Melanie Matchett Wood
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.
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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 …
For $n=3$, 4, and 5, we prove that, when $S_n$-number fields of degree $n$ are ordered by their absolute discriminants, the lattice shapes of the rings of integers in these fields become equidistributed in the space of lattices.
In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic covariants of cubic and quartic forms. This allows one to give precise asymptotic formulae for the number of integers in an interval representable by a binary cubic or quartic form and extends work of Hooley. Further, we give the field of definition of lines contained in certain cubic and quartic surfaces related to binary cubic and 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.
We use the invariant theory of binary quartics to give a new formula for the Cassels-Tate pairing on the $2$-Selmer group of an elliptic curve. Unlike earlier methods, our formula does not require us to solve any conics. An important role in our construction is played by a certain $K3$ surface defined by a $(2,2,2)$-form.
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.
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。
We show that a positive proportion of quartic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof builds on the corresponding result for cubic fields that we obtained in a previous work. Along the way, we also prove that a positive proportion of quartic rings of integers do not arise as the invariant order of an integral binary quartic form despite having no local obstruction.
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 $f(x,y)$.
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。
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…
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。
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 $J$.
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。
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.
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。
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…
We consider representations of real forms of even degree as a linear combination of powers of real linear forms, counting the number of positive and negative coefficients. We show that the natural generalization of Sylvester's Law of Inertia holds for binary quartics, but fails for binary sextics.
In this paper, we consider maximal and irreducible quartic orders which arise from integral binary quartic forms, via the construction of Birch and Merriman, and whose field of fractions is a quartic $D_4$-field. By a theorem of M. Wood, such quartic orders may be regarded as quartic $D_4$-fields whose ring of integers has a monogenic cubic resolvent. We shall give the asymptotic number of such objects when ordered by conductor, as well as estimate the asymptotic number of such objects when ordered by discriminant.
We consider and completely solve the parametrized family of Thue equations \begin{eqnarray*}X(X-Y)(X+Y)(X-λY)+Y^4=ξ,\end{eqnarray*} where the solutions $x,y$ come from the ring $\mathbb{C}[T]$, the parameter $λ\in\mathbb{C}[T]$ is some non-constant polynomial and $0\neqξ\in\mathbb{C}$. It is a function field analogue of the family solved by Mignotte, Pethő and Roth in the integer case. A feature of our proof is that we avoid the use of height bounds by considering a smaller relevant ring for which we can determine the units more easily. Because of this, the proof is short and the arguments are…
On the Quartic Invariant of Odd Degree Binary Forms
研究奇次二元形式的四次不变量的平方自由部分，揭示其与素数幂的关系，并通过AI辅助方法验证关键算术引理，扩展了经典二元三次形式的判别式公式。
Let $\Bbb Z$ be the set of integers, and let $(m,n)$ be the greatest common divisor of integers $m$ and $n$. Let $p\equiv 1\mod 4$ be a prime, $q\in\Bbb Z$, $2\nmid q$ and $p=c^2+d^2=x^2+qy^2$ with $c,d,x,y\in\Bbb Z$ and $c\e 1\mod 4$. Suppose that $(c,x+d)=1$ or $(d,x+c)$ is a power of 2. In the paper, by using the quartic reciprocity law we determine $q^{[p/8]}\mod p$ in terms of $c,d,x$ and $y$, where $[\cdot]$ is the greatest integer function. We also determine $\big(\frac{b+\sqrt{b^2+4^α}}2\big)^{\frac{p-1}4}\mod p$ for odd $b$ and $(2a+\sqrt{4a^2+1})^{\f{p-1}4}\mod p$ for $a\in\Bbb Z$. A…
We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial $3$-torsion fields for a certain one-parameter family of non-CM elliptic curves, we describe a power basis. As a result, we show that the one-parameter family of quartic $S_4$ fields given by $T^4 - 6T^2 - αT - 3$ for $α\in \mathbb{Z}$ such that $α\pm 8$ are squarefree, are monogenic.
The ring of Weyl invariant $E_8$ Jacobi forms
研究外尔不变的\(E_8\)弱雅可比形式环，证明其与二元六次式和二元四次式联合协变量环同构且有限生成，借助已知协变量环生成元获极小生成元基。
We give an explicit necessary condition for pairs of orders in a quartic CM-field to have the same polarised class group. This generalises a simpler result for imaginary quadratic fields. We give an application of our results to computing endomorphism rings of abelian surfaces over finite fields, and we use our results to extend a completeness result of Murabayashi and Umegaki to a list of abelian surfaces over the rationals with complex multiplication by arbitrary orders.
