arXiv subjects

Hourong Qin

Publications and source records attributed to Hourong Qin.

8 recordsLinked to original sources

arXiv2021

The Lang-Trotter Conjecture for the elliptic curve $y^2=x^3+Dx$

Let $E$ be an elliptic curve over $\mathbb{Q}.$ Let $a_p$ denote the trace of the Frobenius endomorphism at a rational prime $p$. For a fixed integer $r,$ define the prime-counting function as $π_{E,r}(x):=\sum_{p\leq x,p\nmid Δ_E,a_p=r}1$. The Lang-Trotter Conjecture predicts that $$π_{E,r}(x)=C_{E,r}\cdot \frac{\sqrt{x}}{{\rm log}x}+o(\frac{\sqrt{x}}{{\rm log}x})$$ as $x\longrightarrow \infty,$ where $C_{E,r}$ is a specific non-negative constant. The Hardy-Littlewood Conjecture gives a similar asymptotic formula as above for the number of primes of the form $ax^2+bx+c$. We establish a relationship between the Hardy-Littlewood Conjecture and the Lang-Trotter Conjecture for the elliptic curve $y^2=x^3+Dx.$ We show that the Hardy-Littlewood Conjecture implies the Lang-Trotter Conjecture for $y^2=x^3+Dx.$ Conversely, if the Lang-Trotter Conjecture holds for some $D$ and $2r$ (for $y^2=x^3+Dx, p\nmid D, a_p$ is always even) with positive constant $C_{E,2r},$ then the polynomial $x^2+r^2$ represents infinitely many primes. For a prime $p$, if $a_p=2r$, then $p$ is necessarily of the form $x^2+r^2$. Fixing $r$ and $D$, and assuming that the Hardy-Littlewood Conjecture holds, we obtain the density of the primes with $a_p=2r$ inside the set of primes of the form $x^2+r^2$. In some cases, the density is $1/4$, which is a natural expectation, but it fails to be true for all $D$. In particular, we give a full list of $D$ and $r$ when there is no prime $p$ for $a_p=2r$.

math.NT↗

arXiv2019

Mahler measure of polynomials defining genus 2 and 3 curves

In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of $L$-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and genus 3 curves. Furthermore, we study the Mahler measures of several families of nonreciprocal polynomials defining genus 2 curves and numerically find relations between the Mahler measures of these families and special values of $L$-functions of elliptic curves. We also find identities between the Mahler measures of these nonreciprocal families and tempered polynomials defining genus 1 curves. We will explain these relations by considering the pushforward and pullback of certain elements in $K_2$ of curves defined by these polynomials and applying Beilinson's conjecture on $K_2$ of curves. We show that there are two and three explicit linearly independent elements in $K_2$ of certain families of genus 2 and genus 3 curves.

math.NT↗

arXiv2016

Lehmer's totient problem over $\mathbb{F}_q[x]$

In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in\mathbb{F}_q[x]$ and $φ(q,p(x))$ be the Euler's totient function of $p(x)$ over $\mathbb{F}_q[x],$ where $\mathbb{F}_q$ is a finite field with $q$ elements. We prove that $φ(q,p(x))|(q^{{\rm deg}(p(x))}-1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3, \; p(x)$ is the product of any $2$ non-associate irreducibes of degree $1;$ or (iii) $q=2,\; p(x)$ is the product of all irreducibles of degree $1,$ all irreducibles of degree $1$ and $2,$ and the product of any $3$ irreducibles one each of degree $1, 2$ and $3$.

math.NT↗

arXiv2016

Non-vanishing Fourier coefficients of modular forms

In this paper, we generalize D. H. Lehmer's result to give a sufficient condition for level one cusp forms $f$ with integral Fourier coefficients such that the smallest $n$ for which the coefficients $a_n(f)=0$ must be a prime. Then we describe a method to compute a bound $B$ of $n$ such that $a_n(f)\ne0$ for all $n<B$. As examples, we achieve the explicit bounds $B_k$ for the unique cusp form $Δ_{k}$ of level one and weight k with $k=16, 18, 20, 22, 26$ such that $a_n(Δ_k)\ne0$ for all $n<B_k$.

math.NT↗

arXiv2011

Higher $K$-Groups of Smooth Projective Curves Over Finite Fields

Let $X$ be a smooth projective curve over a finite field $\mathbb{F}$ with $q$ elements. For $m\geq 1,$ let $X_m$ be the curve $X$ over the finite field $\mathbb{F}_m$, the $m$-th extension of $\mathbb{F}.$ Let $K_n(m)$ be the $K$-group $K_n(X_m)$ of the smooth projective curve $X_m.$ In this paper, we study the structure of the groups $K_n(m).$ If $l$ is a prime, we establish an analogue of Iwasawa theorem in algebraic number theory for the orders of the $l$-primary part $K_n(l^m)\{l\}$ of $K_n(l^m)$. In particular, when $X$ is an elliptic curve $E$ defined over $\mathbb{F},$ our method determines the structure of $K_n(E).$ Our results can be applied to construct an efficient {\bf DL} system in elliptic cryptography.

math.NT↗

arXiv2010

Homological behavior of Auslander's $k$-Gorenstein rings

In this paper we mainly study the homological properties of dual modules over $k$-Gorenstein rings. For a right quasi $k$-Gorenstein ring $Λ$, we show that the right self-injective dimension of $Λ$ is at most $k$ if and only if each $M \in$mod $Λ$ satisfying the condition that Ext$_Λ^i(M, Λ)=0$ for any $1\leq i \leq k$ is reflexive. For an $\infty$-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of $Λ$ are identical. In addition, we show that if $Λ$ is a left quasi $\infty$-Gorenstein ring and $M\in$mod $Λ$ with grade$M$ finite, then Ext$_Λ^i($Ext$_{Λ^{op}}^i($Ext$_Λ^{{\rm grade}M}(M, Λ), Λ), Λ)=0$ if and only if $i\neq$grade$M$. For a 2-Gorenstein ring $Λ$, we show that a non-zero proper left ideal $I$ of $Λ$ is reflexive if and only if $Λ/I$ has no non-zero pseudo-null submodule.

math.RA↗

arXiv2010

Prime ideals in decomposable lattices

A distributive lattice $L$ with minimum element $0$ is called decomposable lattice if $a$ and $b$ are not comparable elements in $L$ there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b), b=\overline{b}\vee(a\wedge b)$ and $\overline{a}\wedge \overline{b}=0$. The main purpose of this paper is to investigate prime ideals, minimal prime ideals and special ideals of a decomposable lattice. These are keys to understand the algebraic structure of decomposable lattices.

math.CO↗

arXiv2010

The structure of decomposable lattices determined by their prime ideals

A distributive lattice $L$ with minimum element $0$ is called decomposable if $a$ and $b$ are not comparable elements in $L$ then there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b), b=\overline{b}\vee(a\wedge b)$ and $\overline{a}\wedge \overline{b}=0$. The main purpose of this paper is to study the structure of decomposable lattices determined by their prime ideals. The properties for five special decomposable lattices are derived.

math.GR↗