On a senary quartic form

Quartic Forms in Many Variables

We show that a quartic $p$-adic form with at least $3192$ variables possesses a non-trivial zero. We also prove new results on systems of cubic, quadratic and linear forms. As an example, we show that for a system comprising two cubic forms $132$ variables are sufficient.

Cubic moments of Fourier coefficients and pairs of diagonal quartic forms

We establish the non-singular Hasse Principle for pairs of diagonal quartic equations in 22 or more variables. Our methods involve the estimation of a certain entangled two-dimensional 21st moment of quartic smooth Weyl sums via a novel cubic moment of Fourier coefficients.

Clifford quartic forms and local functional equations of non-prehomogeneous type

We construct polynomials of degree 4 that can not be obtained from prehomogeneous vector spaces, but, for which one can associate local zeta functions satisfying functional equations.

The Method Of Thue-Siegel For Binary Quartic Forms

We will use Thue-Siegel method, based on Padé approximation via hypergeometric functions, to give upper bounds for the number of integral solutions to the equation $|F(x, y)| = 1$ as well as the inequalities $|F(x, y)| \leq h$, for a certain family of irreducible quartic binary forms.

关于某些局部可解二元四次型族中可解型的比例

On the proportions of soluble forms in some families of locally soluble binary quartic forms

本文研究了局部可解整二元四次型中可解型的比例，基于已有关于椭圆曲线的结果，估计了某些特定子族中该比例的值。

Binary quartic forms with vanishing $J$-invariant

We obtain an asymptotic formula for the number of $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of irreducible binary quartic forms with integer coefficients with vanishing $J$-invariant and whose Hessians are proportional to the squares of reducible or positive definite binary quadratic form. These results give a case where one is able to count integral orbits inside a relatively open real orbit of a variety closed under a group action of degree at least three.

Euclid's Algorithm in quartic CM-fields

In this note we present techniques to compute inhomogeneous minima of norm forms; as an application, we determine all norm-Euclidean complex bicyclic quartic number fields.

A positive proportion of locally soluble quartic Thue equations are globally insoluble

For any fixed nonzero integer $h$, we show that a positive proportion of integral binary quartic forms $F$ do locally everywhere represent $h$, but do not globally represent $h$. We order classes of integral binary quartic forms by the two generators of their ring of $\textrm{GL}_{2}(\mathbb{Z})$-invariants, classically denoted by $I$ and $J$.

Binary quartic forms with bounded invariants and small Galois groups

In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by integral binary quadratic forms $f(x,y)$ of non-zero discriminant. Then, we shall enumerate the $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of all such forms associated to a fixed $f(x,y)$.

Upper bounds for the number of solutions to quartic Thue equations

We will give upper bounds for the number of integral solutions to quartic Thue equations. Our main tool here is a logarithmic curve $ϕ(x, y)$ that allows us to use the theory of linear forms in logarithms. This manuscript improves the results of author's earlier work with Okazaki by giving special treatments to forms with respect to their signature.

Diagonalizable Quartic Thue Equations with Negative Discriminant

The Thue-Siegel method is applied to derive an upper bound for the number of solutions to Thue's equation $F(x,y) = 1$ where $F$ is a quartic diagonalizable form with negative discriminant. Computation is used in this argument to handle forms whose discriminant is small in absolute value. We then apply our results to bound the number of integral points on a certain family of elliptic curves.

三元二次型对与二元四次型的协变及同时对角化

Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms

本文建立了二元四次型与三元二次型对协变间的对应，并给出了任意特征零域上二次型对的标准对角化方法及有理数域上可对角化的精确判据。

On primes represented by quartic polynomials on average

We obtain an upper bound for the distribution of primes in the form $n^4 + k$ up to $x$, averaged over $k$ with small square-full part. As a corollary, we show that for almost all $k$, there is an expected amount of primes in the form $n^4 +k$ up to $x$.

Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves

We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our techniques are quite general, and may be applied to counting integral orbits in other representations of algebraic groups. We use these counting results to prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by their heights, is bounded. In particular, we show that when elliptic curves are ordered by height, the mean size of the 2-Selmer gr…

非伽罗瓦四次CM域上\(\mathrm{GL}_2\)的扭曲西格尔 - 韦伊公式

Twisted Siegel-Weil formulas for $\mathrm{GL}_2$ over non-Galois quartic CM fields

研究非伽罗瓦四次CM域上\(\mathrm{GL}_2\)的扭曲西格尔 - 韦伊公式，通过联系扭曲西塔积分与赫克积分的土居 - 永野提升，证明雅克比 - 朗兰兹对应基变换是同构，还得出博切尔兹形式扭曲CM值相关结论。

奇异二元四次型上的指数和及其应用

Exponential sums over singular binary quartic forms and applications

本文研究奇异二元四次型上的指数和，给出其有限域傅里叶变换的显式公式，并利用初等代数几何与经典不变量理论证明点计数函数的突出出现，进而证明判别式无平方因子且至多含四个素因子的椭圆曲线存在大量2-Selmer元素。

Solubility of Additive Quartic Forms over Ramified Quadratic Extensions of $\mathbb{Q}_2$

We determine the minimal number of variables $Γ^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=4$ over the four ramified quadratic extensions $\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}), \mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10}) $ of $\mathbb{Q}_2$. In all four fields, we prove that $Γ^*(4,K) = 11$. This is the first example of such a computation for a proper extension of $\mathbb{Q}_p$ where the degree is a power of $p$ greater than $p$.

具有无穷多四次点的模曲线$X_0(N)$

Modular curves $X_0(N)$ with infinitely many quartic points

本研究确定了所有具有无穷多四次点的模曲线$X_0(N)$，通过定义诱导二次型的配对，刻画了从这类模曲线到正秩椭圆曲线的有理态射的所有可能次数。

Quartic rings associated to binary quartic forms

We give a bijection between binary quartic forms and quartic rings with a monogenic cubic resolvent ring, relating the rings associated to binary quartic forms with Bhargava's cubic resolvent rings. This gives a parametrization of quartic rings with monogenic cubic resolvents. We also give a geometric interpretation of this parametrization.

加权四次型的一个定量 Hasse 原理

A Quantitative Hasse Principle for Weighted Quartic Forms

本文利用 Hardy-Littlewood 方法，在非奇异局部可解条件下，证明了满足特定加权齐次条件的四次型整数零点计数渐近公式，并改进了 Birch 方法的结果。

Hilbert尖形式的Eichler-Selberg迹公式、四次CM域的类数及其分布

The Eichler--Selberg trace formula for Hilbert cusp forms, the class numbers of quartic CM fields, and their distributions

利用广义Hurwitz类数（由四次CM域类数定义），建立了窄类数为一的实二次域上Hilbert尖形式的Eichler-Selberg迹公式，并研究了类数分布与关系，进行了Hecke算子迹的数值计算。

The proportion of genus one curves over $\mathbb{Q}$ defined by a binary quartic that everywhere locally have a point

We consider the proportion of genus one curves over $\mathbb{Q}$ of the form $z^2=f(x,y)$ where $f(x,y)\in\mathbb{Z}[x,y]$ is a binary quartic form (or more generally of the form $z^2+h(x,y)z=f(x,y)$ where also $h(x,y)\in\mathbb{Z}[x,y]$ is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove that the local density at a prime $p$ is given by a fixed degree-$9$ rational function of $p$ for all odd $p$ (and for the generalised equation, the sa…

四次环与n元二次型对的大野-中川型反射定理

Reflection theorems of Ohno-Nakagawa type for quartic rings and pairs of $n$-ary quadratic forms