Let $M$ be an imaginary quadratic field with the ring of integers $\mathbb{Z}_{M}$ and let $ξ$ be a root of polynomial $$f\left( x\right) =x^{4}-2cx^{3}+2x^{2}+2cx+1,$$ where $c\in\mathbb{Z}_{M},$ $c\notin\left\{ 0,\pm2\right\}$. We consider an infinite family of octic fields $K_{c}=M\left( ξ\right)$ with the ring of integers $\mathbb{Z}_{K_{c}}.$ Our goal is to determine all generators of relative power integral basis of $\mathcal{O=}\mathbb{Z}_{M}\left[ ξ\right]$ over $\mathbb{Z}_{M}.$ We show that our problem reduces to solving the system of relative Pellian equations \[ cV^{2}-\left( c+2\r…
Exponential sums over singular binary quintics
本文利用Waring分解，将二元四次型上指数和的方法推广至奇异二元五次型，给出无特征显式公式。
A Classification of Multiply Monogenic Quartic Orders
本文针对所有四次数域，证明了不属于Bérczes等人定义的两类特定类型的二重单生$\u2124$-序仅有有限多个，完善了此前需附加伽罗瓦群条件的相关结论。
Counting points on a family of degree one del Pezzo surfaces
本文研究了一族由方程y²=x³+AxQ(u,v)²+BQ(u,v)³定义的椭圆面上的有理点，通过Mordell参数化方法，证明了特定子集的渐进行为，利用模形式连接二元二次和四次形式的表示数相关和。
Monogenic Cyclic Quartic Trinomials
本文证明了恰好存在三个形如 $x^4+bx^2+d$ 且 Galois 群为四阶循环群的单生成三项式。
We show that binary forms of degree n less than seven parameterize rings, generalizing a result of Levi on binary cubic forms parameterizing cubic rings, which can be related to results of Bhargava. The question of whether or not a binary form of degree n parameterizes a ring of rank $n$ depends on a symbolic calculation with matrices.
After a brief introduction to the classical theory of binary quadratic forms we use these results for proving (most of) the claims made by Pépin in a series of articles on unsolvable quartic diophantine equations, and for constructing families of counterexamples to the Hasse Principle for curves of genus 1 defined by equations of the form $ax^4 + by^4 = z^2$.
The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied quadratic rings and ideals associated to binary quadratic forms, and found that ideal classes of quadratic rings are exactly parametrized by equivalence classes of integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic rings are parametrized by equivalence classes of integral binary cubic forms. Birch, Merriman, Nakagawa, Corso, Dvornicich, and Simon have all studied rings associated to binary forms of degree n for any n, but it has…
Monogenic Reciprocal Quartic Polynomials And Their Galois Groups
本文针对互反四次整系数多项式，在两种参数情形下，确定了所有具有给定伽罗瓦群的单源多项式。
Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and $g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible, $b\mid a$, and the integers $b$, $d$, $256d-27c^4$, and $\dfrac{256b^3-27a^4}{\gcd(256b^3,27a^4)}$ are all square-free. Using the Montes algorithm, we show that a root of $f_{a,b}(x)$ or $g_{c,d}(x)$ defines a monogenic extension of $\mathbb{Q}$ and serves as a generator for a power integral basis of the ring of integers. In fact, we show monogeneity for slightly more general families. Further, we obtain lower bounds on the density of polynomials generatin…
Monogenic Quartic Polynomials and Their Galois Groups
本文借助 Kappe–Warren 的四次 Galois 群分类，为除循环群外的各类可能 Galois 群构造生成互异四次域的无限单参数单生四次多项式族，并讨论未解决的循环情形。
The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the integers. However, such extensions have always included hypotheses on the rings, and the theorems involve only binary quadratic forms satisfying further hypotheses. We give a complete statement of the relationship between binary quadratic forms and modules for quadratic algebras over any base ring, or in fact base scheme. The result includes all binary quadratic f…
We give a parametrization of the ideal classes of rings associated to integral binary forms by classes of tensors in $\mathbb Z^2\tensor \mathbb Z^n\tensor \mathbb Z^n$. This generalizes Bhargava's work on Higher Composition Laws, which gives such parametrizations in the cases $n=2,3$. We also obtain parametrizations of 2-torsion ideal classes by symmetric tensors. Further, we give versions of these theorems when $\mathbb Z$ is replaced by an arbitrary base scheme $S$, and geometric constructions of the modules from the tensors in the parametrization.
The ring of $D_4$ triality invariants
该研究证明满足特定尖点条件的 D₄ 三元对合不变量环，同构于二元三次型与二元二次型的联合共变式环。
A new parametrization for ideal classes in rings defined by binary forms, and applications
本文通过共正则表示的轨道为二元型定义的环中理想类平方根建立新参数化，并将其应用于界定数域族中2-类群平均大小等若干此前难以解决的分布问题。
We discuss multiplicative properties of the binary quadratic form $a x^2 + b x y + c y^2$ by considering a ring of matrices which is closed under a triple product. We prove that the ring forms a ternary algebra in the sense of Hestenes, and then derive both multiplicative formulas for a large class of binary quadratic forms and a type of multiplication for points on a conic section which generalizes the algebra of rational points on the unit circle.
In this paper we construct binary self-dual codes using the étale cohomology of $\mathbb{Z}/2$ on the spectra of rings of $S$-integers of global fields. We will show that up to equivalence, all self-dual codes of length at least 4 arise from Hilbert pairings on rings of $S$-integers of $\Q$. This is an arithmetic counterpart of a result of Kreck and Puppe, who used cobordism theory to show that all self-dual codes arise from Poincaré duality on real three manifolds.
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