Poo-Sung Park
If a positive definite Hermitian lattice represents all positive integers, we call it universal. Several mathematicians, including the author, found 25 universal binary Hermitian lattices. But their ad hoc proofs are complicated. We give simple and unified proofs.
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A positive definite Hermitian lattice is said to be 2-universal if it represents all positive definite binary Hermitian lattices. We find all 2-universal ternary and quaternary Hermitian lattices over imaginary quadratic number fields.
Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this correspondence is a discriminant preserving bijection between the isomorphism classes of embeddings and integral binary hermitian forms.
Earnest and Khosravani, Iwabuchi, and Kim and Park recently gave a complete classification of the universal binary Hermitian forms. We give a unified proof of the universalities of these Hermitian forms, relying primarily on Ramanujan's list of universal quadratic forms and on the Bhargava-Hanke 290-Theorem. Our methods bypass nearly all of the ad hoc universality arguments required in the original classification.
Even universal binary Hermitian lattices over imaginary quadratic fields
本研究针对虚二次域上的偶泛二元埃尔米特格问题，提出获取所有这类格的方法，并给出相应域上埃尔米特格偶泛性的最优判据。
Given an integral indefinite binary Hermitian form f over an imaginary quadratic number field, 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 infinity.
In this paper we investigate a result of Ueno on the modularity of generating series associated to the zeta functions of binary Hermitian forms previously studied by Elstrodt et al. We improve his result by showing that the generating series are Eisenstein series. As a consequence we obtain an explicit formula for the special values of zeta functions associated with binary Hermitian forms.
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。
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。
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.
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 $L(χ_{d_K},s)$.
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.
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…
Universality of the cokernels of random $p$-adic Hermitian matrices
本文研究 $\mathbb{Q}_p$ 二次扩域整数环上随机埃尔米特矩阵余核的分布，证明其当 $n \rightarrow \infty$ 时收敛至不依赖矩阵选取的普适极限分布，并给出显式公式，解答了 Wood 在 ICM 2022 讲义中提出的公开问题 3.16。
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…
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