky, we constructed a collection of matrices defined by quadratic residue symbols, termed "quadratic residue matrices", associated to the splitting behavior of prime ideals in a composite of quadratic extensions of $\mathbb{Q}$, and proved a simple criterion characterizing such matrices. We then analyzed the analogous classes of matrices constructed from the cubic and quartic residue symbols for a set of prime ideals of $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(i)$, respectively. In this paper, the goal is to construct and study the fi…

Simultaneous zeros of a Cubic and Quadratic form

We verify a conjecture of Emil Artin, for the case of a Cubic and Quadratic form over any $p$-adic field, provided the cardinality of the residue class field exceeds 293. That is any Cubic and Quadratic form with at least 14 variables has a non-trivial $p$-adic zero, with the aforementioned condition on the residue class field. A crucial step in the proof, involves generalizing a $p$-adic minimization procedure due to W. M. Schmidt to hold for systems of forms of arbitrary degrees.

$x^2+a/x$ 模一素数的剩余计数

The residue-counts of $x^2+a/x$ modulo a prime

本文研究了 $x^2+a/x$ 模素数的剩余计数问题，得出一个与三次剩余和二元二次型相关的显式公式。

关于有理幂剩余符号第二补律的一个注记

A note on the second supplementary law of rational power residue symbols

本文针对具有特殊和式形式的素数 $p$，利用负指标多对数函数给出 $q$ 次幂剩余符号 $(q/p)_q=1$ 的显式条件，从而推广第二补律。

A Cubic analogue of the Friedlander-Iwaniec spin over primes

In 1998 Friedlander and Iwaniec proved that there are infinitely many primes of the form $a^2+b^4$. To show this they defined the spin of Gaussian integers by the Jacobi symbol, and one of the key ingredients in the proof was to show that the spin becomes equidistributed along Gaussian primes. To generalize this we define the cubic spin of ideals of $\mathbb{Z}[ζ_{12}]=\mathbb{Z}[ζ_3,i]$ by using the cubic residue character on the Eisenstein integers $\mathbb{Z}[ζ_3]$. Our main theorem says that the cubic spin is equidistributed along prime ideals of $\mathbb{Z}[ζ_{12}]$. The proof of this fol…

一些双二次Pólya域及最简三次与五次域复合域的大Pólya群

Some bi-quadratic Pólya fields and large Pólya groups of compositum of simplest cubic and quintic fields

本文提供了三个双二次Pólya域族，并证明了Shank三次域与Lehmer五次域复合域存在无限多个大Pólya群指数为1的域。

Rational Points on Elliptic Curves y^2=x^3+a^3 in f_{p} where p{\equiv}1(mod6) is Prime

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some results are given on the sum of abscissae of these points. We give the number of solutions to y^2{\equiv}x^3+a^3(modp), also given in ([1], p.174), this time by means of the quadratic residue character, in a different way, by using the cubic residue character. Using the Weil conjecture, one can generalize the results concerning the number of points in F_{p} …

三次有理映射的斥周期点的密度

Density of repelling periodic points for cubic rational maps

针对三次有理映射，证明了其在$\mathbf C_p$上的斥周期点密度猜想，通过反证法结合系数分析，确定了特征为2和3时的情况，得出经典Julia集为空的结论。

Cubes in finite fields and related permutations

Let $p=3n+1$ be a prime with $n\in\mathbb{N}=\{0,1,\cdots\}$, and let $g\in\mathbb{Z}$ be a primitive root modulo $p$. Let $0<a_1<\cdots<a_n<p$ be all the cubic residues modulo $p$ in the interval $(0,p)$. Then clearly the sequence $$a_1\ {\rm mod}\ p,\ a_2\ {\rm mod}\ p,\cdots, a_n\ {\rm mod}\ p$$ is a permutation $s_p(g)$ of the sequence $$g^3\ {\rm mod}\ p,\ g^6\ {\rm mod}\ p,\cdots, g^{3n}\ {\rm mod}\ p.$$ In this paper, we shall determine the sign of this permutation.

An explicit Gross-Zagier formula related to the Sylvester Conjecture

Let $p\equiv 4,7\mod 9$ be a rational prime number such that $3\mod p$ is not a cubic residue. In this paper we prove the 3-part of the product of the full BSD conjectures for $E_p$ and $E_{3p^3}$ is true using an explicit Gross-Zagier formula, where $E_p: x^3+y^3=p$ and $E_{3p^2}: x^3+y^3=3p^2$ are the elliptic curves related to the Sylvester conjecture and cube sum problems.

On the order modulo p of an algebraic number (for p large enough)

Let K/Q be Galois, and let eta in K* whose conjugates are multiplicatively independent. For a prime p, unramified, prime to eta, let np be the residue degree of p and gp the number of P I p, then let o\_P(eta) and o\_p(eta) be the orders of eta modulo P and p, respectively.Using Frobenius automorphisms, we show that for all p\textgreater{}\textgreater{}0, some explicit divisors of p^(np)-1 cannot realize o\_P(eta) nor o\_p(eta), and we give a lower bound of o\_p(eta).Then we obtain that, for all p\textgreater{}\textgreater{}0 such that np \textgreater{}1, Prob(o\_p(eta)\textless{}p) $\le$ 1/p^…