On Universal Binary Hermitian Forms

虚二次域上的二元正规正则埃尔米特格

Binary normal regular Hermitian lattices over imaginary quadratic fields

本研究定义了正定正则埃尔米特格，针对虚二次域上的二元正规正则埃尔米特格展开研究，最终给出了这类格的完整列表，推进了埃尔米特格的分类工作。

虚二次域上通用埃尔米特格的十五定理

The Fifteen Theorem for Universal Hermitian Lattices over Imaginary Quadratic Fields

该研究针对虚二次域上的通用埃尔米特格问题，提出获取所有m对应域上通用格的方法，得出含13等10个数的通用性判定准则，即通用埃尔米特格的十五定理，其与Conway-Schneeberger十五定理的区别在于包含数字13。

Binary Hermitian forms over a cyclotomic field

Let z be a primitive fifth root of unity and let F be the cyclotomic field F=Q(z). Let O be the ring of integers. We compute the Voronoi polyhedron of binary Hermitian forms over F and classify GL_2(O)-conjugacy classes of perfect forms. The combinatorial data of this polyhedron can be used to compute the cohomology of the arithmetic group GL_2(O) and Hecke eigen forms.

From Binary Hermitian Forms to parabolic cocycles of Euclidean Bianchi groups

We study a family of functions defined in a very simple way as sums of powers of binary Hermitian forms with coefficients in the ring of integers of an Euclidean imaginary quadratic field $K$ with discriminant $d_K$. Using these functions we construct a nontrivial cocycle belonging to the space of parabolic cocycles on Euclidean Bianchi groups. We also show that the average value of these functions is related to the special values of $L(χ_{d_K},s)$. Using the properties of these functions we give new and computationally efficient formulas for computing some special values of $L(χ_{d_K},s)$.

Geometry of Integral Binary Hermitian Forms

We generalize Conway's approach to integral binary quadratic forms on Q to study integral binary hermitian forms on quadratic imaginary extensions of Q. In Conway's case, an indefinite form that doesn't represent 0 determines a line ("river") in the spine T associated with SL(2,Z) in the hyperbolic plane. In our generalization, such a form determines a plane ("ocean") in Mendoza's spine associated with the corresponding Bianchi group SL(2,A) in hyperbolic 3-space.

On the arithmetic and geometry of binary Hamiltonian forms

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describ…

Reduction theory of binary forms

In these lectures we give an introduction to the reduction theory of binary forms starting with quadratic forms with real coefficients, Hermitian forms, and then define the Julia quadratic for any degree $n$ binary form. A survey of a reduction algorithm over $\mathbb Z$ is described based on recent work of Cremona and Stoll.

On the Shintani zeta function for the space of pairs of binary Hermitian forms

Throughout this paper, k is a number field. We fix a quadratic extension k_1=k(a_0) of k, where a_0=sqrt(B_0) for a certain B_0 in k^\times/(k^\times)^2. In this paper, we consider the zeta function defined for the space of pairs of binary Hermitian forms. This is the prehomogeneous vector space we discussed in section 2 [1] and is a non-split form of the D_4 case in [4]. The purpose of this paper is to determine the principal part of the adjusted zeta function. Our main result is Theorem (8.15). Our case resembles the space of pairs of binary quadratic forms which we discussed in Chapter 5 [5…