该论文证明了大野-中川型反射定理，涉及四次环、二次型对及相关矩阵数量，还为科恩等人关于此类反射定理无限族的问题提供了答案。

On the number of zeros of diagonal quartic forms over finite fields

Let $\mathbb{F}_q$ be the finite field of $q=p^m\equiv 1\pmod 4$ elements with $p$ being an odd prime and $m$ being a positive integer. For $c, y \in\mathbb{F}_q$ with $y\in\mathbb{F}_q^*$ non-quartic, let $N_n(c)$ and $M_n(y)$ be the numbers of zeros of $x_1^4+...+x_n^4=c$ and $x_1^4+...+x_{n-1}^4+yx_n^4=0$, respectively. In 1979, Myerson used Gauss sum and exponential sum to show that the generating function $\sum_{n=1}^{\infty}N_n(0)x^n$ is a rational function in $x$ and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that …

本原三次、四次与六次L-函数的非零性

Non-vanishing of primitive cubic, quartic and sextic L-functions

该研究在非库默尔情形下，通过适配Gao和Zhao的截断指数方法，改进了三次狄利克雷$L$-函数的非零比例下界，对四次与六次族得到正下界及六次一阶矩渐近公式，所有结果均无条件成立。

Practical solution of some families of quartic and sextic diophantine hyperelliptic equations

Using elementary number theory we study Diophantine equations over the rational integers of the following form, $y^2=(x+a)(x+a+k)(x+b)(x+b+k)$, $y^2=c^2x^4+ax^2+b$ and $y^2=(x^2-1)(x^2-α^2)(x^2-(α+1)^2).$ We express their integer solutions by means of the divisors of the discriminant of $f(x),$ where $y^2=f(x)$.

Power-saving error terms for the number of $D_4$-quartic extensions over a number field ordered by discriminant

We study the asymptotic count of dihedral quartic extensions over a fixed number field with bounded norm of the relative discriminant. The main term of this count (including a summation formula for the constant) can be found in the literature (see Cohen--Diaz y Diaz--Olivier for the statement without proof and see Klüners for a proof), but a power-saving for the error term has not been explicitly determined except in the case that the base field is $\mathbb{Q}$. In this article, we describe the argument for obtaining both the explicit main term and a power-saving error term for the number of $…

Parametrizing quartic algebras over an arbitrary base

We parametrize quartic commutative algebras over any base ring or scheme (equivalently finite, flat degree four $S$-schemes), with their cubic resolvents, by pairs of ternary quadratic forms over the base. This generalizes Bhargava's parametrization of quartic rings with their cubic resolvent rings over $\mathbb{Z}$ by pairs of integral ternary quadratic forms, as well as Casnati and Ekedahl's construction of Gorenstein quartic covers by certain rank 2 families of ternary quadratic forms. We give a geometric construction of a quartic algebra from any pair of ternary quadratic forms, and prove …

三次Siegel模形式与三元四次式的不变量

Siegel modular forms of degree three and invariants of ternary quartics

本研究确定了三次Siegel模形式分次环的结构，明确其由19个模形式生成并找到7个权的齐次参数系，同时建立了三元四次式的Dixmier-Ohno不变量与这些生成元的完整对应。

On parameters of orders of quartic fields and essential pairs -- extended version

Classes of pairs of ternary quadratic forms parametrize quartic rings by a result of Bhargava. We give an algorithm for finding a pair of ternary quadratic forms that parametrize a given order of a quartic field. We examine a new technique, essential pairs, for obtaining parameters of orders of quartic fields from a number field database. Essential pairs for maximal orders of quartic fields are very common and provide a simple means of obtaining an integral basis for the ring of integers.

Manin's conjecture for quartic del Pezzo surfaces with a conic fibration

An asymptotic formula is established for the number of rational points of bounded height on a non-singular quartic del Pezzo surface with a conic bundle structure.

Concomitants of Ternary Quartics and Vector-valued Siegel and Teichmüller Modular Forms of Genus Three

We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmüller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichmüller cusp forms on \overline{M}_g and the middle cohomology of symplectic local systems on M_g. In genus 3, we make this explicit in a large number of cases.

Rational points on Cubic, Quartic and Sextic Curves over Finite Fields

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic curves. In the case where $q$ is a prime number, we give a way to calculate these numbers. As a consequence of these results, we characterize maximal and minimal curves given by equations of the forms $ax^3+by^3+cz^3=0$ and $ax^4+by^4+cz^4=0$.