On the number of equivalence classes of binary forms of given degree and given discriminant

Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary f…

更高组合法则的显式组合恒等式

Explicit composition identities for higher composition laws

本文旨在为Bhargava提出的更高组合法则提供显式的组合恒等式，通过构建判别式保持的双射，将轨道集与理想类集联系起来，揭示投射形式类群的结构。

有限生成域上判别式方程的有效结果

Effective results for discriminant equations over finitely generated domains

本文将关于有限生成整环上判别式方程有限解及有效求解的结果，从整闭整环推广至非整闭整环。

数域上二元二次型的复合

Composition of Binary Quadratic Forms over Number Fields

本文把二次数域理想类群与二元二次型等价类的经典对应推广至窄类数为一的任意数域，显式刻画该对应，并给出全负判别式下全正定型的等价条件。

Quaternary quadratic lattices over number fields

We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space (V,q) with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of (V,q). This yields a good algorithm to enumerate a system of representatives of proper isometry classes of lattices in genera of maximal lattices in (V,q).

Formes quadratiques de discriminants emboîtés

Quadratic forms with embedded discriminants. Integral binary quadratic forms have multiple applications, for example in factorization or cryptography. The Nice family of cryptographic systems makes use of quadratic forms with different discriminants $\pm p$, and $\pm pq^2$ where $p$, $q$ are large primes. This paper shows the precise links between forms with $D$ discriminant and forms with $Df^2$ discriminant, which are crucial in the analysis of the systems Nice and theirs attacks. We also introduce the notion of semi-equivalence of binary quadratic forms, and give some characterizations of s…

二次型等分布中的丢番图对

Equidistribution of Diophantine pairs among the equivalence classes of quadratic forms

本文研究了D(n)-对在二元二次型等价类中的渐近等分布性，并给出了Badesa渐近公式的简明证明。

A novel connection between integral binary quadratic forms and knot polynomials

We establish a novel connection between algebraic number theory and knot theory. We show that the number of equivalence classes of integral binary quadratic forms of discriminant $t^2 - 4$ (for $t\neq \pm 2$) is equal to the number of isotopy classes of links in $\mathbb{S}^3$ with prescribed values (depending on $t$) of three classical link invariants. The equality arises from a natural algebraic correspondence between integral binary quadratic forms (of discriminant $t^2 - 4$ for $t\neq \pm 2$) and isotopy classes of links of braid index at most three. In particular, the class numbers of cer…

Equivalence classes of dessins d'enfants with two vertices

Let $N$ be a positive integer. For any positive integer $L\leq N$ and any positive divisor $r$ of $N$, we enumerate the equivalence classes of dessins d'enfants with $N$ edges, $L$ faces and two vertices whose automorphism groups are cyclic of order $r$. Further, for any non-negative integer $h$, we enumerate the equivalence classes of dessins with $N$ edges, $h$ faces of degree $2$ with $h\leq N$, and two vertices, whose automorphism groups are cyclic of order $r$. Our arguments are essentially based upon a natural one-to-one correspondence of the equivalence classes of all dessins with $N$ e…

平稳下降子与判别式模形式

Stationary Descendents and the Discriminant Modular Form

该研究引入权k的下降拟阵，在权12情形下用其得到所有平稳下降子表示判别式模形式的方式，推导出用椭圆曲线Gromov-Witten不变量表示Ramanujan tau值的闭式，计算借助Sage实现。